Error Bounds for an Inequality System
by
Z iliW u
B.Sc., Xiamen University, 1982 M.Sc., University of Victoria, 1997
A Dissertation Submitted in Partial Fulfillment of the Requirements for the Degree of
DOCTOR OF PHILOSOPHY
in the Department of Mathematics and Statistics
We accept this dissertation as conforming to the required standard
Dr. J. J, Ye,^Supervisor (Department of Mathematics and Statistics)
)r. C. J. Bose, Departmental Member (Department of Mathematics and Statistics)
Dr. R. Illner, Departmental Member (Department of Mathematics and Statistics)
Dr. W. S. Lu, Outside Member (Department of Electrical and Computer Engineering)
:------Dr. Q. jTZRu, Exterral Examiner (Department of Mathematics and Statistics, Western Michigan University)
© Zili Wu, 2001 Uinversity of Victoria
All rights reserved. This dissertation may not be reproduced in whole or in part, by photocopy or other means, without the permission of the author. Supervisor; Dr. Jane J. Ye
ABSTRACT
For an inequality system, an error bound is an estimation for the distance from any point to the solution set of the inequality. The Ekeland variational principle (EVP) is an important tool in the study of error bounds. We prove that EVP is equivalent to an error bound result and present several sufficient conditions for an inequality system to have error bounds. In a metric space, a condition is similar to that of Takahashi. In a Banach space we express conditions in terms of an abstract subdifferential and the lower Dini derivative. We then discuss error bounds with exponents by a relation between the lower Dini derivatives of a function and its power function. For an l.s.c. convex function on a reflexive Banach space these conditions turn out to be equivalent. Furthermore a global error bound closely relates to the metric regularity.
Examiners:
D^/j. J. Ye, Supervisor/Department of Mathematics and Statistics)
Dr. C. J. Bose, Departmental Member (Department of Mathematics and Statistics)
Dr. R. Illner, Departmental Member (Department of Mathematics and Statistics)
Dr. W. S. Lu, Outsiae Mernber (Department of Electrical and Computer Engineering)
Dr. Q. J. Zhir, External Examiner (Department of Mathematics and Statistics, Western Michigan University)
11 Contents
Abstract ü
Table of Contents iii
Acknowledgements v
1 Introduction 1
2 Various Generalized Subdifferentials 9
2.1 The Clarke Subdifferential ...... 9
2.2 Various D erivatives...... 24
2.3 The Michel-Penot Subdifferential ...... 28
2.4 The Proximal Subdifferential ...... 30
2.5 The Fréchet Subdifferential ...... 33
2.6 The Dini Subdifferentials ...... 39
3 Equivalent Formulations of Ekeland’s Variational Principle 44
3.1 The Ekeland Variational Principle ...... 45
3.2 The e-conditions of Takahashi and Hamel with a ...... 47
3.3 Weak Sharp Minima and Error Bounds ...... 50
iii 3.4 A Fixed Point Theorem ...... 55
3.5 The Completeness of a Metric Space ...... 60
4 Error Bounds for Lower Semicontinuous Functions 65
4.1 Error Bounds for Nonconvex Functions on Metric Spaces .... 66
4.2 5o,-subdifferentials...... 72
4.3 Error Bounds for Lower Semicontinuous Functions on Banach
S p a c e s ...... 74
4.4 Error Bounds with Exponents ...... 83
4.5 Error Bounds for Lower Semicontinuous Convex Functions . . . 92
4.6 Error Bounds with Abstract Constraint S ets ...... 99
4.7 Global Error Bounds and Metric Regularity ...... 107
IV Acknowledgements
With most sincere appreciation, I thank Dr. Jane J. Ye for her patience, helps and supervision in my research work, in particular, in the process of writting this dissertation.
Thanks and appreciation are due to Dr. Chris J. Bose, Dr. Reinhard Illner,
Dr. Wu-Sheng Lu and Dr. Qiji J. Zhu for their time in examining my disserta tion and providing me with enlightening suggestions and remarks.
I gratefully acknowledge the Department of Mathematics and Statistics,
University of Victoria, which has provided me with the opportunity and support to work on this dissertation.
Finally, I am greatly indebted to my father, my wife, my brother, my sisters and my daughter for their love, support and encouragement. I would like to extend heartful thanks to Mr. Zhanghong Wu for his impressive instruction and encouragement. Chapter 1
Introduction
Let C be a nonempty closed subset of a normed linear space X and gj : X ^
(~oo, + 00] be extended real-valued functions for i = 1, • • •, r and j = 1, ■ • •, s.
Denote the solution set of an inequality system by
5 := {z E C : /i(z) < 0, -, AM < 0; giM =0, - = 0}-
The set S is said to have a global error bound if it is nonempty and there exists a constant fi > 0 such that
ds{x) := inf{||a; - c\\ : c e S} < ||-FM+|| 4- ||G(a;)|| ) Vx € C, (1.1) where F{x)^ = (/i(z)+, • • •, fr{x)+) E IF with a+ := max{a, 0} for a E R,
G{x) = {qi (x ), • • ■ ,gs(x)) € i?* and || • || is the usual Euclidean norm. The set
S is said to have a local error bound if there exist constants p > 0 and e > 0 such that
dg(a;) < /r(||F(z)+|| 4- ||G(a;)||) Vz E C with ||(F(z)+,G(a;))|| < e.
Apparently if the set S has a global (local) error bound, then functions involved provide a global (local) error estimate for the distance from any point X to the solution set S. Such an estimation is very useful in the study of optimization. Consider the following optimization problem with equality and inequality constraints:
(P) minimize h{x) subject to f{x) < 0, g{x) — 0, x E P ”, where h is Lipschitz of rank L on P", / : P " P and |^| : P" —>■ P are lower semicontinuous. In this situation the feasible set
P := {z E P" :/(% )< 0, ^(z) = 0} is closed and, by the exact penalization, problem (P) is equivalent to the un constrained problem:
(Pa) minimize h{x) + ads{x) subject to z E P" for any a > L (see [17, Proposition 1.3]). The objective function of problem (Pa) involves the distance function which is usually not easy to deal with. However if S has a global error bound then, for some p > 0,
dg(ar) < /^(/(%)+ + l^(z)l) V% E P".
It is easy to check that every solution of (Fa) solves
(Pan) minimize h(x) + aiJ,(f(x)+ + |^(r)|) subject to z E P".
On the other hand, if xq solves problem (Pan)> then for each z E P we have
h(xo) + ads(xo) < h(x) Vz G P. Since h is Lipschitz of rank L,
a(fa(%o) < h(z) - h(zo) < Z,||z; - zo|| Vz E 5^.
Thus ads(xo) < Lds{xo). This implies ds{xo) = 0, that is, xq € S. So Xq solves problem (P). Therefore problems (P), (Pa) and (Pan) have the same solution set. Hence we can solve problem (F) by studying problem {Pap) which maybe simpler since it is unconstrained and does not contain the distance function.
Since Hoffman [32] obtained a result on global error bound for a linear inequality systems on FP, the study of error bounds has received more and more attention in the mathematical programming literature due to many important applications in sensitivity analysis, complementarity problems, implicit function theorem, and the convergence analysis of some descent methods. For more details on error bounds and their applications we see [2, 13, 15, 18, 23, 24, 29,
33, 39, 41, 42, 43, 44, 45, 46, 50, 51, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 65, 66,
67, 68, 72, 83, 89, 90] and the references therein.
Hoffman’s result [32] states that if f{x) = Ax + a and g{x) — Bx + b for some matrices A, B and some vectors a, b of appropriate dimension then there exists some p > 0 depending on A and B only such that
ds{x) < n{\\[Ax + a]+|| + \\Bx + 6||) MxeBA.
There are no special conditions required for a linear inequality system to have a global bound. However for a nonlinear inequality system, additional conditions are usually needed. The Slater condition is one of most often used conditions, which postulates the existence of an a:o E C such that f{xo) < 0 for / : X —>• BP. Under this condition, Robinson [72] obtained a global error bound for an arbi trary bounded closed convex set S' in a normed space; Mangasarian [60] proved that there exists a global error bound when / : i?” —> is a convex differen tiable function and satisfies an asymptotic constraint qualification; Auslender and Crouzeix [2] extended Mangasarian’s result to a nondifferentiable function;
Luo and Luo [50] established a global error bound for a convex quadratic inequal ity system without other conditions; for a continuous convex inequality system on a reflexive Banach space Deng [23] obtained an error bound result under a
Slater condition on the associated recession functions; Deng [24] extended his result to the same system on a Banach space under the Slater condition with the Hausdorff distance assumption.
Some weaker conditions have also been found to characterize a global error bound for a convex function / : R” —> Li [46] showed that a convex differentiable inequality system f{x) < 0 has a global error bound if and only if the system satisfies Abadie’s constraint qualification at each x £ S, that is, the tangent cone of S' at % can be expressed as follows
Ts{x) = {y G R" : y) < 0 for each i with fi(x) = 0}.
Lewis and Pang [42] studied a proper lower semicontinuous convex function
/ : j y —> (—oo, -hoo] and proved that / has a global error bound if and only if for some /x > 0 and for any x E B^ with f{x) — 0 and any normal vector d to
S' at Æ the directional derivative f'{x-,d) satisfies f'{x\d) > /x“^||d||.
It is worth pointing out that the above conditions are all in one class which closely relates to points inside the solution set S. The other class of conditions concerns with various generalized directional derivatives and subdifferentials of functions at the points outside 5^ for which we can use the knowledge of nonsmooth analysis to study error bounds more effectively. To the author's knowledge, it is Ioffe [33] who ffrst obtained a global error bound (as well as metric regularity at a point) for a Lipschitz continuous equality system f{x) — 0 on a Banach space X under the condition that for some /i > 0, 0 < e < +oo, some z E X with f(z) = 0 there holds
IICII* > e d°\ f\{x) 'ix E X with f{x) 0 and \\x - z\\ < e, where d°f{x) is the Clarke subdifferential of / at x. The Ekeland variational principle and the sum rule of the Clarke subdifferential are the main tools in the proof of this result from which we can see that the Clarke snbdifferential in the condition can be replaced with any smaller subdifferentials satisfying the sum rule. Using Ioffe’s method, Ye [89] and Jourani [39] have sharpened the result of Ioffe by replacing the Clarke subdifferential with the limiting subdifferential in i?” and a partial subdifferential in a general Banach space respectively. For a lower semicontinuous system on R^, Wu [83] has used the fuzzy sum rule
(instead of the sum rule) to prove that the Clarke subdifferential in Ioffe’s condition can be replaced with the proximal subdifferential. In a Hilbert space,
Clarke, Ledyaev, Stern and Wolenski [18, Theorem 3.3.1] have weakened Ioffe’s condition using the proximal subdifferential instead of the Clarke subdifferential.
They did not use Ioffe’s method to establish their result (since the proximal subdifferential does not satisfy the sum rule) but the decrease principle (see also Ye [90, Claim]). Ledyaev and Zhu [41] have used the decease principle of the version in a Banach space with a Fréchet smooth Lipschitz bump function to extend their result in terms of Fréchet subdifferential.
Recently Ng and Zheng [66] have proved the existence of a global error bound for a proper lower semicontinuous function / on a complete metric space
X provided that for some /r > 0 and each x Ç. X with f{x) > 0 there exists a unit vector E X such that the lower Dini directional derivative of / at x in the direction hx is less than or equal to — that is,
/ I (3;; ha:)
Their result is based on an equivalent result of Ekeland’s variational principle, that is, the Takahashi theorem, which asserts the existence of minima for a lower semicontinuous function / on a complete metric space {X, d) when / satisfies the condition 0/ Th&ohoaht, that is, for each % E X with infx / < / (z) there exists y E X such that y x and f{y) + d{x, y) < f{x).
Wu and Ye [88] obtained a related error bound result for a proper lower semicontinuous function / on a complete metric space X which states that the solution set S is nonempty and for some /x > 0 and 0 < e < +00 there holds
ds{x) < iJ,f{x)-^ for all a; € Y with f{x) < e provided that the set {x E X : f{x) < e} is nonempty and for each x E X with
0 < f{x) < e there exists y E X such that 0 < /(y) < e and
0 < d(z, %/) < //[/(%) - /(%/)].
As we know, the Ekeland variational principle and the sum rule or the fuzzy sum rule are very important in establishing the above results. One of our pur poses in this dissertation is to reveal the equivalent relations among the Ekeland variational principle, the Takahashi theorem and the above error bound result in chapter 3. We will introduce the concepts of the e-condition of Takahashi and Hamel with a > 0. Then by the Ekeland variational principle we will prove that the e-condition of Takahashi implies that of Hamel. It follows from this relation that the e-condition of Takahashi is sufficient for / to have weak sharp minima. From this we can easily derive the above error bound result for which we will further demonstrate its theoretical application by establishing a new fixed point theorem and proving the Takahashi theorem.
Our second goal is to study sufficient conditions for an inequality system to possess error bounds in chapter 4. For an inequality system on a general metric space we will discuss conditions similar to the condition of Takahashi.
In a Banach space we will introduce the concept of ^^-subdffierential (which satisfies the fuzzy sum rule) and use it to present a sufficient condition similar to Ioffe’s for a lower semicontinuous system. The corresponding result unifies and extends several existing results and is applicable to a system with a closed constraint set. Sufficient conditions for error bounds will also be given in terms of the lower Dini derivative. Such conditions are easy to be translated into those for error bounds with exponents. These conditions are shown to be sufficient and necessary for a lower semicontinuous convex function on a reflexive Banach space to have (local and global) error bounds. We will briefly discuss relations between global error bound and metric regularity at the end of chapter 4.
For the above purposes we will gather the basic concepts of various general ized subdifferentials and their properties in the next chapter.
From now on, unless specified, X and X* denote a normed linear space and its dual space. |jx|| is the norm of a; in X and ||^||* is the norm of ^ in X* defined by
ll^ll* := sup{((,u) : u € X, ||n|| < 1} while B and B* stand for the closures of open unit balls B of X and B* of X* respectively. Chapter 2
Various Generalized S ub different iais
In this chapter we study various generalized subdifferentials and their properties that will be used in this dissertation. We begin with the Clarke subdifferential and its calculus in section 2.1. The close relations among the Clarke subdiffer ential and various derivatives will be revealed in section 2.2. From section 2.3 to section 2.6 we will discuss the Michel-Penot subdifferential, the proximal subd ifferential, the Fréchet subdifferential and the Dini subdifferential respectively.
2.1 The Clarke Subdifferential
It is well known that the Clarke subdifferential is a very important concept in nonsmooth analysis, especially as it relates to optimization (see [6, 7, 8, 9, 10,
11, 16, 17, 20, 19, 25, 27, 33, 38, 47, 70, 71, 74, 76, 89]). It was ffrst deffned for a locally Lipschitz function then for a very general class of functions. 10
Deûnîtion and Properties
Deûnition 2.1 Let C be a nonempty subset of a normed linear space A function / : C —>• i? is said to be Lipschitz {of rank L) on C if for some nonnegative scalar L one has
\f{yi) ~ 7 (2/2)I < L\\Vi — 2/2II Vyi, 2/2 E C.
We shall say that f is Lipschitz of rank L near x iî C — x + ôB fox some 5 > 0.
A function / is said to be locally Lipschitz on a subset 5 of X if it is Lipschitz near every point in 5".
Definition 2.2 Let f : X R he Lipschitz near x. For any vector v in X , the
(Clarke) generalized directional derivative of f at x in the direction v, denoted
7° (a;; n), is defined by
y^x t
The Clarke subdifferential {the generalized gradient) of f at x is the subset of
X* given by
a°7(4 = 6 X* : < 7°(:r; %;) Vu e X}.
With the Clarke subdifferential a wonderful system of theory has been de veloped (e.g., see [16]). We will use the following basic properties.
P ro p o sitio n 2.3 ([16, Proposition 2.1.1]) Let f be Lipschitz of rank L near x.
Then 11
(a) The function v —> f°{x]v) is finite, positively homogeneous, and subaddi
tive on X , and satisfies
|/°(z;u )| < Z,||u||.
(b) f°{x; v) is upper semicontinuous as a function of {x, v) and, as a function
of V alone, is Lipschitz of rank L on X.
(c) r(z;-u ) = (-/)°(a;;t;).
P ro p o sitio n 2.4 ([16, Proposition 2.1.2]) Let f be Lipschitz of rank L near x.
Then
(a) d°f{x) is a nonempty, convex, weak*-compact subset of X* and ||^||* < L
for every Ç in d° f{x).
(b) For every v in X , one has
/°(z; u) = max{((, i;) : ( E 9°/(z )} .
Proposition 2.5 (The mean value theorem) Let x and y be two distinct points in X and f : X ^ R be Lipschitz on an open set containing
[x, y] := {Xx + (1 - A)y : A G [0,1]}.
Then there existsa point u in {x, y) ;= [z, y} such that
/W - /(:c) G (^/(n), - 4 , where the inclusion implies that there exists ^ G d°f{u) such that 12
The Tangent Cone and the Normal Cone
Let C be a nonempty subset of X. The distance function associated with C is dehned by
dc{x) = inf{||x — c|| ; c e C} Vz € X.
Clearly the function dc is globally Lipschitz of rank 1 on X, that is,
|dc(z) -dcW I < Ik-2/11 Vz, 2/ € X.
Thus by Proposition 2.4 we have ||^||* < 1 for each ^ e d°dc{x).
The distance function is a Lipschitz function frequently used in optimization since if a Lipschitz function / of rank L attains a minimum over C at z then for any L > L the function f + Ldc attains a minimum at z, that is, a constrained optimization problem can be changed to a unconstrained optimization problem which may be easier to study. In addition the distance function can be used to introduce the concepts of tangent cone to C at z E C and the normal cone, the geometric counterparts of the Clarke directional derivative and the Clarke subdifferential.
Recall that a set D is a cone xîtD Ç D for all t > 0. Suppose that z E C.
It is easy to check that d^(z; u) > 0 for all u EX. We define the tangent cone to (7 at z to be the set
Tc{x) = {u E X : dc{x; v) = 0} and the contingent cone to C at z to be the set
Kc{x) = {z E X : Ve > 0 E (0, e) and w E v cB st. x -\- tw Ç. C}. 13
The following theorem imphes that the above tangent cone Tc(3:) is inde- pendent of the norm of X.
T h eo rem 2.6 ([16, Theorem 2.4.5]) Let x e C. Then v G Tc{x) iff, for every sequence Xi in C converging to x and sequence ti in (0, +oo) decreasing to 0, there is a sequence V{ in X converging to v such that æ* + Lvi G C for all i.
It follows from Theorem 2.6 that Tc{x) Ç K{x) Vx G C. We say that the set C is regWor at x G C provided Jc(x) = A"c(x).
Definition 2.7 Let f : X (—oo,+oo] be an extended real-valued function.
The effective domain of / is the set
dom f := {x G X : —oo < / ( x ) < -foo}.
We say that / is proper if dom f is nonempty. The epigraph of / : X —)•
(—oo, 4-oo] is the set
ep if := {(x, r) G dom f x R : /(x) < r}.
The function / is said to be regular at x provided epi f is regular at (x, / ( x ) ) .
When / is Lipschitz near x, by [16, Theorem 2.4.9], the function / is regular at X iff / satisfies the following two conditions;
(a) For all v G X,the usual one-sided directional derivative of / at x in the
direction v given by
/(x-t-tn) - /(x) f'(x;v) = lim ^ ^ t-^o+
exists. 14
(b) For all v e X, f'{x; v) = f°{x] v).
P ro p o sitio n 2.8 ([16, Proposition 2.3.12]) Let fi be Lipschitz near x G X for each i £ I = {i — 1, ■ ■ ■, m}. Denote
f{x) = max{/j(rr} : i e 1} and I{x) — {i E I : fi{x) = f{x)}.
We have
Ç co{a°: 2 G /(T)}
(wAere co (fenofea connea; AWi), onj ÿ / i ts regWor of a: /or eocA ê in 7 (a;), f/ien equality holds and f is regular at x.
By Proposition 2.3, the tangent cone Tc(x) is closed and convex. This property allows us to define the normal cone to C at x by polarity with Tc{x) as follows:
7Vc(z) = {(€% ': 2;) < 0 VuG 2b(T)}.
The normal cone Nc{x) has the following properties.
P ro p o sitio n 2.9 ([16, Proposition 2.4.2])
7/c(a;) = cZ{|J dc(a;)}, A>0 where cl denotes weak* closure.
T h eo rem 2.10 ([16, Theorem 2.4.7]) Let f be Lipschitz near x, and suppose
0 0 d°f{x). If C is defined as {y E X : f{y) < f{x)}, then one has
{uG%:r(:c;n) <0}C7b(a;).
If f is regular at x, then equality holds, and C is regular at x. 15
Corollary 2.11 ([16, Corollary 1, p.56]) lef / 6e nmr a; and 0 0
Nc(ar) Ç U Aa°/(z). A>0 If f is regular at x, then equality holds.
P ro p o sitio n 2.12 ([16, Proposition 2.4.4]) ^ C is conrez, (Aen, /or eacA a; €
C, Nc(x) coincides with the cone of normals in the sense of convex analysis, that is,
JVc(T) = {(€ % ': (^,c-a;) <0 VcE C}.
Proposition 2.13 ([16, Corollary, p.52]) Suppose that / is Lipschitz near a; and attains a minimum over C at x. Then 0 G d°f(x) + Nc(x).
Proposition 2.14 ([16, Corollary, p.61]) Let f be Lipschitz near x. Then an element ^ of X* belongs to d°f{x) iff {f, -1) belongs to Ngpi f(x, f{x)).
An Extended Definition of the Clarke Subdifierential
According to Proposition 2.14, the concept of the Clarke subdifferential can be extended to an extended real-valued function.
D efinition 2.15 ([16, Definition 2.4.10]) Let f : X [—oo, -Too] be finite at
X E X. The Clarke subdifferential d°f{x) is the set
a°/W := {f € V : (Ç.-1) 6 f{x))}.
Let C be a subset of X. Then the indicator function of C defined by
I / \ f 0 if z G C 00 ifz $ (C 16 is an extended real-valued function. When z is a point in C, the Clarke subd- ifferential d°tl^c{x) happens to be the normal cone to C at x.
P ro p o sitio n 2.16 ([16, Proposition 2.4.12]) Let the point x belong to C. then
and ipc{x) M regular at x iff C is regular at x.
From Proposition 2.14, the new definition of d°f{x) in Definition 2.15 is con sistent with the previous one for the locally Lipschitz case. Unlike the Lipschitz case, the new set d° f{x) may be empty for the non-Lipschitz case. However if
/ attains a local minimum at x then d° f{x) must be nonempty.
P ro p o sitio n 2.17 ([16, Proposition 2.4.11]) Let f : X [—oo, -Foo] be finite at x £ X and / attains its local minimum at x. Then 0 € d°f{x).
The Clarke subdifferential satisfies the following sum rule and chain rule.
Proposition 2.18 (Sum Rule [16, Corollary 1, p.105]) Suppose that is finite at X and /g is Lipschitz near x. Then one has
a°(A + /2)(T)ca°A(z)-ba°/2(z), and there is equality if fi and are also regular at x.
P ro p o sitio n 2.19 ([16, Theorem 2.3.9]) Consider the composition function 17 where h \ X ET' is a function whose component functions hi {i = 1, ■ • •, n) ore neor z E % onif g : 72" —^ 72 ia 7,(pacA*(z neor A(a;). One Aoa
9°/(z ) Ç c ô { ^ a i( i : E := («i, - ,0:^) E 9°g(/i(z))} i=l
{where œ denotes weak* closed convex hull), and equality holds if g is regular at h{x), each hi is regular at x, and every a of d°g{h{x)) has nonnegative components. {In this case it follows that f is regular at x.)
A function / : X —)■ (—oo, + oc] is said to be lower semicontinuous at x e X provided that
If f is lower semicontinuous at each point x E X, then / is called lower semi continuous. Given a topological space X, this is equivalent to saying that the epigraph epif is closed in X x 72 or that the level set {x E X : f{x) < r} iIS closed in X for every r E 72 (see [30, Theorem 4, p.103]).
Similarly we say that a function / ; X —>■ (—oo, -foo] is weakly lower semi- continuous at z E X provided that for each sequence {xj} weakly converging to x we have
/(z;) < lim inf/(zi). Η»-fCX3
Clearly if / is weakly lower semicontinuous then it is lower semicontinuous. The converse may not be true, but a convex function on a locally convex space is lower semicontinuous if and only if it is weakly lower semicontinuous (see [30,
Corollary 2, p.105]). The norm of X* is weak* lower semicontinuous (see also
[21, Exercise 9, p.128]). 18
Let X be a Banach space and / : X —>■ (— 00, + 00] be lower semicontinuous at a: G dom f. The Clarke-Rockafellar generalized directional derivative of / at
X in the direction u G X is defined as follows:
r(x;0:= lim limsup inf + e->0+ 1 w ev+ eB t y-^x t->0+ where y-Ux signifies that y and f{y) converge to x and f{x), respectively.
Clarke [16] proved that the extended f° and d°f have the same relation as they were in the Lipschitz case.
Proposition 2.20 ([16, Corollary, p.97]) One has d°f{x) = 0 iff f°{x; 0) =
—00. Otherwise, one has
= {^ € X* : ((, n> < u) Vu G X}, and
u) = sup{((, u) : ( G /(a;)}.
R em ark 2.21 Proposition 2.20 implies that if f°{x; 0) ^ —00 then /° (z; 0) = 0 and f°{x] ■) is sublinear and hence convex.
It is easy for us to obtain the following relation from the above concepts.
Proposition 2.22 Let X be a normed linear space and / : X —^ (—00, + 00] be lower semicontinuous. For x G domf, consider the following:
{i) There exist hg G X and p > 0 such that ||hx|| — 1 and f°{x; hx) < —
(ii) There exists p > 0 such that ||^||* > p~^ for all f G d°f{x). 19
Then (i) implies {ii).
Proof. For x G domf if there exist hx £ X with \\hx\\ = 1 and p > 0 such that f°{x; hx) < — then for each f G d°f(x) we have
This implies ||Ç||* > |(^, hx) \ > pT^. This proves that (i) implies {ii). ■
The Subdifferential of a Convex Function
Recall that a function f : X (—oo, +oo] is said to be convex if for every x ,y E X and each A G (0,1) we have
f{Xx + (1 — X)y) < Xf{x) + (1 — X)f{y).
The following proposition implies that the Clarke subdifferential generalizes the subdifferential in the sense of convex analysis.
Proposition 2.23 Let / : AT —> (—oo, +oo] be a convex function on X and
X G domf. Then d°f{x) coincides with the subdifferential of f at x in thesense of convex analysis, that is,
= a/(z) := {(€% *: ( ( , ! / - z ) < /(%/) - /(a ;) V{/ G X } .
Proof. By definition f G d°f{x) if and only if {f, -1 ) G Ngp, f{x, f{^))- Since / is convex, epi f is convex. By Proposition 2.12, ({, —1) G Nepi f{x, f{x)) if and only if
(((, -1), (l/, r) - (z,/(T))> < 0 V(%/, r) G epi /. 20
This inequality is in turn equivalent to that
< /W - /W V^/ e X, that is, f E ^ /(z ). II
Example 2.24 ([70, Example 2.26]) Let X be a Banach space and / : X —^
(—00, +oo) be a function given by
/W = Vz E X.
Then
a°/(z) = a/(z) = {^ E X ' : (f,z) = 11(11, - ||z|| and ||(||, = ||z||}.
To characterize the noninclusion 0 0 d°f{x), we need this expression and the following lemma.
Lemma 2.25 Let X be a normed linear space and C be a nonempty weak* compact subset of X*. Then there exists ( E C such that ||(|j* = dc(0).
Proof. Let {(n} be a sequence in C such that l|(n||* —> dc(0). Then by the weak* compactness of C there exists a subsequence {(«,,} weak* convergent to some ( E C. Since the norm of X* is lower semicontinuous for the weak* topology, the norm of ( satisfies
11(11* < LûmiiLf||(m„ II, = dc(0). K-~^-f-OC
This proves that ||(||* = dc(0). ■ 21
Proposition 2.26 % 6e a BonocA apoce and / : % -4 B 6e IvipacWz near
X. Then the following are equivalent:
(i) 0 ^
(a) There exists ^ in d°f{x) such that
0 < ||(||. = min{||7/||, : 77 e /(a;)} = dg«/(z)( 0).
{iii) There exists fj, > 0 such that ||^||* > for all Ç G d°f{x).
If also X is reflexive, then (z) — {in) are equivalent to each of the following:
(iv) There exist h^ £ X and // > 0 such that ||A^|| = 1 and /°(x; hx) < —
(v) There exist ^ E d°f{x) and Vx E X such that ||^||* — ||%|| and
-zzg) = ((, -% ) = -||(||* - ||i;z|| < 0.
(vi) There exist ^ G d°f{x) and Vx E X such that ||^||* = ||%|| ond, for each
a G (0,1), there exists 6 > 0 such that
/(!/ - W < / W - (all^ll* - ll%ll < / W Vp G a; + 6B & V( G (0, d).
Proof. We prove this proposition by the following route:
(i) (ii) =4> (m) => (z); (zz) => (v) {iv) {in)-, {v) {vi).
(z) => (zz) : Let 0 0 d°f{x). Then
0 < IWII* V 77 G 22
Since, by Proposition 2.4, d°f{x) is nonempty, convex and weak* compact, it follows from Lemma 2.25 that there exists a ^ G d° f{x) such that
0 < 11(11, = min{||T7||. : E = dy/(z)(0).
{a) => (iii) follows immediately by taking p = ||(||“ ^ and (in) => (i) is obvious.
(n) =>■ (r;) ; If statement (ii) is true, then ( is a minimiser of the convex function
/!(;?) = ^lh||2 subject to 7? E /(z).
Applying Proposition 2.13 to C = we have
OEa°/t(() + Arc(().
This inclusion implies that there exists % E d°h(^) such that — % E 7Vc(()-
Since C = d°f{x) is convex and closed, by Proposition 2.12,
;? -(> < 0, %.e., (77, -7;z) < ((, -Uz> V77 E / ( z ) .
According to Proposition 2.4,
= max{(77, -n ,) : 77 E = ((, - % ) = - | | ( | | , - ||7;z|| < 0, where the last equality is due to the fact that in a reflexive Banach space X there holds
a°/i(() = 9A(() = {uz E A : ((,% ) = 11(11* ' ||%|| and ||(||, = ll^^zH}-
{v) => {iv) : If there exist ( E d°f{x) and E A such that ||(||, = ||%|| and
-Ug) = ((, -Uz) = -||(||. - 1|%|| < 0, 23 then, taking n = ||C||r^ and hx = we have
Thus (n) => {iv) holds.
{iv) => {iii) is direct from Proposition 2.22.
(n) 4^ (n%) : Let ^ 6 9°/(z) and n, E % be such that ||^||, = ||nz|| and
/"(a:; -t;i) = ((, -%> = - Ikzll < 0 .
Then, for a E (0,1) and for e = (1 —a)||^||*-||%||, by the definition of f°{x; —%), there exists 6 > 0 such that for each y £ x + 6B and t E (0, Ô) we have
= —II6II* ■ ll%ll + (1 — «)||CI|* ■ \\vx\\
= < 0, from which it follows that
/(l/ - tui) < /W - (allait, - ll^zll < /W E z + & Vt E (0,
Conversely, if ^ E d°f{x) and Vx £ X are such that ||^||* = ||%|| and for each a £ (0,1) there exists <5 > 0 satisfying
/(%/ - < / W - ^«11(11* - 1|%|| < /(%/) Vy E a; + & Vt E (0,6), then
+ ^ _ ^ ||^ ||^ . < 0 V%/ E a; + 6B & Vt E (0, J). 24
By the dehnitions of /°(z; —%) and 9°/(z), we have
— ■ ll%ll < (Ç) ~^x) < f°{x] —Vx) < —o;||Ç|i* • ||% || < 0 Va e (0, 1).
Letting a f 1 gives
-%;i) = ((, -t;,) = - ||( ||, - ||t;z|| < 0.
Remark 2.27 From the proof of Proposition 2.26 we see that if 0 0 d°f{x) and ^ is the element of least norm in d°f{x) then there exists Vx E X satisfying
(vi) in Proposition 2.26. It follows that for each a G (0,1) there exists a 5 > 0 such that for each y E x + SB and each t E (0,5) the point z — y — tvx satisfies
0 < a||2/ - z|| < - /(z)], where |1?||F^-
2.2 Various Derivatives
We recall some concepts of classical derivatives and study their relations to the
Clarke subdifferential.
Definition 2.28 Let F map a normed linear space X to a Banach space Y and
£{X , Y) be the space of continuous linear operators from X to Y.
• X is strictly (Hadamard) differentiable at z E X if there is an operator
DsF[x) E £{X, F) such that for every v E X
Ihn t->-o+ 25
and the convergence is uniform for u in any compact sets. The corre-
sponding operator DsF{x) is said to be the strict derivative of F at x.
• F is said to be Gâteaux {Fréchet) differentiable at a: E X if there is an
o p erato r DF{x) E £ ( X , Y) such that for any n in X
lim + _ oF{x){v) t-»0+ t
and the convergence is uniform with respect to v in any finite (bounded)
sets. The corresponding operator DF{x) is called the Gâteaux {Fréchet)
derivative of F at x.
• F is continuously Gâteaux {Fréchet) differentiable at æ E X if there exists
f > 0 such that F is Gâteaux (Fréchet) differentiable at each point y in
X + 5B and the mapping y - 4 - DF{y) is continuous atx. In particular, F
is Gâteaux {Fréchet) at z E X if F is continuously Gâteaux (Fréchet)
differentiable at x and the mapping y - 4 DF{y) is Lipschitz on æ 4 SB for
some é > 0.
For a function there are close relations among the strict differentiability, the
Clarke subdifferential and other differentiabilities of it.
P ro p o sitio n 2.29 ([16, Proposition 2.2.4]) A function f : X R is strictly differentiable at x and D J{x) = Ç iff f is Lipschitz near x and d°f{x) ~
P ro p o sitio n 2.30 ([16, Proposition 2.3.6]) Let f be Lipschitz near x.
(a) If f is strictly differentiable at x, then f is regular at x. 26
(b) If f admits a Gâteaux derivative Df{x) and is regular at x, then d° f{x) —
{D/W}.
Proposition 2.31 ([83, Proposition 2.17]) A function f : X R is strictly differentiable at x if and only if f is Lipschitz near x, Gâteaux differentiable and regular at x.
With the above propositions we derive the following simple but useful result:
Proposition 2.32 ([87, Proposition 3.1]) Let f be Lipschitz near x € X . Then f is strictly differentiable at x if and only if both f and —f are regular at x.
Proof. If / is strictly differentiable at x, then —/ is also strictly differentiable at X, and hence by Proposition 2.30 they are both regular at x.
Conversely if f and —/ are both regular at x, then by Proposition 2.18 we have
a7W + ô°(-/)(j) = {o} which means that d°f{x) is a singleton since both d°f(x) and d°{—f){x) are nonempty. Therefore / is strictly differentiable at x. ■
The condition for a mapping F : X Y to he Fréchet differentiable at z E
X is equivalent to the assertion that there exists an operator DF{x) E£{X, Y) such that for any e > 0 there exists 5 > 0 such that
\\F{x + h) — F{x) — DF{x){h)\\ < e|]h|] whenever h E X and ||h|| < S
(cf. [70, Definition 1.12]). Hence if F is Fréchet differentiable at x then F is continuous atx. However F is not necessarily Lipschitz continuous at x. 27
Example 2.33 The function / : A —y A given by
_ f ifz fO ( 0 ifz = 0 is Fréchet differentiable at z — 0 with the Fréchet derivative /'(O) = 0. Since for re ^ 0 the derivative f'{x) = 2x sin ^ | cos f'{x) is unbounded near z = 0. This implies that / is not Lipschitz continuous near x — 0.
It is easy to see that pointwise Fréchet (strict) differentiability implies Gâteaux differentiability. The following example shows that its converse may not be true.
Example 2.34 ([18, Exercise 11.20 (c), p. 66]) Let / : ^ A be dehned by
It is easy to check that / is Gâteaux differentiable at (0,0) but that / is not continuous there. And hence / is neither strictly differentiable nor Fréchet differentiable at (0,0).
Although pointwise Gâteaux differentiability does not imply Fréchet differ entiability, a continuously Gâteaux differentiable function is always continuously
Fréchet differentiable.
P ro p o sitio n 2.35 (e.g., see [25]) A function f : X R is continuously
Gâteaux differentiable at x Ç. X if and only if f is continuously Fréchet dif ferentiable at this point. Moreover, these derivatives are identical.
Based on Proposition 2.35, / is simply said to be at x if it is continuously
Gâteaux differentiable at x. Similarly f is at x provided / is Gâteaux at a;. 28
2.3 The Michel-Penot SubdiSerential
The concept of the Michel-Penot snbdiSerential is very similar to that of the
Clarke subdifferential. There are also analogous properties for these two subd ifferentials.
Deûnition 2.36 ([64, Dehnition 1.1]) Let f/ be an open subset of a locally convex topological vector space % and / : f/ — A be a function. The Mtchel-
Penot derivative of / at a; Ç 17 in the direction v ^ X is
r(x-v) := suplim supi a^/(z) := 6 %* : ((, u) < Vu E %}. Proposition 2.37 ([64, Proposition 1.2]) The mapping ff{x; •):«-> f'^{x; v) is sublinear. It is easy to see that if / is Lipschitz of rank L near x then for every v £ X we have /""(r;u) < /°(3:;i;) < (2 .1) This inequality with Proposition 2.37 implies that /""(a;; ï;i) < /^(a;; U2) + L||ui - ugH Vui, U2 E X. Since the above inequality holds with v\ and ug switched, ff{x] •) is Lipschitz of rank L on X. Besides it follows from inequality (2.1) that Ç 9°/(z). 29 However this inclusion may be strict as shown by the following example. Example 2.38 ([16, Example 2.2.3], [12, Example 6]) The function ( 0 ifT = 0 is Lipschitz near 0 and Gâteaux differentiable with ao/(o) = {/(o)} = {0} c [-1,1] = av(o) where the first equality is from the fact that d^f{x) reduces to the Gâteaux derivative when f is Gâteaux differentiable at x. Proposition 2.39 ([64, Proposition 1.5]) Let U be an open subsetof a normed linear space X. Suppose f : U —> R attains its local minimum at x E U. Then 0 e a^/(a:). Proposition 2.40 ([64, Proposition 1.6]) Let U be an open subsetof a normed linear apoce %. Ginen / :[/— one hoa, /o r onp % € [/, u E %, (/ + g)"'(a:;i;) If f^{x] •) is finite and continuous at some point where g^{x; •) is finite, then ^(/ + g)(4 ^ + ^^(a;). It is known that if / is Lipschitz near x then f^{x; •) is Lipschitz continuous. In this case the sum rule in Proposition 2.40 holds no matter whether g’^{x] •) is finite at some point since if g'^{x; v) = +oo for each v E X then for each ^ E d^(f + g)(x) and rj € d^f(x) we always have (( - 7), u) < ^""(3;; u) Vn E X 30 which implies ^ — 77 G d^g{x) and hence Proposition 2.41 ([64, Proposition 1.7]) (7 6e on open an^eet 0 /0 normed Zineor apoce %. For eoch (a;, n) E (7 x % one hoa r(a;; -n) = (-/)^(z; n), ^(-/)(z) = -^/(a;). For more properties of the Michel-Penot subdifferential we refer to [6, 12, 22, 37, 63, 64]. 2.4 The Proximal Subdifferential Definition 2.42 Let / : X —> (—00,4-00] be lower semicontinuous and x E dom f. A vector ^ E X* is said to be a proximal subgradient of / at z if for some M > 0 there exists 5 > 0 such that /(!/) - / ( 4 + M ||p - a;||^ > (^, P - a;) Vp E a; 4- 6B . Another way to say this is that y-^x ||p - rr|p The proximal subdifferential of / at x, denoted by dpf{x), is the set of proximal subgradients of / at a;. Rem ark 2.43 The concept of the proximal subdifferential was first introduced by Rockafellar [74] for lower semicontinuous functions in 72". The reason that 31 “proximal” is used is that in i?” (even in a Hilbert space) the proximal sub differential can be defined through the closest point to a set. For a general Banach space, dpf{x) as defined is called the l-Holder-subdifferential (e.g., see [7]). For simplicity, in spite of slight abuse of terminology, we still call dpf{x) the proximal subdifferential of / on a normed linear space. By definition the proximal subdifferential has the following properties. Proposition 2.44 Let f : X (—oo, -t-oo] attain its local minimum at x. Then 0 e L em m a 2.45 ([47, Corollary 4A.5]) Let / : X —> (—oo, -t-oo] be lower semi continuous and Gâteaux differentiable at x. Then Ç {D /(z)}, where Df{x) is the Gâteaux derivative of f at x. The following example shows that a Gâteaux differentiable function may have no proximal subgradients. Example 2.46 Consider the function f{x) = —|ar|2, x E R . It is easy to see that Df{0) = 0. But dpf{0) = 0. Otherwise, suppose that dpf{0) were nonempty. Then, by Lemma 2.45, dpf{0) = {0}, that is, there exist M > 0 and d > 0 such that \x1^/^ -f Mx^ > 0 Va; € {—S, Ô). 32 This is a contradiction since the inequality fails to hold for any x e (—tW,M2: 0) U (0, M2,' Jz )' Note that in this example /(z) is at 0. This shows that 9f/(z) may be empty even for functions. However if a function / is at x, then dpf{x) is nonempty and coincides with {D/(z)}. P ro p o sitio n 2.47 ([83, Proposition 2.29]) Tef / : {oo} a:. TTien ^ /(a ;). This condition is only sufficient but not necessary. Even for a function f with both dpf{x) and dp{—f){x) being nonempty, the function / may still not Lipschitz near x. Example 2.48 It is easy to see that for the function \ _ f a;^8 i n ^ ifz fO { 0 ifa; = 0 both dpf{0) and dp{—f){0) are equal to {0}. However as we have seen in Ex ample 2.33, the function is not Lipschitz continuous near a; = 0. So / is not at a; = 0. Unlike the Clarke subdifferential and the Michel-Penot subdifferential, the proximal subdifferential usually does not satisfy the sum rule but has the fol lowing properties. 33 Proposition 2.49 JLef ^ % -4^ 72 U {+00 } 6e iotuer gemtcon^inuotw and a; w m (dom/) n (domg) aucA (/ia( J f/(z ) ond 9fg(a;) are 6otA nonemp%. TTien ^f/(a;) + 9fg(a;) Ç ( / + g)(a;). Proposition 2.50 ([83, Proposition 2.43]) Let f,g : X RU {00} be lower semicontinuous. Suppose that dpf{x) is nonempty and g is at x. Then (/ ± p)(z) = /(z) 4 : g(T). T heorem 2.51 ([18, Theorem 8.3, p.56]) Let X be a Hilbert space and f, g : X -4 (—00,+00] be lower semicontinuous. If g is Lipschitz near x G dom f and ^ € 9p(/ + g){x), then, for any e > 0, there exist Xi,X2 E x + eB such that |/(a:i) - /(a;)] < e, |^(a;2 ) - g(a;)| < e and f € dpf(xi) + dpg{x2 ) + eB. 2.5 The Fréchet Subdifferential Definition 2.52 Let C be a nonempty and closed subset of a Banach space X and x & C. For any e > 0, the set of Fréchet e-normals to C at x, denoted by Np{x; C), is defined to be the nonempty set A^(T; C) := e X' : lim sup ^ < e} y{€C)-^x \\y - x\\ for z with C n (x + SB) ^ {z} for any <5 > 0 and Np(x] C) := X* otherwise. In particular, when e = 0, the corresponding set Np{x] C), which is a cone, is called the Fréchet normal cone to C at z and is denoted by Np{x-, C). If X is not in the closed set C, we put Np{x] C) = 0 for all e > 0. 34 Definition 2.53 Let f : X (—00,00] be a lower semicontinuous function and let x G dom /. For any e > 0, the convex set := e X* : ((, -1 ) € ]V^((a;, /(a;)); epz/)} is called the Fréchet e—subdifferential of f at x. In particular, for e = 0, the cor responding dpf{x) is called the Fréchet subdifferential of / at a: and is denoted by The function / is said to be at a; provided that ^f'/(a;) is nonempty. For a Lipschitz continuous function the Fréchet subdifferential is bounded. Proposition 2.54 Tef / : % — (—00 , 00] 6e TipacAitz 0/ L near z. Then, /or onp e > 0, a^/(z)C(L-ke(l-hL))5^. In particular, dpfix) Ç LB*. P roof. Let ^ be in dpf{x). Then, for any Ci > 0, there exists 6 > 0 such that, for (y, u) e epi f with \\y — z|| < 6 and | u — f{x) (< L8, */ - a;) - (n - /(a;)) < (e + 6i)(||2/ - a;||+ | n - /(z) |) from which it follows that <(, 3/ - 4 < (/W - /(4) + (^ + - 3:||+ I /W - /(4 I). Using the Lipschitz condition we obtain (C; y — ^) < (L + (e -f ei)(l -f L))\\y — z||. 35 Hence for any y ^ x the following inequality holds which implies that ||^||* < L + (e + ei)(l + L). Since ci > 0 is arbitrary, ||(||*< j; + e(l + Z,). This proves that the inclusion stated holds. ■ Let / : X —> (—00, 00] be a lower semicontinuous function and letx G dom f. For any e > 0, we denote 4 /(x) := {Ç . X- : > -e}. Particularly, for e = 0, the corresponding dof{x) is denoted by df{x). Ioffe [36] proved that the Fréchet subdifferential dpfix) is the same as df{x). Proposition 2.55 ([36, Proposition 1]) Suppose that f is lower semicontinuous at X and e > 0. If ^ E dpfix), then ^ G dsf{x) with Ô = (e/(l — e))(l + ||^||*). Conneraefy, G /or aome > 0, then ^ G ^/(a;). TTma According to Proposition 2.55 we can further derive properties of the Fréchet subdifferential. Proposition 2.56 Suppose that f is lower semicontinuous atx. Then^ G dpfix) if and only if for each e > 0 there exists d > 0 such that /(*/) > /(z) + */ - r) - ell?/ -T || G T + 36 Hence 0 E dpfix) if and only if for every e > 0 the function /(•) 4- e|| • —x|| attains its local minimum at x. In particular if f attains its local minimum at T (Aen 0 E / (a;). P ro o f. By Proposition 2.55, f G dpfix) if and only if 11^ - :r|| This is equivalent to saying that for every e > 0 there holds ||i/ - a;|| that is, for every e > 0 there exists 5 > 0 such that /(%/) > / W + (^, 3/ - z) - e||3/ - a;|| Vy G a: + JB. Proposition 2.57 Suppose that f,g : X (—oo, +oo] are lower semicontinu ous at X e {domf) n (domg). Then G ( / + p)(z). P roof. The inclusion follows directly from Proposition 2.55 and the following inequality l i ^ i ^ (/ + W - (/ + 3/ - :r) ^ \\y - x\\ lim inf + lim inf Proposition 2.58 6'uppoae f/tof / : % (—00, + 00] w (oiuer aemiconfinuottg o( z E %. T/ien / w (fi^erenfiobie T ^ ond on/;/ i/ 6of/i ond dF(—f){x) are nonempty. Proof. It is known that / is Fréchet differentiable at x if and only if for some ^ e X* and any e > 0 there exists é > 0 such that l / W - / W - ((, 2/ - z ) | < c||2/ - a;|| Vi/ E z + This with Proposition 2.56 implies that Ç E dpfix) and —Ç € dp^—f)(x). Thus the necessity follows. To prove the sufficiency, we suppose that both dpfix) and 9jp(—/)(x ) are nonempty. Then by Proposition 2.57 we have + af(-/)W Ç {0}. Thus dpfix) = {$} and dF{—f){x) — { —^} for some Ç E X *. It follows that i t a i „ f M . r 4 X X j ^ > o II;/ - a;|| and \\y — a;|| From these two inequalities we obtain ||y - 4 that is, / is Fréchet differentiable at x. ■ Corollary 2.59 Suppose that f,g : X -> (—00, +00] are lower semicontinuous at X E (domf) fl {domg) and that f is Fréchet differentiable at x. Then &/W + + p)W. 38 Proof. Suppose that f is Fréchet differentiable at x. Then, by Propositions 2.57 and 2.58, = {0}. Thus + G {o} + a f ( / + g)(T) = ( / + ^)(z) G + ajpp(z). This completes the proof. m It is interesting that some special Banach spaces can be characterized with the fuzzy sum rule of the Fréchet subdifferential. Definition 2.60 A Banach space X is said to be an Asplund space provided every continuous convex function defined on a nonempty open convex subset U of X is Fréchet differentiable at each point of some dense Gs subset of U. If the dual space X* of the Banach space X is separable, then X is an Asplund space ([70, p.22]). Every reflexive Banach space is an Asplund space ([70, p.24]). But not all Banach spaces are Asplund spaces. For example neither P nor l°° is an Asplund space ([70, p.13]). Fabian has characterized an Asplund space interestingly in terms of the Fréchet subdifferential as follows. Theorem 2.61 ([28, Theorem 3]) X is an Asplund space {if and ) only if it is trustworthy in the following sense: for any e > 0 , é > 0 , 7 > 0 , for any 39 functions / i , • • •, /„ : X —> (—00, + 00], n > 2, and for any z E X such that fi w Zotuer and /z, " , A ofiG m a neig/i6orAood 0/ z (Ae following inclusion holds d e ifl + h fn ){z) Ç u{9i?/i(zi) H------h dpfni^n) '■ Zj E z + SB, \fj{zj) — fj{z)\ < 5, j = 1, ■ ■ ■ ,n} + {e + j)B*. 2.6 The Dini Subdifferentials Definition 2.62 Let f : X (—00 , + 00] be lower semicontinuous at a; € dom f. The upper Dini derivative of / at x in the direction v E X is /+(.;») :=üm suph£±tïW M . u—^v t The upper Dini subdifferential of / at x is the set a+/(z) := {( E X* : <(, n) < /+(a;; u) Vu E X}. Similarly the lower Dini derivative of / at x in the direction v E X is /-fe.):==lim m th£±M ziM . ^ U-l-V t t-^0+ The lower Dini subdifferential of / at a: is the set := (^, u) < /-(z; u) Vu E %}. It is immediate from definition that the lower Dini subdifferential has the following properties. Proposition 2.63 Suppose that f is lower semicontinuous at x. If f attains its local minimum at x then 0 E d~f{x). 40 Proposition 2.64 f/iat /, p : % (—00, +00] ore fower gemicontino- 00g o( T € (dom/) n (domg). TAen 9'/(z) + 9"^(z) Ç ^ ( / + g)(z). It is easy to see that if / is Lipschitz near x then /+(a;;n) = /^(a;;n) := lim sup t->o+ t /-(z ; n) = //" (a;; n) := Urn inf 9 -/( 3;) = a^/(a;) :={(€% *:((,u) 9 + /( z ) = 9 ^ / ( 3 ;) := 6 X* : (& u) < /^(3;;u) Vu e %}. By the simpler form of the definitions the following proposition is immediate for a Lipschitz continuous function. Proposition 2.65 Let / : X —>■ (—00, 0 0 ] be Lipschitz of rank L near x, then 9 /( 3;) Ç 9f/(a;) Ç 9f/(a;) Ç 9 -/( 3;) C 9+/(3;) C LB*. Proposition 2.66 Let / : X —> (—00, +00] be lower semicontinuous at x E d o m /. Then / is Gâteaux differentiable at x if and only if both dff{x) and df{—f){x) are nonempty. P r o o f . The necessity is obvious. We only need to prove the sufficiency. Sup pose that both dff{x) and df{—f){x) are nonempty. Then similar to Proposi tion 2.64 we have % /W +%(-/)(a;) Ç % ( / - / ) (a;) = {0}. 41 Thus = {6} and (—/)(z) = for some ^ E %*. It follows that ii^i„f/(^+M .= iW>fc„) v.ex (-4.0+ t - \S, / an d |imsupl(l±5^hLZM <(ç,„) v „ e x t-+o+ i From these two inequalities we obtain f-+o+ ÿ ' ^ that is, / is Gâteaux differentiable at x. ■ Note that for a concave continuous function f the lower Dini sub differen tial dl(—f)(x) is always nonempty since dl(—f){x) coincides with d°{—f){x) (which is nonempty). Hence by Proposition 2.66 we have the following corollary. C o r o lla r y 2.67 Let f : X R be a concave continuous function. Then f is Gâteaux differentiable at x if and only if dff{x) is nonempty. Similar to the Fréchet subdifferential, the lower Dini subdifferential has the following fuzzy sum rule: Proposition 2.68 ([35, Theorem 2]) Let /i, • • •, /n be lower semicontinuous at X E i2”. Then d (/i H h fn)ir) Ç n U H------f 9 fn{Xn) + SB) s>o xjeu{fj,x,s) where [/(/, (^) = {^ E E" : ||t/ - z|| < f, /(y) - /(z) < J}. It is worth noting that the above property does not hold in a general Banach space. To explain this we consider the following example due to Ioffe [35]. 42 Example 2.69 Let X = (7(5), where 5 is a compact HausdorfF topological space with the property that no point of S has a countable base of neighborhoods (as, say, S = {0,1}^ where K is an uncountable cardinal). Then any continuous function on S attains its minimal value on an infinite set and never at a single point. Let /(3:(-)) = ^3:(g). This function is concave continuous, hence Lipschitz, and /-(z(-); h(-)) = /'(a;(-); /^(-)) = /i(a), where 5(æ(-)) = {s G 5 : a;(s) = f{x{-))}. The set S{x{-)) contains at least two distinct points, hence there is an h{-) G C{S) with ||h(-)|| = 1 assuming both values 1 and —1 on 5(x(-)). Then /'-(z(.);h()) = /-(T(-);-fi(-)) = - 1, which implies that d~f{x{-)) = 0 . Now if we take fi = -f{x{-)) and = f{x{-)) then fi is convex continuous, /2 is concave continuous and, for each x e X, d^{fi + f2 )(x) = {0} but D U (^"/i(a:i) + ^"/ 2(a;2) + (^B) = 0 <5>0 XjeU(fj,x,5) since d~f2 {x2 ) is empty for each X2 G X. This implies that Proposition 2.68 does not always hold in general Banach spaces. In the following theorem we use the lower Dini derivative to give a sufficient condition for / to satisfy the inequality ||^||* > for all ^ G d~f{x). This condition is also necessary for a Lipschitz regular function / on a reflexive Banach space. 43 P r o p o s itio n 2 .7 0 JDet % 6e o normed Zineor apoce ond / : % - i (—cc, +oo] 6e lower semicontinuous. For x G domf, consider the following: (i)There exist hx ^ X and /x > 0 such that ||/ia;|| = 1 and /" (z ; hx) < {ii) There exists /x > 0 such that ||^||* > /x~^ for all ^ G d^f{x). Then {i) implies (ii). If also X is a reflexive Banach space and f is Lipschitz near x and regular at x, then (i) and (ii) are equivalent. P r o o f . For x G domf if there exist hx £ X with ||h^;|| = 1 and /x > 0 such that f~{x] hx) < —/x^\ then for each ^ G d~f{x) we have < - / x '\ This implies ||^||* > |(^, hx) \ > /x“h Thus (i) implies (ii). Recall that a locally Lipschitz function / is regular at x iff f°{x; •) = f~{x; •) which is equivalent to d°f(x) = d~f(x). If X is a reflexive Banach space and f is Lipschitz near x and regular at x, then (ii) implies that there exists /x > 0 such that > /x~^ for all $ G d°f{x). By Proposition 2.26 there exists hx EX such that [| 6j;|| = 1 and f°{x; h x ) < — /x '\ that is, / “ (a:; h x ) < — /x“h Thus (ii) => (?) holds. Therefore (i) and (ii) are equivalent. ■ Chapter 3 Equivalent Formulations of Ekeland’s Variational Principle At the beginning of this chapter we review the Ekeland variational principle and the Takahashi theorem. To weaken the condition in the Takahashi theorem we introduce the e-condition of Takahashi and the e-condition of Hamel with a for a proper lower semicontinuous bounded below function / on a metric space (X,d). With the Ekeland variational principle these conditions are shown to be equivalent to each other and weaker than the condition of Takahashi for / to possess not only minimizers but also weak sharp minima. From this relation we derive the corresponding sufficient condition for / to have error bounds. This error bound result allows us to establish a fixed point theorem with a weaker condition than that of the Caristi-Kirk fixed point theorem. Since the Caristi-Kirk fixed point theorem implies the Takahashi theorem which in turn implies the Ekeland variational principle, these theorems are all equivalent to the Ekeland variational principle. As a result, the completeness of a metric space can be characterized in terms of the (-condition of Takahashi. 44 45 3.1 The Ekeland Variational Principle The Ekeland variational principle states that for any point which almost min imizes a given function there is a “ nearby point ” which actually minimizes a slightly perturbed function. This principle usually reads as follows: Theorem 3.1 (Ekeland’s variational principle [25]) Let {X, d) be a complete metric space and f : X (—oo, -t-oo] be a proper lower semicontinuous function bounded from below. If u is a point in X satisfying /(«) < iÿ / + G for some e > 0, then for every A > 0 there exists a point x in X such that (i) /W < /(n). (a) d{u, x) < A. {in) For ally ^ x in X, one has /W + z) > /(z). The proof of this theorem is based on a device due to Bishop and Phelps [4]. Along the line of Ekeland’s proof of the variational principle given in [27], Takahashi [79] established the following result: Theorem 3.2 (Takahashi’s theorem [79, Theorem 1]) Let {X, d) be a complete metric space and / : X —> (—oo, -foe] a proper lower semicontinuous function 46 bounded below. If for each x E X with infx / < f{x) there exists y E X such that y ^ X and /(%/) y) < /(%), (/ten (Aere o poW To € % attcA /(To) = inf{/(T) : T E %}. From Theorem 3.2 Takahashi derived the Caristi fixed point theorem ([14, Theorem (2.1)’]) and the e-version of the Ekeland variational principle which are well known to be equivalent (see[69]). Theorem 3.3 (Caristi’s fixed point theorem) Let {X, d) be a complete metric space and let f : X (—oo, +oo) be a lower semicontinuous function bounded below. Let T : X ^ X be a mapping satisfying d(T,TT) T h e o r e m 3.4 (Ekeland’s e-variational principle) Let {X, d) be a complete met ric space and let f : X —>• (—oo, +oo] be a proper semicontinuous function bounded below. Lete > Q be given and u E X be such that / W < j n f / ( T ) + €. Then there exists a point v E X such that f{v) < f{u), d{u, v) <1 and f{w) > f{v) — ed{v, w) Vw ^ V. On the other hand, Hamel [31] proved that the Ekeland variational principle implies the Takahashi theorem. Therefore these three theorems are equivalent 47 one another. And hence like the Ekeland variation principle, the Takahashi theorem is an important nonconvex minimization theorem. However it is worth noting that the condition in the Takahashi theorem is obviously too strong. Even for the simple function f(x) = y/x we are not able to use this theorem to explain the existence of its minimum since this function does not satisfy the condition of Takahashi. 3.2 The 6-conditions of Takahashi and Hamel with o Let (%, d) be a metric space and / : % —> (— 0 0 , + 0 0 ] be a proper lower semi- continuous function bounded below. Denote inf / := inf {/(a;) : a; € %} and We define the e-condition of Takahashi and the e-condition of Hamel with a for a proper lower semicontinuous bounded below function / on a metric space (%, d) as below. Definition 3.5 The function / : X (—00, -t-oo] is said to satisfy the e- condition of Takahashi with a if for some 0 < o, 0 < e < -t-oo and each x Ç: X with infx / < f{x) < inf% f + e there exists y G X such that y x and /(%/) 4- ad(a;, %/) < /(a;). 48 We say that the function / satisfies the e-condition of Hamel with a if for some 0 < a , 0 < e < 4 - 0 0 and each z E % with inf% / < /(z) < infx / + e there exists z € ^ such that /W + z) < /(a;). In particular for the case e = 4 -oc the e-condition of Takahashi and the e-condition of Hamel with a are respectively called the conditio» o/ ThAoAagfii and the con dition o/ Hofnet with CK. It is clear that, for any 0 < ei < eg, the eg-condition of Takahashi with a implies the ei-condition of Takahashi with the same a and the eg-condition of Hamel with a implies the ei-condition of Hamel with the same a. For any 0 < e the e-condition of Takahashi and the e-condition of Hamel are respec tively weaker than the condition of Takahashi and the condition of Hamel. For example the function f{x) = y/x satisfies the e-condition of Takahashi an d the e-condition of Hamel with a for any a > 0 and 0 < e < but it does not satisfy the condition of Takahashi nor the condition of Hamel. Furthermore the e-condition of Hamel always implies the e-condition of Takahashi. Next result asserts that in a complete metric space the converse is also true. Theorem 3.6 ([84, Theorem 2.6]) Let (X, d) be a complete metric space and / : X -4 (—0 0 , -t-oo] a proper lower semicontinuous function bounded below. For 0 < e < -t-oo, the function f satisfies the e-condition of Takahashi with a > 0 if and only if f satisfies the e-condition of Hamel with the same a. P r o o f . The sufficiency is obvious so we only need to prove the necessity. 49 Firstly for the case 0 < e < + 0 0 we denote Z = {z 6 % : /(z) = infx/} an d Ma (a;) :={%/€%: / ( y ) 4- 2/) < /W } Vz € X. Then it suffices to prove that th e set Ma (æ) n Z is nonempty for each x e X with inf% f < f{x) < inf% f + e. Let X e X with inf% / < f{x) < inf% / + e be fixed. Since / is lower semicontinuous, the set Ma(x) is nonempty and closed. Hence the restriction (/, Ma(z)) of / to Ma(z) is also proper lower semicontinuous and bounded below. Thus, by Ekeland’s variational principle, there exists x G Ma(x) such that / W + a d ( 2/,z) > /(Z^) V|/ G Ma(z) with 1/ f T. In fact this inequality holds for all y in X with y ^x. Otherwise we suppose that there were a point u G X\Ma{x) such that /(u) + < /(z). Note that the following inequality y(T) + ad(z,z) < /(a;) holds. From these two inequalities and the triangle inequality it follows that / satisfies the inequalities fix) < inf / + e and f{u) + ad{x, u) < f{x). The latter implies that u G Ma (x). This contradicts the assumption about the point u. Therefore we have /(y) + ad(y, z) > /(z) Vy G % with y # 50 In addition the point x G Ma{x) must be a minimize: of the function /, that is, X Ç: Z. Otherwise suppose that x were not in Z then we would have iÿ / < < inf / + By the assumption there exists a point ÿ G X with ÿ ^x such that /(^) -t-ad(^,^) < /(r). But this is impossible since we have already the inequality /(p) + > /(z). Therefore the set Mg (a;) O X is indeed nonempty. Next we suppose that / satisfies the condition of Takahashi with a. Then for each 0 < e < +00 the function / satisfies the e-condition of Takahashi with a. Thus / satisfies the e-condition of Hamel with a. This implies that Z is nonempty. For each x £ X with infx / < f{x), if f{x) < 4-00 then infx f < f{x) < infx / + e for some 0 < e < -t-oo. In this case we can find z e Z such that /(z ) -|-0!d(z,z) < /(z). If f{x) = + 0 0 then this inequality holds for each z G Z. Therefore / satisfies the condition of Hamel with a. ■ 3.3 Weak Sharp Minima and Error Bounds For a proper lower semicontinuous and bounded below function / : X —>■ (—0 0 ,4-00], we say that / has weak sharp minima if the set Z of minimizers of 51 / is nonempty and there exists a > 0 such that < /(z) — W / Vz E % where dg(a;) = mf{d(z, z) : z E Due to the equivalence stated in Theorem 3.6, we can use the e-condition of Takahashi to replace the condition of Takahashi in the Takahashi Theorem to get more conclusions. Theorem 3.7 ([84, Theorem 2.7]) Let (X,d) be a complete metric space and f \ X {—oo, -f-co] a proper lower semicontinuous function bounded below. If for some 0 < a and 0 < e < -f-oo the function f satisfies the e-condition of Takahashi with a, then the set Z of minimizers of f is nonempty and for every X E X with infx / < f{x) < infx / + e and each z E Z there holds / W - / W > A(^z(a;). P r o o f . From Theorem 3.6 and its proof we see that the set Z is nonempty and for each x E X with in fx / < f{x) < infx / + e there exists z E Z such that /(z) -|-W(a;,z) < /(T) which with the simple inequality /(z) 4- adz(T) < /(z) -I- ad(a:, z) implies that, for each x E X with infx / < f{x) < infx / + e and each z E Z, the inequality /(z) -I- adz(z) < /(:r) holds. m 52 Exam ple 3.8 Consider the function /(%) = \/z on [0, +oo). Of course it is easy to see that / attains its minimum at z = 0. We can also use Theorem 3.7 to explain th e existence of minima of this function. In fact since / satisfies e-condition of Takahashi w ith a for any a > 0 and 0 < e < (4o^) by Theo rem 3.7, there exists xq G [0, Too) such that /( x q ) = inf[o,+oo) /■ Recall that a lower semicontinuous function / : X —> (—oo, -Too] has a local error bound if for some 0 < e < -t-oo there exists /r > 0 such that dg(x) := inf{d(x, s) : s E 6"} < /r/(x)+ Vx G X with /(x) < e where S = {x £ X : f{x) < 0} and /(x)+ = max{0,/(x)}. The function / is said to have a global error bound if the above inequality holds for all x G X. Now using Theorem 3.7 we derive the following sufl&cient condition for a lower semicontinuous inequality system to have error bounds. This is motivated by [66, Lemma 2.3] (under the condition 5 7^ 0) which is now a special case of the following result with e = 4-00. T h e o r e m 3.9 ([ 88 , Theorem 2.5]) Let (X, d) be a complete metric space and / : X —> (—00, -t-oo] be a proper lower semicontinuous function. Denote S {x £ X : f{x) < 0}. Suppose that for some 0 < p, 0 < e < 4-00 the set /~ ^(—00, e) is nonempty and for each x £ / “^(O, e) there exists a point y £ X such that 0 < f{y) < e and 0 < d(x,ïf) < /i[/(x) - /(;/)]. 53 7%e» 6' w and (fg W < / / / W + Vz e / " X - o o , e). Proof. Let // > 0 and 0 < e < +oo be given. Then /(•)+ is a lower semi continuous function bounded below with S — {x Ç X : f{x)^ — 0} and infx /+ := inf{/(a:)+ : a; G X} > 0. By Theorem 3.7, it suffices to prove g = Z : = {z E X :/W + = i ÿ A } , that is, infx /+ = 0. This must be true. Otherwise suppose infx /+ > 0 then for any z G X we would have some e > 0 such that z G / “^(O, e) for which there exists 2/ G X such that 0 < /(p) < e and 0 < d(z,p) < -/(p)] ffiom which it follows that 0 < /(p)+ < /(z)+ = infx /+- This is a contradiction. « Remark 3.10 Note that the nonemptiness of S in Theorem 3.9 is not a part of assumption but a part of conclusion. Since S' = {x G X : f{x) < 0} is nonempty, in fx /+ = 0. Hence if f satisfies the condition in Theorem 3.9 then /+ must satisfy the e-condition of Takahashi with a = jjT^. Applying Theorem 3.6 to the function /+ we see that for each x G / “^(O, e) there exists a point y G /~ ^(—oo, 0] such that 0 < d(z,%/) < 54 For a locally Lipschitz function / on a reflexive Banach space, if the set f^^[0, e] is compact for some 0 < e < +00 and satisfies that 0 0 d°f{x) for each X G / ”^[0, e] then / must have a local error bound. Corollary 3.11 o re/leMue Bonoch apace ond / : X — (—0 0 , + 0 0 ] be lower semicontinuous. Suppose that for some 0 < e < 00 the set /~^[0, e] is nonempty and compact, f is locally Lipschitz on / “^[O, e] and satisfies 0 0 d°f[x) for each x G /~^[0, e]. Then S is nonempty and there exists p > 0 such dg(T) < Vz E /"X-oo,c). P r o o f . Suppose that 0 ^ d°f{x) for each x G f~^[0, e]. Then for such x by Proposition 2.26 and Remark 2.27 there exist > 0 and //^ > 0 such that each point y G /^^(O, e) n (re + S^B) has a point z G / “^(O, e) n (z + d^B) satisfying 0 < lip - z|| < - /W]. Since / “ ^[O, e] is compact, there exist finitely many Xi G / “ ^[0,e], 6^ > 0 and A^i > 0 (1 < i < m) satisfying the above property and Let p — max{pi '■ I < i < m}. Then for each x G /~^(0,e) there exist y G / “ ^(O,e) such that 0 < Ik - p|| < //[/W - /(p)). Thus, by Theorem 3.9, S is nonempty and (fsM < A(/W+ Va: E 55 Rem ark 3.12 Note that to guarantee the existence of local error bound the set e] in Corollary 3.11 is not allowed to be replaced with / “^(O,e]. An easy counter example is the function f{x) = x G R. Even though, for any e > 0, f'{x) — 2x^0 for each x G /~^(0, e], the function / has no error bounds. 3.4 A Fixed Point Theorem Corresponding to the e-condition of Takahashi, we use Theorem 3.9 to present a sufficient condition for a multivalued mapping to have a fixed point. T h eo rem 3.13 ([84, Theorem 2.8]) Let {X, d) be a complete metric space and let f : X (—oo, -f-oo] be a proper lower semicontinuous function bounded below. Let T : X X be a multivalued mapping such that for some 0 < a, 0 < e < -t-oo and each x E X with infx / < /(^) < infx / + e there exists X G Tx satisfying ad(a;,z) < /(æ) - /(z). Then there exists an Xq E X such that G Tzo inf / < /(a:o) < i^ / + e. Moreover Z = Q {To G % : To e TTo, inf / < /(To) < W / -b e}. E>0 ^ P r o o f . Suppose that t ^ T t for all t G X with infx f < /(t) < infx / + e. Then for the function F{x) := f{x) — infx / and for each x E F~^(0,e) there exists a point x eTx satisfying 0 < o:d(T, T) < /(T ) - /(T ), 56 th a t is, 0 < ad(a;, f ) < F(a:) — F(z), By Theorem 3.9 there exists Xq Ç. X such that F{xq) = 0, or /(xq) = iuf% /. Thus for such an Xq there exists x^ e Txq such that 0 < ad(zo,^) < /(zo) - /(^ ) < 0. This is impossible. W e denote = {zo e % : ro G Tzo, iM / < /(zo) < inf / + e}. It is easy to see that Ç Z. On the other hand, if Z is empty then Z = CicyoZe- If Z is nonempty then it follows from the assumption that for each xq e Z there exists xq e Txq such that ad(To,^) < / W) - /(^) < 0. This implies that = xq. Thus xq G Txq for each xq G Z, that is, Z Ç D^yoZe. Therefore Z = HeyoZe- ■ Remark 3.14 Under the condition of Theorem 3.13, there exists a sequence {xn} such that G and lim„_>+oo/(^^n) = infx/- The set of minimizers of / can be determined by the set of fixed points of T. Upon taking e = +oo and a = 1 in Theorem 3.13, we obtain the famous Caristi-Kirk fixed point theorem [14, Theorem (2.1)’ ] (see also [1, Theorem 4.12] or [69, Theorem Cj). 57 Theorem 3.15 (Caristi-Kirk’s fixed point theorem). Let {X, d) be a complete metric space and let f : X (—00, +00] be a proper lower semicontinuous function bounded below. Let T : X X be a multivalued mapping such that for each X E X there exists x e T x satisfying < /(a;) - Then (here eaists an Zo E % anch (ha( a;o G Tzg. The Caristi-Kirk fixed point theorem can easily be used to prove the Takahashi theorem. Proposition 3.16 ([84, Proposition 2.10]) Let {X,d) be a complete metric space and let f : X ^ (—00, -Too] be a proper lower semicontinuous function bounded below. If f satisfies the condition of Takahashi with a = 1, then the Conafi-jTirh ^ e d poinf theorem imphea the Tohohoah* theorem. P r o o f. Suppose that for each x E X with infx / < /(a;) there exists y E X such that y ^ X and /(p) + d(a;,p) < /(a;). Define the mapping T : X X hy Ta; = {p E % : p ^ a; and /( p ) + d(a;, p) < / ( z ) } for a; € X with infx f < /(a;) and Tx = {x} îo i x E X with f{x) = infx /• It is obvious that the mapping T is multivalued and satisfies the inequality d(x,x) < f(x) — f(x) Vx eTx\/x E X. 58 By Theorem 3.15 there exists zo in X such that Zo € Tzo. Such an zo must be a minimizer of / since for each T E X with infx / < /W we always have T 0 Tz by the dehnition of T. This completes the proof. m There have already been various interesting proofs for Ekeland’s variational principle (see [5, 16, 25, 27, 79, 77]). However for the completeness of this chapter we use Takahashi theorem to prove the Ekeland variational principle as in [79]. Proposition 3.17 fief (X, d) a complete metric apace and let / : X — (—oo, +oo] be a proper lower semicontinuous function hounded below. Then the Takahashi theorem implies the Ekeland variational principle. P ro o f. For any fixed u e X with f{u) < infx +e, let Xo = {T E X : /(z) < /(u) - ^d(T,u)}. Then Xq is nonempty and closed since / is lower semicontinuous. And hence Xq is a complete metric space. In ad d itio n , for each T E Xo, ^d(z,u) < /(u) - /(T) < /(u) - iÿ / < e from which we have f{x) < f{u) and d{x, u) < A. It remains to prove that for some X E Xo the inequality /W + > /(a;) holds for all y ^ X in X. Suppose that for every x E Xq there exists z E X such that z ^ x and /(z) < /(x) - ^d(x,z). 59 Then Y(f(z, u) < z) + t() X X X < [/W -/W ] + [/W -/W ] = / W - / W and hence z € Xq. Applying Theorem 3.2 to the restriction of the function / to the complete metric space (Xq, ^d) we obtain a point xq e Xq such that f{xo) = infxo- This is a contradiction since, by the assumption, for such an Xo there exists zo € Xo such that /(zo) < /(To)- " Remark 3.18 Up to now, based on the Ekeland variational principle (Theo rem 3.1) we have proved that the e-condition of Takahashi with a > 0 implies the e-condition of Hamel with the same a (Theorem 3.6). As a result, the e- condition of Takahashi with a > 0 is sufficient for a function to possess weak sharp minima (Theorem 3.7). With this result we have established Theorem 3.9 which states that for the function /+ the e-condition of Takahashi with a > 0 guarantees the existence of error bounds for /. The error bound result allows us to obtain a fixed point theorem (Theorem 3.13) which extends the Caristi-Kirk fixed point theorem (Theorem 3.15). Furthermore we have proved the Takahashi theorem from the Caristi-Kirk fixed point theorem (Proposition 3.16) and the Ekeland variational principle from the Takahashi theorem (Proposition 3.17). Therefore the theorems involved are equivalent to each other. 60 3.5 The Completeness of a Metric Space As we have seen, the Ekeland variational principle and the Takahashi theorem are both established on a complete metric space. It is interesting that these two theorems are useful to characterize a complete metric space. In fact Sullvan [78] has used the Ekeland variational principle to characterize the completeness of a metric space while Takahashi [79] has presented an equivalent statement for a metric space to be complete. Theorem 3.19 ([78, Theorem 2]) Let (A, d) be a metric space. Then X is complete if and only if for every proper continuous and bounded below function / : A (—oo, +oo] and every e > 0 (here is o point v € A satis^iny (i) /(%;) < inf A: / + e (ii) /(w ) + ed(v, w) > /(v ) Vw v. Theorem 3.20 ([79, Theorem 3]) Let (A, d) be a metric space. Then the fol lowing are equivalent: (i) A is complete. (ii) If for every uniformly continuous function / ; A —>■ [0, +oo) and every X e X with infx / < f{x) there is a point y G A such that y ^ x and /( y ) 4- d(ar, y) < / ( z ) , then there exists xo E X such that f{xo) = inf{/(z) : z G A}. 61 We conclude this chapter with the following characterizations of a complete metric space in which (ii) and (in) are established according to Dr. Jane Ye’s suggestion. Theorem 3.21 Let (X, d) be a metric space. Then the following are equivalent: (i) X is complete. (ii) If for every uniformly continuous function / ; X —> [0, +oo) there exist aome 0 < /r ond 0 < e < + o o aucA (Ae e) ia nonem pty ond for each point x € /~^(0, e) there exists a point y € / “ ^[O, e) satisfying 0 < d(z, p) < - /(%/)], then S = {x e X : f(x) =0} is nonempty and daW < p/(a;)+ Va; e e). (Hi) If for every uniformly continuous function / : X —)■ [0, + oo) there exist some 0 < a, 0 < € < +oo and multivalued mapping T : X X such that each point x e X with infx / < f(x) < infx / + e has an x E Tx satisfying od(a;,z) < /(a;) - then there exists an Xq Ç: X such that Xq e Txq and < /W < mf/ + €. 62 (iv) If for every uniformly continuous function / : X —?► [0, + 00) there exist gome or > 0 omf 0 < e < +00 aucA (Aof eacA poinf z E X toifA infx / < /(z ) < infx / + c Aog o point p E X gotig/pinp p ^ 3; ond /(p ) + orj(a;, p) < /(a;), then there exists Xq E X such that f{xo) = inf{f(x) : x E X}. Proof. Since the implication (z) => (ii) is immediate from Theorem 3.9, it suffices to show (ii) => {in) => (iv) => (z). {ii) {in) : Let / : X —)■ [0, + 00) be uniformly continuous. Suppose that a; 0 Tz for all z E X with infx / < /(z) < infx / + e. Then for the function f(a;) := /(a;) - infx / and for each a; E e) there exists a point z E Ta; satisfying 0 < o:d(a;,^) < /(a;) - /(z ), that is, 0 < ord(a;,z) < F(a;) — F(z). By {ii) there exists xq E X such that F{xq) = 0, that is, /(xq) = infx /• Thus for such an xq there exists x^ eTxq such that 0 < ord(xo,%) < /(xo) - /(^ ) < 0. This is impossible. {in) {iv) : For a uniformly continuous function / : X -> [0, 00+ ), suppose that for each x E X with infx / < f{x) < infx / + e there exists p E X such that y ^ X and /(p) + ad(x,p) < /(x). 63 Deûne the mapping T : X -f % by Tz = {1/ E % : %/ 9 ^ T and /(^) + ad(z, %/) < /(%)} for each z E % with inf;c / < /(a;) < infx / + e, Ta; = {%} for each a; E % with /(a;) = infx / and Ta; = {z^} for each z E X with infx / + e < /(z) and for some fixed z^ E X with f{x^) < infx / + e- It is obvious that the mapping T is multivalued and satisfies the inequality ad{x,x) < f{x) — f{x) Vz E Tz Vz E X with inf / < f{x) < inf / + e. By (iii) there exists xq in X such that Xq e Txq. Such an xq must be a minimizer of / since for each z E X with infx / < f{x) we always have z 0 Tz by the definition of T. {iv) => (i) : Let {z„} be a Cauchy sequence in X. Then the function f : X R defined by /(z) = lim d(z»,z) Vz E X n->+oo is uniformly continuous and infx / = 0. Now for any fixed 0 < e < +00 and any z E X with 0 < /(z) < e there exists m e N such that d(zm, z) < 2/(z) and 3d(zn,Zm) < /(z) Vn > m. Thus 3f{xm) < /(z). And hence z^ # z and 3/(zm) + < /(z) + 2/(z) = 3/(z). 64 This implies that / satisfies the e-condition of Takahashi with a = |. By (iv) there exists xq e X such that f{xo) = inf% / = 0, that is, lim d(xn,Xo) = 0. n —>+oo Therefore (i) follows. Chapter 4 Error Bounds for Lower Semicontinuous Functions This chapter is devoted to global and local error bounds for an inequality system f{x) < 0 whose solution set is S {x £ X : f{x) < 0}. In section 4.1 sufficient conditions (similar to the e-condition of Takahashi with a) will be discussed for a function / on a metric space {X, d) to have error bounds. We will then introduce the d^,-subdifferential in section 4.2 which unifies many generalized subdifferentials including those studied in chapter 2. This concept is used in section 4.3 to express sufficient conditions for an inequality system on a Banach space comparing with the lower Dini directional derivative. In section 4.4 we derive a relation between / “(æ; •) and (/^)“(æ; •) to study error bounds with exponent. For a lower semicontinuous convex function we prove in section 4.5 that the existence of local error bound is equivalent to that of global error bound and that in a reflexive Banach space conditions presented in sections 4.1 and 4.3 are all equivalent to saying that /+ satisfies the e-conditions of Takahashi and Hamel with a. Since we mainly consider lower semicontinuous functions, 65 66 we can apply results in the above sections to a lower semicontinuous inequality system with closed constraint set to obtain corresponding conditions for error bounds in section 4.6. In the last section we will also investigate some important relations between global error bound and metric regularity. 4.1 Error Bounds for Nonconvex Functions on Metric Spaces For nonconvex functions Ng and Zheng [66] have obtained several interesting results about the existence of global error bounds. One of them is the following sufficient condition for a global error bound for a proper function on a metric space. Proposition 4.1 ([66, Lemma 2.2]) Let (X, d) he a metric space and f : X (—oo, +oo] a proper function; let p > 0 and 0 < p < 1 be constants. Suppose that S := {x E X : f{x) < 0} is nonempty and that for each x G / “^(O, +oo) = {%/ G X : 0 and -/(z/)]- Then, for each x e X, ds{x) < pf{x)+. P roof. It suffices to show that, for each x G / “ ^(O, +oo), ds{x) < pf[x). For any fixed x G / “^(O, +oo), let Zo = z and suppose that Xq,--- ,Xk have been 67 chosen in -foo) such that, for each I and By the given condition, there is x^-t-i € / “^[O, +00) such that < Pdg(zt) < and d(z&,a:t+i) < If /(zt+i) = 0 then /c+1 fe+l daW < d(To, zt+i) < = ju/(a;). i=l «=1 Thus we may assume that /(z^+i) > 0 and hence inductively we obtain a sequence {a;„} Ç /'^(O, + 00) such that 3:0 = T, dg(zn) < /)"dg(z) and d(3:n-i,:Cn) < - /(a:»)] for all n e N, where N is the natural number set. Hence, for n e N, n da(z) < d(z, Zn) + dg(%n) < ^ d(zi_i, 3;^) + p"dg(3;) 2=1 n = A* + /a"dg(a;) < /r/(z) + /)"dg(3:) i=l as f{xn) > 0. Since 0 < p < 1, the inequality ds{x) < nf{x) + p^ds{x) implies that dg(3:) < » Applying Proposition 4.1 to the function defined by A (a:) = / N + V'g. (z) V 3; 6 % where is the indicator function of the set S'e {x Ç: X : f{x) < e}, we can easily derive the corresponding sufficient condition for error bounds. 68 Theorem 4.2 (%, (f) 6e o metric apoce onj / : % (—0 0 , + 00 ] o proper function. Suppose that the set S := {x G X : f{x) < 0} is nonempty. If for some constants //>0,0 (4.1) p) < - /(p)], (4.2) then (tg(a;) < ///(%)+ Vz e / " X - 00, ^)- Note that the assumptions in Theorem 4.2 are very weak in that the space X is only required to be a metric space and f is only a proper function. If the space X is a normed linear space and the function / is lower semicontinuous, then one can replace condition (4.2) with an inequality involving the lower Dini derivative. We need the following result of the mean value type theorem given by Ng and Zheng [66, Lemma 2.1]. L em m a 4.3 ([66, Lemma 2.1]) Let X be a normed linear space and f : X (—00 ,+ 0 0 ] be a proper lower semicontinuous function; let x € dam f, h E X with ||h|| = 1 and t > 0. Assume that there exists 6 E R such that for each a E [0, t), f£(x + ah] h) < S. Then f{x + th) — f{x) < tS. 69 Proof. For any 0 < 6 < +oo, let = sup{0 < 8 < t : /( z 4- g/i) — /(z) < s(J + e)}. By the lower semicontinuity of f, we have /(æ + tfh ) - / ( z ) < + ^)- (4.3) We must have = t. Otherwise, suppose te < t. Then x + teh G dom f and so f£{x + teh; h) < S hj assumption. Consequently there exists t' G {te, t) such that / (z + t'h) — / (z + teh) < {t' te) (<5 + e). This and (4.3) imply that f{x + t'h) — f{x) < t'{5 + e). It follows from the definition of te that t' < te, which is a contradiction. This shows that /( z + th) - /(z) < t((^ + e). Letting e 0, we have f{x + th) — f{x) < tS. » As Ng and Zheng showed in [66, Theorem 2.4] that inequality (4.2) in The orem 4.2 can be replaced with a condition in terms of the lower Dini derivative for the case e — -foo, we use Lemma 4.3 to prove the corresponding result for the case 0 < e < -foo. T heorem 4.4 ([88, Theorem 2.3]) Let X be a normed linear space and f a proper lower semicontinuous function on X; let 0 < e < -foo, 0 < p < -foo and 70 0 < p < 1. Suppose that S is nonempty and that for each x € f ^(0, e) = {y G X ; 0 < f{y) < e} there exist tx > 0 and hx Ç: X with \\hx\\ — 1 such that + tzAz) < p(fg(a;) oW (a; + /ii) < -pT^ Vf € [0,2%). (4.4) Then < p[/(z)]+ Vz E /"X-oo, Proof. For the case e = +oo, let x E / “ ^(O, +oo) be fixed. We will show that there ?/ E +oo) such that dg(i/) < pdg(a;) and ||r - 3/|| < p [/(a;) - /(%/)]. If f{x + txhx) > 0, then we can take y := x + txhx since, by assumption (4.4) and Lemma 4.3 (applied to = —p“^), we have Ik - I/ll = f z < p[/(a;) - /(a; 4- f^hz)]. If f{x + txhx) < 0, let Sx = sup{0 < t Then f{x + Sxhx) < 0 and f{x + thx) > 0 for all t E [0, Sz). By the lower semicontinuity of f,Sisa, closed set. It follows from x ^ S that ds{x) > 0. Thus there exists s'^ E (0, Sx) such that dg(z + s^hz) < Ik + - (z + Sih%)|| = s - 8^ < pdg(a;) and x+s'^hx E Too). Note that f£[x+thx] hx) < —pT^ for each t E [0, 8^]. As in the first part of our proof, the point y := x + s'^hx has the desired properties. 71 Next we consider the case 0 < e < + 00. For any n > 1 with n G iV, let where & = {a; E X : /(a;) < (1 - and is the indicator function of S'». Then for a; G F^^(0,+ 0 0 ), by the assumption, there exist > 0 and hr G X with 11 hr 11 = 1 such that (z + fh r; h r) < Vt G [0, tr)- By Lemma 4.3, the above inequality implies that /(z + thr) - /(z) < Vt G (0, tr), that is, / ( z + t h x ) ^ / (a:) — t f x ^ < (1 — —)e Vt G (0, t x ) ■ Consequently (z + th r; h r) = hr) Vt G [0, tr). Therefore applying the result of this theorem for the case e = +00 to the lower semicontinuous function one has < /fFi.(z)+ Vz G F ^X - 0 0 , + 0 0 ). Since 1 < n G iV is arbitrary, the desired result is proven. ■ For any proper function on a normed linear space we give one more sufficient condition for error bounds which is very strong but guarantees the nonemptiness of 72 Theorem 4.5 % 6e o nonTiej fmear apoce and / : % — (—0 0 , + 0 0 ] 0 proper /rinc^io». 5'uppoae /or aome cona(on( /i > 0, 0 < e < + 0 0 / ^ ^ ( —0 0 ,e) ta nonemp(p o n j/o r eoc/i a; E /"^(O,e) (/lere eziata /tz G % \\hx\\ = 1 such that A>0 A Then S is nonempty and dg(a;) P r o o f . The set S must be nonempty. Otherwise if S were empty then there would be a point x € / “^(O, e) and a unit vector hx E X such that for each A > 0 we have f{x + Xhx) < f{x) — Taking A = pf{x) yields f{x + Xhx) < 0. This contradicts the assumption S' = 0. From the above proof we also see that for each x e / ”^(0, e) there exists a unit vector hx E X such that f{x + Xhx) < 0 for A = iif{x). It follows that (faW < IIAhzll = /i/(z). This completes the proof. m 4.2 Qj-subdifferentials To present sufficient conditions for error bounds for lower semicontinuous func tions on Banach spaces in terms of various generalized subdifferentials including those in Chapter 2, we introduce the concept of 9^,-subdifferentials for lower semicontinuous functions in this section. 73 Deûmtion 4.6 Let % be a Banach space and / : X — (— 00, + 00] be lower semicontinuous at z E dom f. A subset of X*, denoted by d^f{x), is called a d^-subdifferential of f at x if it has the following properties: (wi) 9wg(z) = 9w/(z) if p = / in a neighborhood of z. {0J2 ) 0 6 dojf{x) when / attains a local minimum at x. (wg) If / is convex and Lipschitz of rank L near r, then Ç LB*. (W4) The fuzzy sum rule holds, that is, if g : X (—00, +00] is convex and Lipschitz near x, then for any ^ E dfff + g){x) and any 6 > 0 there exist Z i,3:2 E z + (^B such th a t /(zi) 6 /(z)4-6Bi, ^(3:2) € g(T)+JBi, and ^ E + where Bi = (—1, 1). The concept of an abstract subdifferential with the sum rule can be traced back to Ioffe [34]. Thibault and Zagrodny [80] gave a definition similar to Ioffe’s which in turn has been generalized by Jourani and Thibault [40] with the fuzzy sum rule replacing the sum rule. We use the ô^j-subdifferential to generalize the above ones and to include more generalized subdifferentials in nonsmooth analysis. (For other kinds of subdifferentials we refer to [81].) The following subdifferentials all turn out to be 9w-subdifferentials. Example 4.7 Let X be a Banach space and / : X —)■ (—00, +00] be lower semicontinuous a t z E dom f. By Definition 2.15, Propositions 2.17, 2.4 and 2.18 the Clarke subdifferential d°f{x) satisfies properties (wi)-(w 4). Similarly 74 by Definition 2.36, Propositions 2.39 and 2.40, and the relation d^f{x) Ç d°f{x) for locally Lipschitz function / we see that the Michel-Penot subdifferential is also a ^^^-snbdifierential. Example 4.8 Let X be an Asplund space and let / ; X -> (—oo, +oo] be lower semicontinuousa t z E dom f. Based on Propositions 2.55, 2.56, 2.54 and Theorem 2.61, the Fréchet subdifferential dpfix) is a 9^,-subdifferential of / at X. Example 4.9 Let H he & Hilbert space and f : H (—oo, +oo] be lower semicontinuousa t æ E dom f. It follows from Definition 2.42, Proposition 2.44 and Theorem 2.51 that dpf{x) satisfies properties (wi), (wg) and (W4 ). Besides if / is Lipschitz of rank L near x then dpf{x) Ç d°f{x) Ç LB. Thus dp f{x) satisfies property (wg). Therefore dpf{x) is a ^-subdifierential of f at x. Example 4.10 Let (—00, + 00] be lower semicontinuous a t a; E dom f. By Definition 2.62, Propositions 2.63, 2.65 and 2.68 we see that the lower Dini subdifferential d~f{x) is a ô^-subdifferential of / at x. However as we indicated in Example 2.69, Proposition 2.68 is not valid in a general Banach space X. So d~ f{x) is not always a 5^-subdifferential of / at a: E X. 4.3 Error Bounds for Lower Semicontinuous Func tions on Banach Spaces Consider a simple inequality system /(a;) < 0 , 75 where / is a locally Lipschitz function defined on R. If the set 5' := {z E A : /(a;) < 0} is nonempty, then the inequality dg(z) < holds automatically for any X 6 5 and any > 0. To look for a su@cient condition for this inequality to hold for some p > 0 and any point x E R\S, we can take one point xq E S such that f(xo) = 0 and f{y) > 0 for any y G {xq , x ] = {txo + {l-t)x :te[0, 1)}. By the m ean value theorem (Proposition 2.5), there exist z € {xq, z] and ^ E d°f{z) such that /(z)-/(zo) = ^-(z-zo) from which /(z)+ = ||^|| • ||z —zo|| > ||^||dg(z). Therefore if ||Ç|| > holds for some p > 0 and any ^ E d°f{x) for each x E R\S, then ds{x) < /j,f{x)+ holds for any x E R. For a lower semicontinuous function / defined on a Banach space X, will the existence of a positive c o n stan t ju, such that ll^ll* > E Vz E also implies the existence of a global error bound? Wu and Ye [ 86 , Theorem 3.1] have given an affirmative answer. The following result is a refinement of [86 , Theorem 3.1]. Theorem 4.11 Let X be a Banach space, / : X -> (—oo, +oo] he lower semi continuous and S {x E X : f{x) < 0}. Suppose that for some 0 < e < +oo the set / “ ^(—oo, e) is nonempty and there exists p > 0 such that 11(11* > E 9w/(z) 76 /o r any z 0 < /(ar) < e (or /or gome aro € 5" anj ony a; ||z — Zo|| < e and 0 < f[x) < +oo). Then S is nonempty and (fg(z) < /i/(a;)+ Vz e % /(z ) < e ( or ||z - zo|| < Proof. We only need to prove that ds{x) < ///(z)+ Vz G X with /(z) < e ( or jjz — zo|| < -). Suppose that there were u such that f{u) < e {ov u E xq + (|)-B for the case 0 < e < +oo) and dg(n) > /i/(n)+. Then n ^ and hence 0 < /(n) < c. Besides we can choose t > 1 (and a > 0) such that dg(n) > W(n) := ( and ||n - Zo|| < (4.5) JL -T OL L Thus /(n)+ = /(u) = 7 (t/x)"^ and hence /(«)+ < W /(i;)+ + 7(tp)"'\ V^X Note that the function /(•)+ is lower semicontinuous and bounded below. Applying Ekeland’s variational principle (Theorem 3.1) to /(•)+ with e = 7 (t/x)"^ and A = 7 , we find x EX satisfying / w + < /(^ )+ , (4.6) | | z - n | | < 7 (4.7) and /(n)+ + (t/i)-^h(n) > /( z ) + Vn 6 X , (4.8) 77 where h{v) := ||u — x\\. From (4.5), (4.7) and (4.8), we have T E %, a; ^ S' and 0 < /(z ) < +oo. (4.9) On the other hand, (4.8) implies that the function /(u)+ + (t/^)-^h(n) attains its minimum on X at x. Hence, by property (wg) in Definition 4.6, 0 E ^w[/(a;)+ + (f/f)"^h(a;)]. (4.10) Since / is lower semicontinuous and 0 < f(x), there exists > 0 such that 0 Thus, by property (wi) in Definition 4.6 and (4.10), 0 E 9w(/ + (t//)"^fi)(a;). (4.11) Let (5 ;= min{/(z), (1 — e — /(u), ae(2 + a)~^}. Then by property (W4) in Definition 4.6 and (4.11) there exist xi and xg both in x +SB such that /(z ) - 6 < /(a;i) < /(a;) + and 0 e 9w/(zi) + 9w((W^/i)(:z:2) + The inequalities mean that Xi E x + SB and 0 < f{xi) < + 00. The inclusion, by property (wg) in Definition 4.6, implies that there exists ( E gw/(a:i) 78 such th a t II6II, < W + (^ < ((//)-' + (1 - which contradicts the assumption since from (4.6) we have 0 (or ||zi — Zoll < ||zi — z|| 4- ||z — m|| + l|u — Zo|| < + 7 H- 2 + a a e , , , e _ (1 + o)e ,, , - 2T Ï + + + - 2 + a 2 + CK Rem ark 4.12 It is worth noting that in Theorem 4.11 the nonemptiness of S is a part of the conclusion not a part of the assumption, w hich is different from the previous results including [86, Theorem 3.1]. In addition, in [86, Theorem 3.1] the inequality ds{x) < nf{x)+ holds only for all x G X with f{x) < e/2. We tried to show it to be true for all z E X with f{x) < e but did not succeed until Dr. Qiji Zhu pointed out recently that it is really true as long as the Ekeland variational principle is completely used. According to Theorem 4.11, Examples 4.8 and 4.9, we have the following corollaries. Corollary 4.13 Let X be a Hilbert space, / : X —>■ (—00, +00] be lower semi continuous and S' := {z G X : /(z ) < 0}. Suppose that for some 0 < e < +00 the set /"^(—00, e) is nonempty and there exists // > 0 such that ||(||. > V( G ap/(z) 79 for any x with 0 < f{x) < e (or for some xq € S' and any x with \\x — Zo|| < e and 0 < f{x) < +oo). Then S is nonempty and dg(a;) < Vz € X /(z ) < e ( or ||a; — zo|| < Remark 4.14 In terms of the proximal subdifferential in a Hilbert space, Clarke et al. [18, Theorem 3.3.1] indicates that the inequality ds(x) < iaf{x)+ holds if X is sufficiently near Xq and 0 < f(x) is sufficiently small (for more discussion about Clarke et al [18, Theorem 3.3.1] we see Ye [90, Claim]). Corollary 4.13 guarantees the inequality to be true if x is sufficiently near xo (or 0 < f{x) is sufficiently small for a nonnegative function /). Corollary 4.15 Let X be an Asplund space, / : X —>■ (—oo, +oo] he lower semicontinuous and S := {x E X : f{x) < 0}. Suppose that for some 0 < e < Too the set / “^(—oo, e) is nonempty and there exists /r > 0 such that /or ony a; 0 < /(z ) < 6 (or /or aome zo E S and a; with [[a; — a:o|| < e and 0 < f{x) < T oo). Then S is nonempty and ds{x) < pf{x)+ Vx € X with f{x) < c ( or ||x — xo|| < -). Rem ark 4.16 Corollary 4.15 coincides with an error bound result of Ledyaev and Zhu [41] which was established in a Banach space with Fréchet-smooth Lipschitz bump functions while such a space is an Asplund space. Upon using Proposition 2.22 and Theorem 4.11, we obtain relatively stronger sufficient conditions for global and local error bounds. 80 Theorem 4.17 % 6e o BonocA apace, / : % — (— 0 0 , + 0 0 ] 6e Zotcer aemi- continuous and S {x E X : f{x) < 0}. Suppose that for some 0 < e < + 0 0 the set f~^{—oo,e) is nonempty and for some p. > 0 and each x E X with 0 < f{x) < e {or \\x — Zo|| < e and 0 < f{x) < + 0 0 ) there exists hx E X with ||Aj;(| = 1 such that Then S is nonempty and (ZaW < A(/(3;)4- Vz e % wiZA / ( z ) < e ( o r ||z - zo|| < Rem ark 4.18 If % is a reûexive Banach space and / is Lipschitz on e) for 0 < e < + 0 0 , then, by Proposition 2.26, Theorem 4.17 is equivalent to the corresponding result of Theorem 4.11 for = d°. We recall th a t in a general Banach space the Dini subdifferential is n o t always a ôt^-subdifferential. Thus we can not derive similar results to Theo rems 4.17 from Theorem 4.11. However it is still possible for us to present similar sufficient conditions. In fact, based on Theorem 3.9, Wu and Ye [88] have obtained corresponding results in which the condition is weaker and the nonemptiness of S still comes as a conclusion instead of an assumption. T h e o r e m 4.19 ([88, Theorem 2.6]) Let X be a Banach space and / : X —)■ (—00, + 00] be a proper lower semicontinuous function. Let 0 < e < 4-00 and 0 < p < 4-00. Suppose that the set / “^(—oo,e) is nonempty and for each x E /"^(O, e) there exists hx E X with \\hx\\ = 1 such that f~{x; hx) < —p~^. Then 81 5" nonempty omj (faW < At/(a;)+ Va; 6 ^)- P r o o f. By Theorem 3.9, it suffices to show that for any A > 1 and x € / “^(O, e) there exists a point y E /~^[0, e) such that 0 < Ik - y|| < A/i[/(a;) - /(y)]. Let A > 1 be fixed. For each x E / “^(0,e), by assumption, there exists hx E X with \\hx\\ = 1 such that f^{x;hx) < — Let x E f~^{0,e) be fixed, and Un -> hx and -> 0+ be such that U r n A ri->+oo tn Then for sufficiently large n we have f{x + tnUn) ~ f{x) ^ __ 1_ 11 '^n 11 From this and the lower semicontinuity of / it follows that 0 < < /k + < /k) - < e for sufficiently large n. For any such n, taking y — x + t„u„, we have 0 < (niknil = Ik - 3/11 < A/i[/(z) - /(?/)]. This completes the proof. ■ T h e o r e m 4.20 Let X be a Banach space and f : X (—oo, Too] be a proper lower semicontinuous function. Let 0 < e < +oo and 0 < p < Too. Suppose 82 that the set f ^(—oo,e) is nonempty and the Gâteaux derivative Dfix) exists eocA z € e) ||D /(z)||* > TTien S' is nonempty and (fgW < ///W+ Vz e /"X-oo, P r o o f. Suppose that for some 0 < e < +oo and 0 < p < +oo the Gâteaux derivative Df{x) exists at each x G e) and satisfies ||D/(z)||* > According to Theorem 4.19 we need to show that for any fixed A > 1 and each X e f~^{0, e) there exists h^ E X with |)ha;|] = 1 such that f'{x; hx) < —(A/r)” h Let A > 1 be fixed. Then, for each x 6 /~^(0, e), since ||D/(z)||* > p^^, there exists Vx E X such that ||%|| = 1 and /'(z; = Df{x)(vx) > (A/r)“ L Taking hx — — % we have f'{x; hx) < —(A/r)~L By Theorem 4.19 the set S is nonempty and there holds the following inequality < A/i/(z)+ Vz G y^(-oo,e). Letting A —>• 1'*' in the above inequality yields (faW < Vz G /-^(-oo,e): Rem ark 4.21 Note that a Gâteaux differentiable function may not be Lipschitz continuous (see Examples 2.33 and 2.34). Theorem 4.20 can not be obtained by Ioffe [33, Theorem 1 or Corollary 1.1] which is only applicable to a Lipschitz continuous function. 83 4.4 Error Bounds with Exponents For a proper extended-valued function / on a metric space X, we say that / has a local (global) error bound with exponent /3 > 0 if for some 0 < e < -foo (e = -f oo) there exists p > 0 such that dg(z) < Vz e % with /(z) < e. Error bounds with exponents were initially studied by Lojasiewicz [48, 49] for an analytic equality system on iî”. Luo and Pang [52] extended his result to an analytic inequality system. For a polynomial system Luo and Luo [50] proved that error bounds with exponents exist. For other inequality systems we see [82] and [65] and the references therein. Simply replacing / by the function [f+]^ in Theorems 4.2 and 3.9, we obtain the following sufficient conditions for error bounds with exponents. Theorem 4.22 Let (X, d) be a metric space and f : X (—oo, -Foo] be a proper function. Suppose that the set S is nonempty. If for some constants p > 0,0 < p < 1 and 0 < e < +oo and for each x € f~^(0, e) there exists y e / “ ^[O, e) such that and then 84 With the stronger condition that X is complete and f is a proper lower semi continuous function satisfying only (4.13), S is automatically nonempty and the concfMawn (4.14) Aofck. Note that Theorem 4.22 extends [65, Theorem 1] from a weakly lower semi continuous function / on a reflexive Banach space X to a proper lower semi continuous function on a Banach space. In order to derive the corresponding sufficient conditions for error bounds with exponents in terms of the lower Dini derivative of function /, we first give a chain rule for the lower Dini derivative of f^. L em m a 4.23 ([88, Lemma 3.2]) Let X be a normed linear space and / : X (—oo, -j-oo] be a lower semicontinuous function; let Q < < -f-oo, 0 < ^ < -Too and 0 < e < -Loo be constants. Then for x G / “ ^(O, e) and h E X with ||/i|| = 1 the following are equivalent: (z) y-(a;;h) < (zz) (/^)-(z;/z)<-/z-^^. Moreover if (i) or (zz) holds then (/^)-(a;; h) = h). Proof. Let x G / “ ^(O, e) and h E X with ||h|| = 1. Suppose that inequality (z) holds. Then there exist sequences in X and in R such that -L h and -> 0“^ as n —> +oo and n^+oolim \ \ / 85 Since 0 < f(x) < e, for sufficiently large n, we have tn which with the lower semicontinuity of f implies that the following properties hold for / : (1) lim^_++oo[/(z + - /(r)] = 0. (2) f{x + tnUn) — f{x) < 0 for sufficiently large n. as M —> +CX3. Therefore from which it follows that (/)-(.; A) < n -+ Ito + oo tji H This proves the implication (i) => (ii). Conversely let inequality {ii) be true and Un h and —>• 0+ be such that lim - .£ M = (f)-(r,h) < - g . n-^+ co tn p From this inequality and the lower semicontinuity of / with 0 < f{x) rk k ) < lim /(^ + ^ - n - > + 0 0 86 that is, inequality (z) follows. Note that in the proof of implications (z) (zz) and (zz) => (z) we have (/^)-(a;;/z) < ,9/^"'X:c)/-(a;;/z) and /-(a;;/z) < ;g-Y^-'^(z)(/^)-(%;/z). Therefore no matter whether (z) or (zz) holds we always have (/^)-(z; h) = ^/^"X:c)/-(a;; h). Remark 4.24 We can use and (/^)i respectively to replace / and (/^) in Lemma 4.23 as long as we take = /i in the proof. Replacing the function / in Theorems 4.4 and 4.19 with [f+Y zmd applying Lemma 4.23 and Remark 4.24, we obtain the following sufficient conditions for error bounds with exponents. T h eo rem 4.25 ([88, Theorem 3.3]) Let X be a normed linear space and f : X —> (—oo, +oo) be a lower semicontinuous function; letO < fa < +oo, 0 < p < 1 and 0 < /3 < 4-co be constants and 0 < e < H-oo. Suppose that S is nonempty and for each x E /~^(0, e) there exist tx > 0 and hx ^ X with \\hx\\ = 1 such that dg(z + 4hz) < pdg(%) (4.15) 87 and /r (a; + + t/^z) Vt G [0,4). (4.16) TTien (^a(a;) < ^ [/(a;)+ ]^ Va; G (4 1 7 ) If X is a Banach space and f satisfies /-(z;/ii) < -/i-^/^-^(a;) then S is automatically nonempty and the conclusion (4.17) holds. Rem ark 4.26 It is worth pointing out that Theorem 4.25 extends [65, Theorem 3] and [65, Corollaries 3 and 4] in which X is a reflexive Banach space and f is weakly lower semicontinuous and directionally continuous atx in the direction ha, (th at is, limf_,.o4- /(a; 4- th i) = /(a;)) w ith /^^(a;; h^) < -/^ -^ ^ -^ (a ;). Let / be a quadratic function on i?" defined by f{x) = x*Qx + ifx + c, where Q i s a real n x n symmetric matrix, 6 G R " and c E R with a ; * denoting the transpose of x. Since Q is symmetric, there exist an invertible matrix A, integers k and m with 0 < k < m < n such that r 4 0 0 \ A ^ Q A = 0 ~ ^ m —k 0 V 0 0 0/ where and Tm-t are the A x & unit matrix and (m — t) x (m — A) unit matrix respectively. For each a; = (zi, " , z») 6 .R", k m n /(Ac) = ÏZ + + c i=\ i=k+l î = l = + Z Z + T i=l i—k+l î= m + l where Q = and r = - ZLi ^ + Z%=t+i %- + c. Dehne k m n = " Z Z ^ VæeR". i=l i=k+l i = m + l If we denote = {si E R" : /(8i) < 0}, 52 = {S2 E R" : f{As2 ) < 0} and 53 = {«3 E R" : 4>{s3) < 0} then Si = AS2 , S2 = A^^Si, S2 = S3 — Co and S3 = S2 + cq where cq = |(ci, • •• ,Ck, —c^+i, ■ • •, -Cm, 0, • • ■, 0). Thus for any x £ R" we have dgg(a;) = dg,(a; + co) and dg^a;) = dgg(a; - Co), where ||v4|| is the norm of A as an operator from R" to R”. It follows from these relations that an error bound for f(x) holds if and only if an error bound for /(Ac) holds if and only if an error bound for Proposition 4.27 Jvef k m n H Vz E i=l i=fc+l i—m+1 with m > 0 and S = {x e X : (p{x) < 0}. Then either S is empty or it is nonempty and there exists p > Q such that Vz E % (4.18) for /? = 1 or 1. Proof. Note that for any z E iî" we have V<^(z) = (2Zi, ' ' ' 1 2Zfe, 2Zfc.).i, ■ ■ • , 2Xjyi, c^+l ' ' ' Cn) and m n llW(z)|| = (4^ z?+ *=1 i=m+l We prove the result for the following six cases of which one and only one must happen to f: C a s e 1 Cjg 7^ 0 for some m < io < n \ Since for any x E X with 0(z) > 0 we have ||V^(z)|| > \ci^\ > 0, by Theorem 4.20, S is nonempty and inequality (4.18) holds with p = |cio|“ ^ and ,9 = 1. C a s e 2 Cj = 0 for each m < i < n and r < 0 : For any x E X with 0(z) > 0 we have the following inequalities fc m ^Z j > ^ z? - r > |r| > 0 i=l i=k+l from which it follows that ||V(z)|| > |r|5 > 0. Thus, by Theorem 4.20, S is nonempty and inequality (4.18) holds with p = \r\~^ and /? = 1. 90 C a s e 3 Cj = G for each m < i< n , r = 0 and 0 = k < m : In this case, S = and inequality (4.18) holds automatically for any n > 0 and /3 = 1. Case 4 Ci = 0 for each r = 0 and 0 < t < m : For any a; E % with (^(ar) > 0 let Then It follows from Theorem 4.25 that S is nonempty and inequality (4.18) holds for /r = 2 and ,9 = ^. Case 5 Q = 0 for each Tn S — 0. C a s e 6 Ci = 0 for each m A r a;? + - > 0. i = i ^ Taking = (a;i, - - -, zt, %+i, - - -, 2/m, a:m+i, ' " , a;,i) such that E U t+ i %/? = Z L i + r, we have 4>{y) ~ 0. This implies that S is nonempty and k dg(a;) < ||a; - 2/|| < *=i Thus It is easy to see that pi is a positive constant. For each x G R^\S with Y !^ k+ i let m k ^ = ( ÏÏ fi = (^a;-+r)2-( i = k + l i = l 91 and hx = -(0, • • • , 0, Xk+I, ■ • • ,Xjn, 0, • • •, 0). Then ( > (^)a, > 0 (since a; ^ 5"), ||hg|| = 1 and k m j. k (j)(x + txhx) = yi a;? + r — ^ x^{l + —)^ = '^^xf + r — {t + = 0. 1=1 *=&+! ^ i=l Thus m j. -(j){x) = d g (z) < ||a; - (z + (4.20) (2r)2 Therefore, taking fx = max{/ii, (2r)"?} and combining (4.19) with (4.20), we obtain dg(z) < At[<6(z)]{. Va; € % with /? = 1. ■ Remark 4.28 The result of Proposition 4.27 was first obtained by Ng and Zheng (see [65, Theorem 4] and [66, Theorem 5.1]). We have used our result (Theorems 4.20 and 4.25) to give an easier proof. 92 4.5 Error Bounds for Lower Semicontinuous Con vex Functions A nonconvex function may have a local error bound but no global error bounds. For example, it is easy to see that the function = I ^ Ü III > has no global error bounds even though it has a local error bound. However, for a proper convex function, the existence of a local error bound always implies th at of a global error bound. Proposition 4.29 ([88, Proposition 4.1]) Let X be a normed linear space and / : X —> (—oo, +oo] he a proper convex function. Then, for some f j , > 0 and 0 < e < + 00 , the following statements are equivalent: (i) S is nonempty and ds{x) < p/(æ)+ for each x E f^^{—oo,e). (n) dg(z) < /Lt/(a;)+ /or eocA a; E X. Proof. The implication (ii) => (i) is trivial. To show (i) => (ii) we suppose that S is nonempty and ds(x) < pif(x)+ Vz E X with f(x) < e. It suffices to prove that the above estimate holds for a l l z E X with e < f(x) < + 0 0 a s w e l l . Let X E f~^[e, +oo). Then for any n E N there exists x E S such that ||z -z|l < ds(x) + 93 Taking A := and y — Xx + (1 — X)x, we have \\y~m = A||a;-^|| and, by the convexity of f, f{y) < Xf{x) < | which implies ds{y) < fxf{y)+. Hence ds(x) < li^ _ 5|t = l ! l ^ = ^ [(faW + ^ - It;/ - a;||] ^ Ma(2/) + ^ M/W++^]^ MA/(z) + - A - A 1 2 = /z/(a;)+ + — = |/^+ —]/(%)+. This implies that ds{x) < fif(x)+ holds since n E TV is arbitrary. ■ For a continuous convex function / on a reflexive Banach space X, under the condition that the set S be nonempty, Ng and Zheng showed that there exists /j, > 0 such that ds{x) < y,f(x)^ for each x e X ii and only if ||^||* > fi~^ for each Ç 6 df{x) and each x E X with 0 < f{x) < + 0 0 (see [66, Theorem 3.3]). Recall that for a proper lower semicontinuous convex function / the subdifferential df{x) coincides with the Clarke subdifferential d°f{x). This fact allows us to use Theorem 4.11 to extend their result to a proper lower semicontinuous convex function on a Banach space. In fact we can use Proposition 4.29 to obtain additional equivalent statements about error bounds as follows. T heorem 4.30 ([88, Theorem 4.2]) Let X be a Banach space and / : X —> (—00 , + 00] be a proper lower semicontinuous convex function. Then for some constant p > 0 the following are equivalent: 94 (i) ||Ç||* > IX ^ for each ^ G df{x) and each x e f ^(0,+ 00). {ii) For some 0 < e < +00 the set / “^(—oo,e) is nonempty and ||^||* > for each Ç G df{x) and each x G /~^(0, e). (m) S is nonempty and there exists 0 < e < +00 such that ds{x) < /xf{x)+ /o r eacA a; G /" ^ (—00, c). (m) da(z) < /i/(a;)+ /or eocA a; G X. Proof. For each x G /~^(0, e), by Proposition 2.23, d°f{x) = df{x). We will implicitly use this relation in the following proof of implications (i) => (ii) => (izi). (i) => (ii) : This is immediate. (ii) => (iii) : If 5 is nonempty and there exists 0 < e < +00 such that IlCll* > for each Ç G df{x) and each x G /~^(0, e), then taking d° as an abstract subdifferential d^i in Theorem 4.11 we have ds{x) < fxf{x)+ Va; G X with f{x) < e. (iii) (iv) follows directly from Proposition 4.29. It remains to prove (Zt;) (i). Let X be such that 0 < f{x) < +00 and ds{x) < pf{x). Then ds{x) > 0 and for any ^ G df{x) we have ' II2/ - :c|| > - ( ( , :/ - a;) > - [ / W - / W] > / W Vï/ G ,9. Taking inferiors of both sides of the inequality for all y over S we obtain 11(11* ' > /(a;), 95 from which we have Therefore the inequality desired follows. m It is easy to check that a function / : X —> (—00, +00] is convex if and only if for each x G domf and each v E X the function is nondecreasing in (0,1]. Thus if / is convex then the lower Dini derivative f£ (x; •) coincides with the directional derivative f'{x\ ■). When / is convex and each point in a neighborhood of S has a closest point in S, the sufficient conditions for the existence of error bounds given in Theorems 3.9 and 4.19 become necessary as well. Proposition 4.31 ([88, Proposition 4.3]) Let X be a normed linear space and / : X —> (—00, +00] a lower semicontinuous convex function. Suppose that for some 0 < e < + 0 0 each point x G / “ ^(O, e) has a closest point in S := {x E X : f{x) < 0}. If for some p > 0 and each x E / “ ^(O, e) there holds then /or each z G e) (i) there exists y E / “^(O, e) such that 0 < ||a; — y|| < p[f{x) — f{y)] and (ii) there exist > 0 and h^ E X with ||hx|| = 1 such that /'(a: + thz; h^) < Vt G [0, tg,). P r o o f . Given x E / “ ^(O,e), let æ be in 5 such that ||z — x|| = ds{x). Taking tx = ||a: — x|| and hx = — x), then x — x + txhx- Obviously for each 0 < t < tx we have ds{x — thx) — t. This implies that 0 < f{x — thx). In 96 addition, by the convexity of /, f{x — thx) < (1 — —)f W + 'rf (^) < Thus by the assumption t < /i/(f - (hi) < - (hi) - /(z)] VO < ( < (i_ Rewriting gives the following inequality S L Æ l É î I < _ 1 vo<(<<.. ( For each 0 < ( < (i, by the convexity of / again, /(z + (hi) - /(z) ^ - /W ^ /(z)-/(æ-(ihi) ( - (i (z ~ ~ h For any 0 < ( < tx, taking y = x + (hi, we see from the above inequality that G /"XO, e) and 0 < ||z - ^|| < /i[/(z) - /(i/)]. This proves ((). To prove (ii) we note that the point x is also a closest point in S to the point X + thx for each t G (0, tx). Thus, for each 0 < g < (i — ( with t G [0, tx), as in the above discussion, we have f{x-\- thx + shx) — f{x + thx) ^ , ----- 1 S l i from which we obtain {ii). ■ Note that a convex subset of a locally convex space is closed if and only if it is weakly closed (see [21, Corollary 1.5, p.126]) and that for a lower semicontinuous convex function / on a Banach space X the set 5 — { 2: G X : f{x) < 0} is closed 97 and convex. If X is reflexive and S is nonempty then, by Alaoglu’s Theorem, each point x G X\S has a closest point in S. As a result of Theorem 4.30, Proposition 4.31, Theorems 3.9 and 4.19, the equivalent statements about the existence of error bounds can be summarized as follows. Theorem 4.32 ([88, Theorem 4.4]) Tef % 6e a re/Ieziue Banoch apace and / ; X —> (—oo, +oo] be a proper lower semicontinuous convex function. Then for some p > 0 the equivalent statements {i) — (iv) in Theorem f.SO are all equivalent to any one of the following: (v) For some 0 < e < +oo the set /~^(—oo,e) is nonempty and for each X G / “ ^(O, e) there exists a point y G / “ ^[O, e) such that 0 < ||a;-p ||< p [/(T )-/(%/)]. (vi) For some 0 < e < 4-oo the set /'"^(—oo, e) is nonempty and for each A > 1 and each x G / “ ^(O, e) there exists a point y G /"^[O, e) such that 0 < ||z - p|| < Ap[/(a;) - /(%/)]. (vii) For some 0 < e < + oo the set f~^{—oo,e) is nonempty and for each X G / “ ^(O, e) there exist > 0 and hx E X with ||hj,|| = 1 such that f (a; + hz) < G [0, (vin) For some 0 < e < Too the set / “^(—oo, e) is nonempty and for each x G e) there exists hx E X with ||ha,j| = 1 such that f'(x; hx) < Proof. For the case 0 < e < +oo the implications {Hi) => (v) => (vi) {in) follow directly from Proposition 4.31 and Theorem 3.9 while {Hi) =4- {vii) => 98 (viii) => (iii) follow from Proposition 4.31 and Theorem 4.19. Similarly for the case e = + œ by Proposition 4.31 and Theorems 3.9 and 4.19 we have “(iv) => (v) (vi) => (iv)” and “(iv) => (vii) => (viii) => (iv)” ■ ■ Rem ark 4.33 Even though Theorem 4.32 is established in a reflexive Banach space, the results are still valid if % is a general Banach space and S is compact since in this case each point in X\S has a closest point in S and Proposition 4.31 is applicable. As an application of Theorem 4.30 or 4.32, the following example is used to illustrate that not all convex functions have error bounds. The function in this example appeared in [75] and was subsequently used in [3] and [42]. By its subdifferential we prove that this function has no error bounds even though it is convex. Example 4.34 Consider the closed proper convex function ^ if 3:2 > 0 /(Zl,Z2) = ^ 0 if = Z2 = 0 oo otherwise. Obviously S = {(0, 0)} and for each n e N with 0 < f(xi,n) < oo the subd ifferential df(xi,n) = {(2^, —^)}. For fixed xi and any 0 < e < oo we have 0 < f(xi,n) < e when n is large enough and (2^, — ^ ) -4 (0,0) as n Too. Consequently, by Theorem 4.30 or 4.32, there can not exist /r > 0 such that, for all n, 99 4.6 Error Bounds with Abstract Constraint Sets The result in Theorem 4.11 for a single inequality system can easily be extended to a system including equalities, inequalities and an abstract constraint x E C as follows. Theorem 4,35 ([86, Theorem 3.3]) Let C be a closed subset of a Banach space X and each ]%| : X —>• (—oo, +oo] be lower semicontinuous for i = 1, - ■ ■ ,r ond j = 1, - - , g. Denote g := {z E C : /i(a;) < 0, - /r(a:) < 0; = 0, - -, = 0} /(a ;) = m a x { /i(z ), A (a:); |pi(a;)|, |p«(:z:)t}- Suppose that for some 0 < e < oo the set C C\ /~ ^(—oo, e) is nonempty and for some fx > 0 we have Kll, > whenever f G d^if + 'ipc)(x) for any x E C with 0 < f{x) < e (or for some Xo E S with ||z — Toll < e and 0 < f(x) < Too). Then S is nonempty and dg(j;) < At/(a;)+ < //(||F(z)+|| + ||G(z)||) for any x E C with f{x) < e ( or ||a; — rro|| < f), where F(:r)+ = (/i(z)+,- -,/r(a;)+) ond G(a;) = (^i(z)+, - - -,^,(3:)+). P r o o f . By Theorem 4.11, it suffices to check that / is lower semicontinuous. 100 For any z € X, denote = /^(z) for % = 1, - - - , r and = |gi_r (a;) | for i = r + 1, ■ • ■ ,r + s. Then, for each 1 < z < r + s, Fi{x) is lower semicontinuous, liminf/(y) = lim inf maxiFTy) : 1 < i < r + s} y~^x y-~¥x L _ j > limWF;(r/)>Fi(a;), and hence limjmf/(y) > f{x) Va: G X, which implies that f is lower semicontinuous. ■ R em ark 4.36 (z) We have proved Theorem 4.35 based on Theorem 4.11 while Theorem 4.11 can be obtained from Theorem 4.35 by taking C — X, r = 1 and s = 0 in it. Therefore they are equivalent to each other. (ii) Theorem 4.35 has extended Ioffe [33, Theorem 1 and Corollary 1.1] from a Lipschitz equality system to a lower semicontinuous inequality system. It is also an extension of Wu [83, Theorem 4.19] in which X = i?”, r = 1, s = 0, e = +oo and dp. Theorem 4.35 is stated in terms of any O^j-subdifferential, however, to sim plify checking the conditions, we often try to use smaller ^^^-subdifferentials (such as the proximal subdifferential in a Hilbert space and the Fréchet subdif ferential in an Asplund space) or some ô^j-subdifferentials with better properties (for example the Clarke subdifferential). Besides in Theorem 4.35, only \çj\ is required to be lower semicontinuous no matter what gj is. These points are illustrated in the following example. 101 Exam ple 4.37 Consider the function g : A —y ^ given by / \ f 1 — |a;|, if Æ is a rational number ; ' I —1 + |a;|, if z is a irrational number. Take C = R. Then S = {x E R : g{x) = 0} = {—1,1}, ipc{x) = 0 Vx e -R and th e function is Lipschitz of rank 1. It is easy to find dp\g{x)\ = {—1} for x < —1 or 0 < z < 1, dp\g(x)\ = {1} for — 1 < z < 0 or 1 < x and For any x E C with g{x) 0, since + = c{-i,i}, we have ||^|| = 1 for any ^ G dp{\g\ + xl)c){x). Thus, by Theorem 4.35, ds{x) < \g{x)\ = |1 - |a;|| Vx E R. Rem ark 4.38 Note dg(0) = 1 = |g(0)| in this example. So /r — 1 is the small est constant such that the above inequality holds for any x in R. Besides to use Theorem 4.35 to find a global error bound, we can not use the Clarke sub differ ential since if we choose it as a 5a,-subdifferential then — d°g(0) — [—1,1] and it is impossible to find a // to satisfy the condition in Theorem 4.35. Let y be a real normed linear space and F : X x Y (—oo, -hoc] be lower semicontinuous. For any fixed y E Y, when F{x, y) is considered as a function 102 of the hist variable, (z,^) denotes Q^-snbdiSerential of F(at z. U pon applying Theorem 4.35 to th e function F{-, y) we have the following result. Theorem 4.39 ([87, Theorem 3.4]) Let X be a Banach space and Y be a real normed linear space. Let C be a nonempty closed subset of X x Y and F : X X Y (—oo, Too] satisfy that for each y E Y the function F{-, y) is lower semicontinuous. Assume that for y E Y the set S'(^) := {a; 6 % : (z,2/) € C ond F(z,2/) < 0} is nonempty and that there exist p > 0 and 0 < e < oo such that for any x E X with (x, y) E C and 0 < F{x, y) < e. Then we have dg(ÿ) (z) < 1/)+ Vz E X with (%,%/) E C ond F(a;, ;/) < e. Proof. For y E Y in the assumption, denote /(-) := F(-, 3/) an d C (i/) := {a; E X : (a;, %/) E C } . Upon applying Theorem 4.35 to the solution set ^(y) = {z E C(i/) : /(a;) < 0} we obtain the inequality desired. ■ Definition 4.40 Let X be a Banach space and T be a real normed linear space. The partial subdifferential, in x with respective to y, on X is any operator % satisfying the following properties: For any lower semicontinuous functionf : X x Y —> (—oo, + oo], any locally Lipschitz function g : X x Y (—oo, Too], any x E X and any y EY, 103 (A) !/) Ç and ^) = 0 if /(a;, %/) = + 0 0 ; (P2) dxg{x, y) coincides with the partial subdifferential in the sense of convex analysis whenever g{-, y) is convex, that is, !/) = {( e X* : ((, n - z) < g(n, %/) - g(z, %/) Vn 6 %}; (P3) 0 € dxf(x, y) whenever a; is a local minimum for / with respective to y, (P4 ) dxf{x,y) = dxh{x, y) whenever / coincides with h around {x, y); (fs) + %/) Ç + Obviously a partial sub differential must be a ^^/-subdifferential. However since we use the fuzzy sum rule in the definition of ^^/-subdifferential instead of the sum rule as in that of the partial subdifferential, 9^,-subdifferentials in clude more subdifferentials in nonsmooth analysis than partial subdifferentials. For example for the case F{x,y) — f{x) My £ Y the proximal subdifferential dpF{x, y) = dpf{x) is a 9^,-subdifferential but not a partial subdifferential. As a result, [39, Theorems 2.4] follows from Theorem 4.39 by taking C = X xY, — dx and e = -t-00 in it. T h eo rem 4.41 ([39, Theorem 2.4]) Let f : X xY —> (—00, -t-00] be an extended real-valued function such that for each y G Y, /(•, y) is lower semicontinuous. Suppose that there exists // > 0 such that da^/{œ,ÿ)(0) > for each x E X and y e Y with f{x, y) > 0. Then dg(y)(z) < ?/)+ Vz G X ond 3/ E Y 104 In the remaining part of this section we suppose that X is a real normed linear space. Motivated by a note of a referee of Deng [24, Corollary 2], we present the following condition to guarantee the existence of a global error bound for a general inequality system. Theorem 4.42 ([86, T heorem 4.1]) C 6e o nonempty anbaet o/ a norm ed linear space X and f : X (—oo, -boo] be a proper function with C n {dam f) ^ 0. Suppose that there exist a unit vector u in X and a constant p > 0 such that, for each A > 0, T + e C .n d sup (4.21) A>0 A for each x E C with 0 < f{x) < + o o . Then S ~ {x E C \ f{x) < 0} is nonempty and ds{x) < fjLf{x)+ V z e C. Proof. Applying Theorem 4.5 to the function f + tpc completes the proof. ■ Remark 4.43 Theorem 4.42 is a new version of [86, Theorem 4.1] in w hich S assume to be nonempty. Besides C in Theorem 4.42 must be unbounded since for z E C with f{x) < -foe and any A > 0, z + An must be in C. Recall that for a nonempty closed convex subset C of X, the recession cone of C, denoted by C°°, is the set C°° — {x E X : Ç (0, +oo) & { z j Ç C s.t. lim /r* = 0 and lim piXi = rr} i-^OO i—>-00 According to Rockafellar [73, Theorem 2A(c)], C°° can equivalently be expressed as 105 = {T 6 % : C + {z} Ç C}. For a proper lower semicontinuous convex function / : % -4 (—oo, +oo], since its epigraph is a closed convex subset of X x R, we can use the recession cone of epi f to define the recession function of f, denoted by f°°, that is, Similar to Deng [23, 24], we use the recession function to give the following sufficient condition for a global error bound. Corollary 4.44 ([86, Corollary 4.2]) Let C be a closed convex subset of a normed space X and each fi : X -)■ (—oo, +oo] be a proper lower semicontinuous convex function for i e I = {1, ■■■ ,r}. Denote S = {x E C : fi{x) < 0,i E 1} and f{x) := max{fi{x) : i E I}. Suppose that there exist a unit vector u E C°° and a constant /a > 0 such that ff^{u) < for each i E I. Then S is nonempty and, for any 1 < p < +oo, dg(z) < < ju|]F(a;)+)|p Va; E C, where F{x)+ = (fi(x)+, • • •, fr(x)+) and ]] • jjp denotes the p —norm on RL. P r o o f . Since S = {x EC : f{x) < 0}, we only need to check that the conditions in Theorem 4.42 are satisfied for C and /. Firstly, by Rockafellar [73, Theorem 2A(a)], the inclusion u E C°° implies that X + Xu m u st be in C for each x E C and any A > 0. Besides according to Rockafellar [73, Corollary 3C(a)], for each i E I, A>o A 106 So if then, for any A > 0, /i(z + An) < /i(z) - A/^"^ Vz 6 dom Hence, for any a; € dom / and any A > 0, /(a; + An) < /(a;) - A//"\ In particular, for each x E C with 0 < f{x) < +oo, A>0 A Therefore, by Theorem 4.42, S is nonempty and, for any 1 < p < +oo, dg(a;) < p/(a;)+ < p ||f (z)+||p Vz E C. Remark 4.45 Note that each fi in Corollary 4.44 is a lower semicontinuous and convex function on a real normed linear space. So it is an improvement on Deng [23, Theorem 2.3] in which X is a reflexive Banach space, each fi is continuous and convex for i = 1, • • •, r and S is assumed to be nonempty. Besides Deng [24, Corollary 2] can be obtained as a special case of Corollary 4.44 where p = 1 and fi is a continuous and convex function on a Banach space X. Furthermore Corollary 4.44 not only extends Jourani [39, Theorem 3.3] but also proves that condition (i) in it is redundant. For easy reference we quote Jourani [39, Theorem 3.3] in the following: Proposition 4.46 ([39, Theorem 3.3]) Let f : X (—oo, +oo] be a proper lower semicontinuous convex function and C be a closed convex subset of X 107 such that C n dom f 7^ 0. Consider the solution set S = {x E C : f{x) < 0}. Suppose that (i) for all X ^ C n dom / with x ^ S, + V'c)(a;) = a / W + ]V cN ; {ii) there exist p > 0 and u G with ||it|| = 1 such that f°°{u) < Then (faW < Ai/(z)+ Vz e C. 4.7 Global Error Bounds and Metric Regularity In Deng [24] close relations between global error bounds and metric regularity are revealed for a continuous and convex inequality system. Most of them turn out to be true for a lower semicontinuous convex inequality system and some of them can further be refined. To show this we recall the concept of metric regularity and introduce that of uniformly metric regularity. Definition 4.47 Let f be an extended real-valued function on X, (7 be a subset of X, and S — {x E C : f{x) < 0} be nonempty. The system / N < 0, T e C (4.22) is said to be metrically regular at a nonempty set 5i Ç 5' if there exist constants 6 > 0 and p{5) > 0 such that ds{x) < p{S)f{x)+ \/x e C with ds^{x) < S. 108 When = {z } Ç S, we simply say that the system (4.22) is metrically regular at z. In particular, the system (4.22) is said to be uniformly metrically regular at S if it is metrically regular at each point of S with the same 6 > 0 and //((^) > 0. Obviously for any 0 5'i Ç ^"2 we have (a;) > dg, (z) for any z E so if the system f{x) < 0, z E C is metrically regular at S2 , then it must also be metrically regular at Si- The following result states the relations between global error bounds and metric regularity for a lower semicontinuous inequality system. Theorem 4.48 ([86, Theorem 5.2]) Let X be a normed linear space and f : X —> (—00, +00] be a lower semicontinuous function such that S = {x G X : f{x) < 0} is nonempty. Consider the following statements: (а) There is a constant p > 0 such that dg(z) < pf{x)+ for any x G X. (б) The system f{x) < 0 is metrically regular at any nonempty set Si Ç S. (c) The system f{x) < 0 is metrically regular at S. (d) The system f{x) < 0 is uniformly metrically regular at S. (e) The system f{x) < 0 is metrically regular at each point of S. Then the following implications hold: (i) (a) => (b) (c) (d) => (e). (ii) If f is convex, (a) <=> (b) (c) <=> (d). 109 (zii) //S' w compacf, (6) <=> (c) (d) 4*^ (e). ^GMce w conrez onj 5^ ia compocf (Aen (a) 4^ (6) <:> (c) <:> ((f) <=> (e). Proof. Since the implications (a) => (6) => (c) {d) #- (e) in (i) are obvious, it suffices to show {d) => (6) for (i), (d) => (a) for {ii), and (e) => (b) for {in). {d) => {b) for {i): We suppose that statement {d) is true. Then there are constants 6 > 0 and jj,{5) > 0 such that for each z E S, ds{x) < fi{S)f{x)j^ whenever ]|x — z|| < S. Hence for any nonempty subset Si of S we have ds{x) < jj,{S)f{x)+ for any X with ds^{x) < I since for such an x we can find a point Xi G such that \\x — xill < Ô. This proves that statement (6) holds. {d) (a) for'(iz) : Suppose that / is convex, that (d) holds and that 5 > 0 is the constant in the definition of uniformly metric regularity. Then, since S is closed, ds{x) > 0 for any x G X\S. Thus for any fixed x G X\S and any e > 0 there exists x E S such that \\x — x\\ < ds{x) + e. If ||x — z|| < ô, we have already the inequality ds{x) < n{S)f{x)+. If ||x —x|l > S, taking A := p4gjj and y = Xx + {1 — X)x, we have \\y — x|| = A||a; — æ|| = 6 , which implies ds{y) < ii{6 )f{y)^. Besides, by the convexity of /, f{y) < Xf{x). Hence ds(x) < = = A A ^ dg(z) + e - ||i/-T|| ^ ds(i/) + 6 “ A “ A 110 - A - A = X<^)/W+ + ^ < /i(a)/(T)+ + W + This means that statement (a) is true since e > 0 and x are arbitrary. (e) => (6) for {in) : Let (e) hold. Then for each s & S there exist > 0 and Hs> ^ such that Vz € AT with ||g — z|| < By the compactness of S there exist Sj € 5, <5j > 0 and /Tj > 0 (1 < i < m) such that S' Ç + (^iB) and dg(z) < Vz e AC with ||z — 8i|| < 36^, where x + ôB is the open ball with center x and radius 6 > 0. Taking S = min{J* : i = 1, • • •, m} and /x = max{/Xj : z = 1, • • •, m}, then for any nonempty subset Si of S and x E X with dg, (z) < ô there exists y E Si such that ||z — y\\ < 2Ô and \\y — Si|| < Si for some 1 < z < m. Such an X satisfies ll^; — Sj|| < ll^: — y \ \ + \ \ y — Si|| < 2(5 + 5* < 3(5*, so (fgiW < < /x/(z)+, that is, the system f{x) < 0 is regular at S%. Thus (b) follows. ■ I l l Rem ark 4.49 Deng [24] proved the implications (a) (6) <=> (c) for a con- tinuona convex system on a Banach space and the implication (e) => (a) when also % = R" and 5" is bounded. Theorem 4.48 has extended these results to a lower semicontinuoussystem and contains the new equivalent statement {d). Furthermore Theorem 4.48 is applicable to a lower semicontinuous exten d ed real-valued function f defined on a closed convex subset <7 of X to obtain an equivalent result whose statement is the same as in Theorem 4.48 with the set {a; E X : /(a;) < 0} and the inequality "dg(a;) < for any z E X" replaced by {x E C : f{x) < 0} and “ds{x) < fif{x)+ for any x E C” respec tively. In the rest of this section, we use Theorems 4.11 and 4.48 to give some sufficient conditions for a lower semicontinuous system to be metrically regular a t sets. Proposition 4.50 ([86, Proposition 5.3]) Let X be a Banach space and f : X ^ R he lower semicontinuous. Assume that S = {x e X : f{x) < 0} is nonempty and that there exist p > 0 and 0 < e < oo such that, for each z G S, whenever Ç E d^jfix) for any x E X with 0 < /(x ) and ||x — z\\ < e. Then the system /(x) < 0 is metrically regular at S. If f is in addition convex, then there is a constant p > 0 such that dg(x) < Vx E X. 112 Proof. According to Theorem 4.48, it suSces to show that the system /(z) < 0 is metrically regular at S. By Theorem 4.11, the inequality holds for each z e S and any x £ X with \\x — z\\ < that is, the system f{x) < 0 is uniformly metrically regular at S. Hence, by implication (i) of Theorem 4.48, the system f{x) < 0 is metrically regular at S. ■ The following proposition indicates that if the solution set S is compact and contains no stationary points for 9o,-subdifferentials with some limiting property then the system is metrically regular at the solution set S. Proposition 4.51 ([86, Proposition 5.4]) Let X be a Banach space and f : X R be lower semicontinuous. Assume that S — {x E X : f{x) < 0} is nonempty and compact and that for each z E S, 0 ^ duf{z) and d^jf satisfies (hot ^ z, G 9w/(zn) ond & -4 Then the ag/gtem /(z) <0 is metrically regular at S. If also f is convex then there is a constant /r > 0 such that dg(T) < Va; € A. P r o o f. Based on relations in Theorem 4.48, we only need to prove statement (e) in Theorem 4.48. Let z G S' be fixed. Then by Theorem 4.11 it is enough to show that there exist p > 0 and e > 0 such that 11( 11, > V( G 9 w/(z) 113 for any x with ||z—z|| < e and 0 < f{x) < + 0 0 . In fact if this were not true, then there would exist sequences and {^n} such that z» —> z, E 9^/(3:») and Çn —> 0. But this would lead to 0 € dcjf{z) which contradicts the assumption. ■ We now consider a convex system which also includes an abstract constraint. In the following proposition we prove that the generalized Slater condition is sufficient for metric regularity. Proposition 4.52 ([86, Proposition 5.5]) C 6e o closed and conuez subset of a Banach space X and fi : X R be locally Lipschitz and convex for i e I = {1 , • • •, r}. Let N U L be a partition of the index set I such that fi is nonlinear for each i G N and linear for each i € L respectively. Denote /(z) = max{/i(a;), |/,(z)| : % E N ,; E I,}. Suppose that there exist E S' : = { r E C : < 0,% G = 0 , j E 1 } such that fi(x*) < 0 for each i E N and {—Vfi(xo) : i E L} is C-linearly independent, i.e., for Xi E R {i E L) the inclusion — ^ XiXfi{xo) E Nc{xo) implies Aj = 0 Vi E L. iÇiL Then there exist positive numbers S and p such that (i) II6II* > VÇ E df{x) + Nc{x) for any x E C with ||a; - zo|| < S and 0 < /(z); (ii) ds{x) < juy(z)+ for any x E C with ||z — Zo|| < f, that is, the system f{x) < 0 ,xEC is metrically regular at xq. 114 Moreover if also X = and {—Xfi{x) : i E L} is C-linearly independent for each X E S, then for any bounded subset Ù Ç there exist J > 0 and fx > 0 such that /or any a; € C n (O 4- 6B), that is, the system f{x) P r o o f. Since / is Lipschitz near x and ipc is finite at x and both functions are convex, by Propositions 2.18 and 2.16, ^ (/ + Vc) W Ç a°/W + = a/(a;) + WcW. Hence, by applying Theorem 4.11 to the function f+ ip c with d^j = d°, statement {ii) follows from statement (*). So for statements (i) and (ii), it suffices to prove statement (i). Suppose that statement (i) were not true. Then there would exist sequences {a:jk} Ç C and Ç ^/(zk)+jVc(a;*) such that a;* -4 To, 0 and 0 < /(a;*,) for each k. By Propositions 2.8 and 2.19 for each Xf. there exists a set of numbers such that Aj") > 0 Vi e N , ^ A|*) + lA ^ I = 1, & ^ I ] A|*')a/i(Tk) + ^A|*^V/i(a;k) + Nc(Tk) and «en - /(art)) = 0 Vi € TV, A|*)(|/i(a;&)| - /(a;&)) = 0 Vi E T. Since each sequence is bounded by 1 for each i, we can assume that A'^^ —> A; for each iENUL as k—> Too. We denote the index of binding constraints at Tq by I(xo) = {i E TV : fi(xo) = 0}. Taking the limit as k oo 115 gives Ai > 0 V 2 € 7(zo), Ai = 0 Vi E jV\/(a:o), ^ Ai + ^ |A i| = 1 and ieJv «ez, 0 6 ^ Xidfi{xo) + XiVfi(xo) + Nc {xq) i€/(zo) where the inclusion follows from the fact that d°fi{xk) is the subdifferential of fi at Xk and Nc{xk) is the cone of normals in the sense of convex analysis. Since by assumption {—V/i(zo) : i € L} is C-linearly independent, this inclusion implies that there is at least one iq e I{xo) such that Xi^ >0, from which we would obtain a contradiction. In fact if we use the above Aj to define the following function ^(l/) = 1] W +1] + V'cW, iEl(xo) i&L then g is convex and by the sum rule of subdifferentials (in the sense of convex analysis) we have 0 E \9 /i(z o ) -t- ^ AiV/i(a;o) 4- IVc(a;o) = ^^(zo), which means that g attains its global minimum at Xq. Therefore this together with the continuity of g yields 0 = g(a;o) < p(a:') < < 0. This is a contradiction. Now suppose that X — RP'. Let d > 0 be the positive number stated in {i). Then for any fixed bounded set we can take e > 5 such that ÇI + SB Ç 116 Xo + By Theorem 4.35, it suffices to show that there exists /x > 0 such that ll^ll > E df{x) + Nc{x) for any x E C with <5 < ||z — SqII < e and 0 < /(z). Suppose that there exist sequences {xk} Ç C and Ç df{xk) + Nc{xk) such that S < \\xk — ^o|| < e, 0 < f{xk) and 0 as A: —>■ +oo. Since {xfe} lies in a compact set, we can assume that Xk converges to some point X E C with (^ < ||T — xoll < e. Taking limit for ^k E df{xk) + Nc{xk)-, we have 0 e 9/(x) 4- Nc{x) Ç d{f + i>c){x) by the sum rule of subdifferentials in the sense of convex analysis. This means that / attains its global minimum over C at ^ since / + %Ac is convex. Note that / is continuous and /(z t) is positive. So 0 = /(a;*) > /(^) = lim > 0. K-> + 00 Thus X E S. But by statement (i) there exist positive numbers 6 and /x such th a t ll^ll > € a/(z) + Nc(z) for any x E C with ||a: — ^|| < 5 and 0 < f{x). This contradicts the properties of subsequences {rr*,} and ■ Remark 4.53 In Proposition 4.52, the Slater condition fi{x*) < 0 fox i E N is important to guarantee (i) and (ii) to hold. Without this condition, (i) and (ii) may fail. 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