DERIVATIVE CHARACTERIZATION
OF
CONSTRAINED EXTREMA OF FUNCTIONALS
A SURVEY
Mohamed El-Hodiri
University of Kansas
Research Paper No. 34
AUGUST 1970
RESEARCH PAPERS IN THEORETICAL
AND
APPLIED ECONOMICS
DEPARTMENT OF ECONOMICS THE UNIVERSITY OF KANSAS LAURENCE, KANSAS INTRODUCTION
One of the basic axioms of economic analysis is the axiom of ration-.: ality, idea,of postulating that economic behavior results from a process of optimization. As it is, at best, difficult to directly test the axiom, economists have been interested in characterization theorems of optirniza- tion theory. In this paper we review the theorems that characterize opti- mality by way of derivatives. First we formulate a very general optimiza- tion problem. Then we present characterization theorems for three types of problems: Finite dimensional, variational and problems in linear topo- logical spaces. In each case we present theorems for equality - inequal- ity constraints. The theorems in each case are: first order necessary conditions, first order sufficient conditions, second order necessary conditions and second or~er:,sufficienTy&osditions. The scheme of repre- sentation is as follows: Statements of theorems are followed by remarks. referring the reader to the earliest, known to us, proofs of the theorems.
In some instances, slight generalizations of some theorems appear here for the first time, an indication of necessary modifications to existing proofs are indicated. A case is "solved" if proofs for all the four types of characterization theorems exist. The only "unsolved" case is that of problems in linear topological spaces with inequality and with equality - inequality constraints. For this case, we present two conjectures about second order conditions that are analcgousto the equality constraint case. 2
We now state the "general" optimization problem:
Let A,B,C,D be real linear topological spaces. Consider the func- tions :
Let P be a partial ordering of By let >- be defined on C - as usual - by way of a convex cone and let el e2 e3 and e4 be the neutral element of 33' addition - zero elements - of A,B,c and D respectively. The problem may (1) be stated as follows .-
A a Find E A such that f (xf is P-maximal (2) subject to:
In case B is the real line, P is the relation ">"- defined for real numbers and we have a problem of scaler optimization. In case B is finite dimen- sional we have..a finite-vector maximization problem. In general, charac- terizations of solutions to finite-vector maximization problems may be derived from characterizations of solutions of scaler maximization prob- lems. We shall restrict our presentation to scaler maximization prob3ems.
As an application of scaler maximization theorems we may "solve" a part- ticular finite-vector maximization problem, namely for the case where P is taken to be a aret to'^) ordering of B. FQ' infinite-vector maximization problems, a scalerization theorem appears to be the most appropriate inter- 3
mediate step to derive characterization theorems. Such a theorem was
proved by Hurwicz [20]. The method, for finite-vector m3ximization, con-
sists(') of simply observing that the problem is equivalent to an in-
finite number of scaler maximization problems. However, we shall only be
concerned with scaler maximization problems.
2. FINITE DIMENSIONAL PROBLEMS
In this part we let A = E", B = E', C = E~ and D = E', where E", E~,Em II. E Euclidian spaces of dimensions n, one, m and R respectively. For
theis part we reformulate the scaler maximization problem as follows:
Problem 1. Find x E E~ such that f (x) >- f for all satisfying B ga(x) = 0, a = 1,..., m, h (x) 20,- B =1,..., R . .- We further state some properties of the constraints that will be used
in discussing the theorems in this section. Some of the nomenclature,
designated by " " is Karush's C231.
0 D.1) Definition. Effective constraints. 'Let xbe a point that satisfies hB 0 B (x) >- 0, = 1,.. . , m. Let r (R) be the set of indices B such that h ($1 = 0. The constraints with indices B E r i:) 'kill be referred to as the 0 effective constraint6 at x.
D.2) Definition. "Admissible Diyectiop'! Let h be differentiable. We say that
X is an admissible direction if X is a non-tridal. solution to the inequalities: 0 D.3) Definition. "A curve issuing from x in the direction X." By that we
mean,. an n-vector valued function, §(t), of a real variable t such that d E(0) = $ and 5 '(0) = - = A'. dt S(t) Itzo
0 D.4) Definition. admissible arc issuing from x in the direction A" is
0 an arc issuing from x in the direction X such that h (S(t)) >- 0.
0 D.5) Definition. "Property Q" -for inequality constraints, is satisfied at x.
iff: For each admissible direction A, there exists an admissible arc
issuing from 8.
D.6) Definition. The rank condition for inequality constraints. We say that
the rank condition h is satisfied at $ iff 1.) is differentiable and 2) the B rank of the matrix cR. I, f3 E r ($) , equals the number of effective con- 1 straints.
D.7) Definition. The rank condition for equality_- inequality constraints is -- - -- 0 satisfied at x iff 1) the functions g and h are differentiable and 2) the
0 ag" 9 constraints at x, where EOL.i = -ax. I x =x: L .> ,.: . ..) D.8) Definition. The rank condition for equality constraints is satis-
fied at if the rank of the matrix is a. [E?I1 Finally wa define local and global constraint maxima:
D.9) Definition. x is said to be a local solution to problem 1. iff:
A ,. ,. There exists a neighborhood of x, N(,x), such that f(x) >- f(x) for all ,. ? r N(;) satisfying g(*) = 0 and h(x) 2 0. ,. ,. ,. D.lO) Definition. x is said to -.be a global solution to.problem 1. iff f(x) 3 f(x) for all x satisfying g(x) = 0 and h(x) > 0.
2. 1. First order necessary conditions. +. Theorem -1. If f, g -and h -are contjnuously differentiable and if x --is a
global solution to problem 1, then there exists a vector (A0' v,p) = (Ao, 1 m L v v ,e ,pa) 0 such that, 1 > 0 and ; ,. . . , ,. . . 0 - 1, eB>0,- ,OhB (;) =o..
2) Fx- 0 = 0, where F0 = X f + w. g + p. h, Fx0 is the vector of partial 0
A derivatives of F with respect to the component's of x evaluated at k = x.
n For the case of equality constraints, i .e. the set {x I h (x) 0 )=E , (6) Theorem 1 was proved by ~aratheodor~(~)C93 and Bliss C71. For the case n of inequality constraints, i.e. the set {x I g (y) = 0) = E , theorem 1 -
except for the non-negativity of p in condition 1 of the conclusion - was
proved by ~arush'~)[231. The non-negativity of p may be proved by using a separation theorem (8 . This was, essentially, the crux of Fritz John's
[22] proof who has treated a more general problem of an infinite number of
inequality constraints. For the equality-inequality constraints the proof
involves writing equality constraints as inequality constraints (g lo and - g (x) >- 0) and applying Karush's or Fritz Jone's theorem.
Theorem 2. -If, in addition to the assumptions of theorem 1 we have either.(9)
. . a) Property Q for inequality constraints, D. 4, and the rank condition for .. .. equality constraints, D. 7, are satisfied at x.
I a ) --The rank condition ~-for .. equality -inequality constraints is satisfied
A -at x. Then the conclusions of theorem 1 follow with X 0. ---L--I)-
For equality constraints the theorem was proved as a corollary of theorem 1 by ~liss(lO' [7]. For the case of inequality constraints the theorem was proved by Karush (I1) C231. The proof, for equality-inequality constraints, may be accomplished by converting inequality constraints to equality constraints and obtaining the theorem as a corollary to theorem 1.
This was presented by Pennisi (13) C271. A direct proof was presenied by
Hestenes (l" [l9].
2.2. First Order Sufficient Conditions.
Theorem 3. If: 1) f. E and h are differentiable 2) The conclusions of
A ,. A theorem 1 hold with X_ > J at a point x with g(x) = 0, h (x) > 0 3) " .
Either (3.a) FO, of theorem 1, is concave or (3.b) G! is linear and h is ,. concave. Then x is a rloh31 solution to ~roblem1.
The theorem follows from the fact that a concave function lies above its tangent plane. The implications of this fact was utilized by Kuhn-
Tucker C241 in the proof of their equivalence theorem. The present the- orem may be proved by applying Lemma 3 of Kuhn and Tucker C241 to FO. 2.3 Second Order Necessary Conditicns. Theorem 4. If: 1) --f, g and h have second order continuous partial de-
A rivatives. 2: x is a solutLon-- to problem 1. 3) The rank condition for
equality - inequality ccastraints , 9.5. . I.... -- 5 .I--- . ..-!:. 1 m 1 R Thenthere exis-? multipliers ( v,p) = (v ,. . :, v ; LI ,. . . , LI ) such that:
For the equality constraints, thc! theorem was proved by Caratheodory (15) [9] and ~Liss(~~)['7].For inequality constraints the theorem was proved
by Karush (17) [231. For the equality-inequality case, the theorem was (19) proved by Pennisi (lR) [27], under the, dispensible , stipulation that
the number 02 no~l-~sromultipliers attached to effective inequality con-
straints is at most one.
2.4. Second order sxffi,.ent conditions.
Theorem 5,. If: l? f, g c7d h have continuoas second order derivatives.
A A A G! at :; (x) (x) 0. 2) The conclusio~~s theorem 2 holds with g = 0, h >- - 3)~heconclusion of theoren. 4 hclds with strict inequality for r, f 0.
A Then x is a local solution to probl~n1.
For equality constra~nt; f;ic ti !?orex tris proved by Bliss' 20 )[ 7 I. Caratheodory (21) [9] assumes, c~ri;o~cly,that the ra~kcondition holds. For in-
equality constriints the theorec wzs proved by ~arush(~~)[23].A very closely related theorem was proved by ~ennisi'23)~ 27 I, where the restrict -
^' > 0 for-indiceg 4 with ions on n are augmented by requiring that C.h.11"i-
p0 > 0. Theorem 5 may be proved by applying the sufficiency theorem for
equality constraints by using Karush Is device' 24)~23I. A direct proof was
provided by He~tenes(~~)[191.
3. A VARIATIONAL PROBLEM
In view of the vast literature on the calculus of variations and
the wide accessability of such literature, this part will be very brief.
Many interesting topics will be left out, e.g. problems with retarded ar- (26 guments . For a survey of variational problems with equality constraints,
the reader is referred to Bliss [6], which we shall take as a point of de-
parture for this part. We formulate the Bolza-Hestenes (27) problem in the
Calculus of variations as follows: Let T be a subset of the non-negative
half of the real line. Consider the class of piecewise smooth functions
x(t) defined on t and having values in E" together with their derivatives k.
The problem is :
A ^AA A A A -ern 2.Find t tl, k(t), ~(t),k(?), x(t3 = 3 that maximizes J0 [XI = 0 0
L1 0 x, x x It f (t, A) d t + rQ(to, tl, (to ), (t1 )) subject to 0 + B (3) ip (t, X, A) = 0,a = 1,...,. 21, B - (4) 4 (t, x, 2) >- 0 , B = R 1 + 1,... ,L,
with t < t 0 1'
As was noted, by Berkovitz C51 and Hestenes C171, the above
problem is equivalent to the problem of optimal control(28). The re-
sults of optimal control theory are derivable from the results that (29) we shall present.
3.1. First order necessary conditions.
Theorem 6. If: 1) 1
ferentiable. as functions of real variables. 2) Z is a solution to Dro-
- and where the index desig-
0 A nates indices B with 4 (2) = 0, i '= 1,.. . , n.
Then there exist a constant vector A , q,o) = (A- ; q1 ,. . . , qm ; o1 , . . , us)
1 R A A and a vector function p = (p ,..., p ) defined on Ct , t 1 such that:
1) There does not exist t E Ct tll with LAn, q, W, PI = 0, also (ho,q,o) A'. 0.. 3) P (t) is piecewise continuous, continuous at points of continuity of
- A k,p t >- 0, pB (t) +OL i31 = 0
5) -d = , i = 1,..., n, where F = A o f0 t TCY qCYfa t pB 0' , Fk IX dt i i
a' A A 7) d G +[ !F-fxiFx. ) dt t Z FA dxi(t )lY'l = 0 is an identity & dxi i Y1i y y=o
,. A .-. A A .-. ,. E$ 8) E (t, X, X, k) = F (t, X, X) - F (t, X, k) - C (X - k.) FA 1 i 1 2 ru i B
A whenever (t, x, A) satisfy the.constraints (1) - (6).
For fixed end points (i.e. t x (t end x (t are conslants), 0, 0 1 (30-31) Valentine [29] proved that conclusion: 5 E 8) are necessary- with conclu- B sion 3) holding for p except for the non-negativity of p . The non-neg- - ativity of pB was proved by applying a Clebsch (32) type second order nec-
essary condition' 33). Valentine Is method consisted of converting differ-
ential inequality constraints to equations by subtracting from each, the
square of a derivative of an added variable. Then he derived his results as applications of characterization theorems for the problem of Balza with equality constraints (34). For the general problem we may convert the in- equality constraints (2) and (6) to equality constraints by introducing - - - - three new sets of variables ya1 (t), Y: (t) and y; (t as follows : - - - - - ci B y, (tl) free, G: (t) = 0, y2 (ty ) free and $Y=3 0, yY(t3 1 ) free, and consider
0 ..equivalen~prcblen of maximizing J subject to constraints (11, (31, (5) . ------1 -a 1 -.G 1 and: (2 ) j0 = J~ - (yl (tl)12 = 0, (4) = 6' - = 0 and (6 ) = - - ci 2 qa - (y3 (tl)) = O. We then get, in addition to Valentine's conditions,
condition 7) of theorem 6. Noting that condition 6) may be obtained from
5) and 7) we would have all of the conditions of the theorem. A direct ..
elegant proof of theorem 6 was provided by Hestenes [18].
We now discuss a condition that guarantees that A in theorem 6 is 0 non-zero. The case where the multipliers are unique, choosing X = 1, is 0
what is known in the literature as the normal case. Although conditions
for normality are very hard to verify in applications, we shall present one
of these conditions in this section (35)
* D.11) Definition. Normality. Z is said to be normal iff conclusions 1) - 5) and
7) of theorem 6 hold with ho = 1 and the multipliers q,p,w are unique.
D.12) Definition. Admissible variations. Consider a point 2 and a vector valued u b u - - - functions E (t) = (C1 (t),.. . Sn (t)) ; 0 1,. .. , m t s, where m = m t . 1 - - the number of constraints J~ that hol4 as equations at 2, effective at the - - end points of Z, and where 5 = S, + the number of constraints GY that are - - 1 mts effective at g. E(t) = (5 (t),..., 5 (t)) is said to be admissible
variations iff:
1) 5. (t) are differentiable on Ct 1 0 ' ill, - - - 3 s a L? *' = c for a with 4p (Z) = 0, where m0 = - 2.2) 4- (SO) = ? 4x Si + l 4x Si 'i li 1 - - and where , 4,a and 4; are defined similarly. i i i
D,13) Definition. The rank condition. The first rank condition is said to be
0 satisfied at 2 if there exists a set of admissible variations 5 , and arbi-
u 0 - - trary constants T -ri. 0.1,. , m+s, such th~tthsmatrix 0 ' . .
L . I - - a - m t s, where c1 = x (to) - fa (Io, X(to),x(to))) T,u '! 1 o i(t ) 0
- - ' 0 - S? (tl) t 1:' (, 5; (t) + r: ;.(t)) d tl, 6 denotes indices of t Xi i 1 0 - - effective constraints J [XI at Z, " = 1,..., m + s, - 3 - - - u - (tl)l, 7 denotes indices - of $Y that are effective at 5, 0 = 1,. . . , m + s, and where subscripts
If denote ~artialderivatLves with res~ectto indicated variables and " - - above an expression indicates that it is evaluated at X.
Theorem 7. If f, g, 4 and 1) are differentiable then the rank condition, - - definition D.12., at S is necessary and sufficient for the normality of 3.
In the absence of inequality constraints, the theorem was proved (36 by Bliss [6]. Reformulating the problem with added inequality constraints,
as indicated above, the theorem is obtained as a straight forward a~plication
of Bliss's theorem.
3.2. First Order Sufficient Conditions.
Theorem 8. If 1) f, g, b and $ are differentiable, and concave. 2) Con-
clusions 1) - 5) and 7: of theorem 6 are satisfied with: 2: i)
h > o, qY :iY y 0 2. ii) > a, > o, = I,... , ml, y = l,... , sl, at a point 2 ,. ~~--- that satisfies the constraints (1) - (6). Then Z is a global solution to
The theorem was moved by Mangasarian C261 for the canonical, optimal control, problem with fixed to and t 1' Theorem 8 may be proved by
repeating the steps of Mangasarianls proof for problem 2. As Man-
gasarian notes, in the absence of equality constraints condition 2. ii)
of theorem 8 is not needed, not is it req-uired if the equality constraints
are linear.
3.3. Second Order Necessary Conditions.-
Theorem 9. (Jacobi - Myer - Bliss) If: lj f, g, h and + have continuous
A A second order derivatives. 2) 3 is a solution to ~roblem2 and 3) Z is
normal. Then there exist multipliers as in theorem 6 with X = 1 such that: 0
't, -A * * +I, (.I. F 5.5. + 2 .I. F 5.5. + .Z. F. t.t.)d t < , for 1, X.X 1 ] I,] X.? 1 ] l,] X.? 1 J - tn 1j 1 j 1 j
(T, 5) = (To, T~,El (t);... , En (t)) 0 satisfying
^B B (i) @ (5) =O, Q- (5) = 0 .I
with the terms in (i)and (ii) defined as in (2) of D.12 and as in D.13
5 5 with (T, <) as amatrix with- one rov! (T , 5 ) = (r, <)), and where: 2 0 A d G (2; 'r, 5) is the sxond differential of G at 2 with (r, 5) as in- crements, a single subscript denotes a first derivative and a double subscript denote second (mixed partial) derivatives with " - signifying evaluation at Z . For equality constraints the theorem was proved by ~liss(~~)[61
and [81!~~)1.1 adied i:lequality constraint the theorem may be proved by
applying Bliss's theorem to our problem, after converting it to a problem
with equality constraints as we indicated in the discussion of theorem 6.
Theorem 10. (The Clebsch condition). If the hypotheses of theorem 9 are
satisfied then 1. F. . II.II. < 0 for II = (II , nn) f 0 satisfying i,] x,x, 11 - ,,... - ? iBn. =o,B=~,..., LI x, II.1 = 0, B are indices of inequality i=l x, 1 iB
constraints that are effective at Z- -a C ,aA 11. =0,=. ml,i~lli = 0 a indicates constraints ;fk: 1
A J~ effective at Z.
Theorem (10) was proved by Valentine '39'~~9~for fixed end points.
Valentine's method of proof can, easily, be applied to prove our theorem.
3.4. Second Order Sufficient Conditions
For the purposes of this section, we have to define a weak local
solution of problcm 2.
A D.13. Definition. Weak Local Solution. We say that Z is a weak local solution
A if JO [a] >- JO [a] for all 3 satisfying the constraints (1) - (6) with
1) (X-X, -1I I < E -or sone E > 0, where 1 1 *II denotes the Euclidian norm.
Theorem (Pennisi) If 1) f, g, 4 and + have continuous second order 16.
Partialderivatives. 2) Conclusions 1 - 5 and 7 of theorem 6 are satisfied at a point Z that satisfies the constraints 1 - 6. 3) The matrix
A of theorem 6, has full rank. 4) The form Q (Z; r, 5) of @A theorem 9 is negative definite under constraints (i) and (ii) i ' (in the statement of theorem 9) for (T, 5) -) 0. Then there "exists an
A E > 0 such that 2 is a weak local solution of ~roblem2, in the sense of '\ definition D. 13.
Assuming that X = 1 and that at most one inequality constraint 0
A is effective at Z, Valentine (40) proved an analogous theorem in C291.
Penni~i(~~)[27]proved a sufficiency theorem without the assumptions of
Valentine. Pennisi's theorem is stronger than theorem 11 in the sense that Q is negative on the subset of variations (-r, 0) of theorem 11 which in a+dition (to (i) and (ii)) of theorem 9) satisfy some inequal- ities for those constraints that are effective but have zero multipliers.
One way to prove theorem 11 is to convert the problem into one with equality constraints and apnly Pennisi's theorem. Another way is to note that theorem 11 is corollary of Pennisi's theorem, after modifying the latter to take care of the additional constraints.
I
4. A PROBLEM IN LINEAR TOPOLOGICAL SPACES.
Let A, C, D be real Banach spaces and let R denote the real line. Consider f: A - R : A -t C and h: A -t D. The problem we study here is :
g(x) and h(::) L €14 where the Problem 3. t:r:Lr-!ize f(::j subject to = 0 3 inequality 0 and 8 are as defined in the introduction. 3 4 Dealing with dizferentiable functions we note that there are num- (42) erous equivalent vr2.y~ of defining derivatives in linear spaces. We shall use Frechr:tls definition and mean ":'rechc.t differentiable" when ('13 we say tha-t a function is differenthblc.
4.1. First Order Necessa~%yConditions
A Theorem 12. If f, g and h are diffcrcntiable and if x is a solution to problem 3 then there exists a constant X > 0 and linear functionals 0 - 2 2 C+R, x2: D + R such that 1) >- 0, [a2, h(;)l = 0, where [e , hl denotes the value of -.the functional k2 at h(r). 2) For any 1 i 2 2 y E C, y2 E D, the triple (A,, [PA , yl!, [a , y ] / 0 3) F1 (x) = 0,
1 2. A where F = X .f t [R , gl + [R , hl and F' (x) = d F (x, 5) with 5 as the n
"increment" in the definition of the differential.
~1,~1;1.2.: -?.J,TJ ;--,? 1 JL,;, 6.;-- ... -1-.,. *.-:-I theorem 2.1 in Duboviskii and . .
Milyutin("4)[l~~.
Conditions that guarantee that X > 0 and that the functionals 0 1 2 R and R are unique are referred to in the literature, pertaining to problem 3, as regularity conditions and as constraint q-ualifications.
We shall now list these conditions and present some sufficient condi-
tions for them to hold.
(4.1.1) Regularity Conditions:
(R.l) (Gapushkin [14]): For equality constraints, x is said to (R.1) regular
if for every 5 E A with g1 (X, 5) = e3, 5 we have: There exists a
[O ,l] + A such that ~(0)= g(T~(t))8 function of a real variable t, V: 2, = 3
for t E [0, 11, V' (t, T) exists for t E C0,ll and V' (0, -r) = 5.
(R. 2) (Hurwicz [20]) : For the inequality constraint, x is (R. 2) regular iff:
For any 5 s A with 5 0 such that x = + 6 implies h' 5) + h (x) 1 e4, 1 x (x,
we have: There exists a function of a real variable t, V: [O, 11 + A
such that :
(i) V' (t, T) exists fort E CO, 11 - (ii) x = V (0)
(iii) h (V (t)) >- 04, t E CO,ll (iv) V' (0, r) = 5, r > 0.
(R.3) (Gapushkin C141): 2 is said to be (R.3) - regular iff: For any 5 E A
with g1 (;, 5) = 0 and h + h' O4 we have: There exists a 3 (x, 0 2
function of a real variable t; V: [O, .11 + A such that (a) v (0) = x
(b) g (V (t)) = 03, h (V (t)) > e4, t E [O,ll
(c) V' (t, T) exists for t E C0,lI
(d) v' (0, T) = 5, T > 0-
Remark 1: (R.3) is a specialization of Gapushkin's regularity condition
which is a uniform regularity condition(46). (R.1) is a further special-
ization for the case of equality constraints.
Remark 2: Recall that the inequality in constraint 2) of problem 3 is
defined in terms of a closed convex cone, say, K2. Let K = {e K,, 3 I&, where (0 1 is a cone that contains only 0 Let 8 = 03Q04 and let 3 3 '
G: A -t C D be the "pair" valued function < g, h >. Then we may write
constraints 1) and 2) in the form G(x) >- 8 where ">"- is in the sense of K. With that formulation, (R.2) becomes a regularity condition for equality-
inequality constraints.
4.1.2. Sufficient Conditions for Re~ularitv:
We now present some conditions that imply regularity. We present
some sufficiency lemmas for equality constraints and some sufficiency lemmas
for equality - inequality constraints. These last lemmas are, of course,
sufficiency lemmas, for R.1 and R.l regularity. However, they may be strengthened by s~scializingthe conditions when we are concerned with
'(R..l) - regularity or (R.2) - regularity. Before stating these condi-
tions we introduce some notations.
0.1) The constraint set N = {x E A: g (x) = O3 and h (x) >- e4].
0.2) Let I I 11, the norm of the space A, a sohere in A with centw at ;and - radius 6 will be denoted by y (x, 6) and y (x, 6) = {x: I I x - x I In 5 ti)-
0.3) The set D; = E A: F' = 03} is a subspace of A. Let P- be the 1F (x, 5) X projection operator with P- A = D-. X X ..... 0.4) Let 2 be a linear space, we denote by 2 the space of line are functionals
defined on Z.
0.5) Denote by N the set N =U. y(x E) 6 6 XEN O' - 0
ncw l',st the conditions which we use in the statements of suffi-
ciency theorems for regularity.
(S.1) The function hi (x, 5) and g' (x,S) are continuous and bounded on N .
(S.2) g' (x, 5) maps A onto C. Furthermore, the snace A may be written as the
direct sum of D- and another subs~aceE- i.e., A = D- 4l E- where the pro- X x ' X X jection operator (see 0.3) P- is bounded, i.e., there exists a positive X constant M such that P- < M. I I x I I A- .'. .I. .'. (S.2)'The s;,ace A" may be written as the direct sum of two subs~acesR: and S" X X .'. .5 ... .'. .I.,. - .L.. .'. ... -.... i.e., A" = R- W S-, where R- = {g' (a, F1) = 0, 5" E c"} and \,,here g1 is X X X 1 (48) ... *.. the conjugate operator of g' and where S- is a subanace of A". Further- ... X ...,. more, the projection operator Q- with Q- A = R- is bounded. X X X (5.3) Given 6 > 0. For any x E y(x, 6) ,q M, the snace A can be written as .#. the direct sum of A=DxEJx and the projection operators P x with P x A" = Rx
are bounded and satisfy the Linschlitz condition, i.e. , I / Px ( I <- M and
X1' X2 6 y (x, 6)nN, where
P,: P,: and M arc positive constants. 1 .'. .'. .I. (5.3)' For any x E A(;, 6) nN we have A" = R;@S~ where the projection
I.. -7- operators Q with Cx A = R are bounded and Linschitzian. x ' x '
, -9. -1. -1. -9. (S.4) 11 g '' (X, 5:) I/ ) M 11 51 11, for any F; E c*', where M > 0.
(S.4)' 11 gt (X, n) (1 M1 I I 4Iln, for any q E E-x ' where M 1 0. t - (S.4)" For any y E C, the equation g (x, b) = y has a solution b(y) with
is a positive constant. 11 b(y) I IA M2 /I y I IC where M 2
(5.5) There exists E. r A with / 1 5 / IA K such that: (i) gr (x, 5) = e3.
1 (ii) [L, (h (x) + h (x, ?))I 2- P, where L is a non-negative linear functional with I I L I I = 1 and where P and K are positive constants.
We now present sufficiency conditions for regularity. These
1el11111ctefol l.ow from Gapushkin's theorems L-141 on uniform regularity of N.
Lemma 1. For equality constraints-, !S,3) => (R.1). -.
The lemma follows from theorem 2 of Gapushkin [28].
1 Lemma 2. If A is reflexive then (S.3) => (R.1) for equality constraints.
This follows from lemma 1 (as corollary to theorem 2 of Gapushkin [14].
I It Lemma 3b. S.l, S.2 and either S.4, S.4 or S.4 2 (R.1) for equality
constraints. This follows from theorem 3 of Gapushkin C141 and its corollaries. Lemma 4: In the presence of equality and inequality constraints -any of the following conditions is sufficient for (R. 3 ) :
(i) The equality constraint satisfies (R. l), and (s. 1) and (s. 5) are
satisfied .
(ii) (S.l), (S.3) and (S.5)
(iii) A isreflexive- (S.1), (S.3) and (S.5)
1 (iv) S.,2 4 and (S.5)
1 (v) (S.l), (S.21, (S.4) and (S.5)
1 r (vi) Sly2 4 and (S.5).
The lemma follows from theorem 4 of Gapushkin C141 and from his
remark at the end of section 4 of [141.
4.1.3. First Order necessary conditions for the regular case
A Theorem 13. If, in addition to the assumptions of theorem 12, x is
(R.3) - regular then the conclusions of theorem 12 follow with XA > 0 " and k1 and e2 are unique (taking h = 1). 0
For the case of equality constraints, the theorem was proved directly(49)
by Goldstine C151, ?~tilizlag(S.2) without assuming that A is the direct
sum of S.2.(50) The theorem was proved directly by ~urwicz(~~)[20],and
it follows from theorem 5 of Gapushkin, who restricts A and C to be
reflexive. 4.2. First Order Sufficient Conditions.
Theorem 14) If 1) The functions f, g and h are differentiable, 2) The
A conclusions of theorem 12 are satisfied, with An > 0, at a point x that satisfies the constraints of problem 3, and if either: 3.n) The functional
F of theorem 12 is concave, 3.b) The equality constraint g is linear and
A f and h are concave. Then x is a global solution of problem 3.
For an outline of the proof of this theorem see the proof of theorem' v.3.3. of Hurwicz [20] where he utilizes the fact that the difference be- tween the values of a concave func'cional, say J (XI, at two different points is less or equal to the differential, i.e., J(xl') - J(xf) c J1(x" - x')).
Guignard [16], using cl constraint qualification, proves theorem 14 with pseudo-concavity of f and h replacing assumption 3 of the theorem (in the absence of equality constraints).
4.3 Second O~derNecessary Conditions. Conjecture 1. If 1) the ,. functions f, g, and h have second order differentials, 2) x is a solution to problem 3 and 3) x is regular. Then the upper bound of F" (x, 5) is non-positive, for 5 with 11 5 [IA = 1 that satisfy a) g' (x, 5) = 0, b) If ,. ,. h is effective i.e., if h(x) = 0 then h' (x, 5) = 0, where F is as defined in theorem 12.
FOP equality constraints the conjecture was proved by ~oldstein(~~)[l~].
4.4; Second Order Sufficient conditions. 24.
Conjecture 2. If 1) f, g and h have second differential, 2) The con- ,. clusions of theorem 12 are satisfied at a point x that satisfies constraints
A 1) and 2) of problem 3, 3) The point x is regular and 4) The upper bound
6 of F" (x, 5) is negative for I: with I I 6 I I = 1 satisfying a) g1 (x, 6) = 0
A A and b) if h is effective at x then h' (x, 5) = 0, where F is as defined in theorem 12. Then there exists a neighborhood N in A such that * f (x) >- f(x) for x E fin N.
This conjecture was proved by ~oldstein(~~)[51for the case of equality constraints. 25. FOOTNOTES
This general statement of the problem is due to Hurwicz C201.
We shall restrict our attention to maximization. Characterizations of solutions to minimization problems follow trivially from maximization theo~ens. i In that case, x is said to Se Pareto superior to y if f (x) 2 A fL (y) for i = 1,.. . , r (r is the finite dimension of B) , and x is Pareto optimal (p - maximal) if there does not exist a point y E A satisfying g(y) 02, h(y) = 0 which is Pareto superior to x 2 3 Suggested to the autho~by Hurwicz.
Theorem 2, sect!-on 187 pat 11.
Theorem 1.1.
Theorem 3.1.
Separating the linear sets: +. B fi Si < 0 and 1 > 0. t fii i - See Dubovskii Milyutin 1101 for an extensive discussion.
T1.- condi.i-jons -:.:lxt kcllci.r -.?e aLternatiire forxs of tSc constraint qualification, see Ar~ow-I-furwicz, Uzawa C11 for other forms and for relations among various forms of the constraint qualification.
In remarks following theorem 1.1.
Theorem 3.2. B D Following Karush C231, by writing h (x) >- 0 as h (x) - (z8l2 = 0 and solving the problem in the space of n + m - vectors (x, g).
Theorem 3.1.
Theorem '0.1, Chapter 1.
Theorem 3, section 212, Part 11.
Theorem 1.2.
Theorem 5.1.
Corollary to theorem 3.2. See my notes [Ill, where Pennisi's theorem is proved by directly applying the second order necessary conditions for equality constraints using Karush's device (footnote 12).
Theorem 1.3.
Theorem 4, section 213, Part 11.
Theorem 6.1.
Theorem 3.3.
See [121.
Theorem 10.3, chapter 1.
See El'sgol'c [121, Halanay C171 and Ewing [13].
See Hestenes [18].
As stated in these two papers.
See Berkovitz C51 and Hestenes El81 for the necessary transformations.
First Necessary Condition I, Page (412).
Second Necessary Condition 11, Page (414).
See section 3.2 in this paper.
Corollary 3.4.
These .theorems may be found in Bliss's paper [6].
See Berkovitz [5], section VIII, theorem 3, for alternative sufficient condition for normality.
Section 9, Page 693.
Sections (24) - (26).
Theorem 80, P. 228, the statement and proof here are more complete than they are in [61.
Corollary 3:4 P. 9.
Theorem 10.2, section 10. Theorem 2 -1.
See Averbukh and Smolyanov [2] and [3].
See Vainberg [281 and Liusternik and Sobolev [25] for an exposition of calculus in linear sDaces.
Duboskii & Milyutin [lo] utilize the fact that the set of "variations1' that give the maximand value greater than the ma could not intersect with the sets of "variations" that satisfy the constraints. By variations they mean differentials at x. Since these sets are de- fined by linear inequalities and equations they are convex. Using a separation theorem they derive what they call the Euler equation. Writing the Euler equation in terms of differentials of the maximand and constraints we obtain conclusion 3 of theorem 12.
For finite dimensional spaces, this is equivalent to the rank condition.
Section 2, page 592, in the sense that the condition holds for all points of the constraint set.
This is Gapushkin's [14] notation.
See Kantorovich [211 Chapter XII, nage 476.
Without using theorem 12, theorem 2.1.
See Liusternik and.7Sobolev C251, (page-204) fcr a proof that this part of (S.2) is dispensable and for an elegant proof of theorem 13 for equality constraints.
Theorem V.3.3.2 (page 971, see remark 1 in 4.1.1 of this paper.
Theorem 2.3 (page 147). REFERENCES
Arrow, Hurwicz and Uzawa: Constraint qualification in Maximization problems. Naval Research Logistics Quarterly, Vol. 8 (1961) 175-191.
V.I. Averbukh and O.G. Smolyanov: The Theory of Differentiation in Linear Topological Spaces. Uspekhi Mt. Nauk 22: 6(1967) 201-260; English Translation in Russian Math. Surveys 22: 6(1967) 201-258.
V.I. Averbukh and O.G. Smolyanov: The Various Definitions of the Derivative in Linear Topological Spaces. Uspekhi Mat. Nauk 23:4 (1968) 67-116. English Translation in Russian Math Surveys 23:4 (1968) 67-113.
Claude Berge: Espaces Topologique, foncions multivoques, dunod, Paris 1959. English Translation: Topological Spaces, MacMillan, New York 1963.
Leonard D. Berlcovitz: Variational Problems of Control and Pro- graming. Journal of Mathematical Analysis and Applications, 3(1961) 145-169.
Gilbert Ames Bliss: The Problem of Lagrange in the Calculus of Variations. American Journal of Mathematics, 52 (1930) 673-743.
Gilbert Ames Bliss: Normality and Abnormality in the Calculus of Variations. Transactions of the American Mathematical Society,
Gilbert Ames Bliss: Lectures on the Calculus of Variations. University of Chicago Press, Chicago 1946.
C. Caratheodory: Variationresechnung and Partielle Differential- gleichungen Erest Ordnung. B.G. Teuber, Berlin 1935. English T;-anslation, Holden-Day , San Francisco, California, Vol. I 1965, Vol. I1 1967.
A.Ya. Dubouskii and A.A. Milyutin: Extremum Problems in the Presence of Restrictions. Zh. Vychisl Mat. Fiz. 5 (1965) 395-453. English Translation, U.S.S.R. Computational Hathematics and Math Physics, 5 (1965) 1-80.
Mohamed El-Hodiri: Notes on Optimization Tt.eory, Part I: Finite Dimensional Problems. Research Papers in Theoritical and Applied Economics, No. 21, University of Kansas 1969.
L. E. El'sgo? I,.;Qualitative Methods in Mathematical Analysis, Moscow 1955, English Translation as Volume 12, Translations of Mathematical Monographs, American K~thematicalSociety. Providence, R.I. 1964. George M. Ewing: Calculus of Variations with Applications, Piorton & Co., Mew York 1969.
V.F. Gapushkin: On Critical Points of Functionals in Banach Spaces, Mat. Sbornik 64 (1964) 589-617. English Translation in Three Papers on Optimization Theory, PEREC Report, Purdue University, Dept. of Electrical Engineering 1966.
Herman H.- Golstine: Minimum' Problems in the Functional Calculus. Bul. American Math. Society 46 (1940) 142-149.
M. Guignard: Generalized Kuhn-Tuker Conditions for Mathematical Programming Problems in a Banach Space. SIAM Journal on Control 7 (1969) 232-241.
A. Halanay: Optimal Control Systems With Time Lag. SIAM Journal on Control 6 (1968) 215-234.
Magnus Hestenes: On Variational Theory and Optimal Control Theary, SIAM Journal on Control 3 (1965) 23-48.
Magnus Hestenes: Calculus of Variations and Optimal Control Theory, John Wiley E Sons, New York 1966.
Leonid Hurwicz: Programming in Linear Spaces, in Arrow, Hurwicz and Uzawa (eds) "Studies in Linear and Nonlinear Programming1'. Stanford University Press, Stanford, California 1958.
L.V:Kantorovich and G.P. Akilov: Functional Analysis in Normal Linear Spaces, Moscow 1959, English Translation, Pergamon Press 1964.
Fritz John: Extremum Problems with Inequalities and Subsidiary Conditions. In K.O. Friedrick; O.E. Neugebaur and J.J. Stokes (eds) "Studies and Essays: Courant Anniversary Volume", Interscience Publishers, New York 1948.
William Karush: Minima of Functions of Several Variables with In- equalities as Side Conditions. University of Chicago Master's Thesis 1939.
H. W. Kuhn and A. GI. Tucker : Nonlinear Programming, in "Proc. Second u Berkeley Symposium on Math. Stat. and Probability, J. Neuman (ed.), University of California Press, Berkely 1951.
L. Liusternik and V. .->bolev: Elements of Functional Analysis, Mos- cow 1951. English R,&lslation, Ungar Publishing Co., New York 1961. 26) O.L. Mangasarian: Sufficient Conditions for the Optimal Control of Nonlinear Systems. SIAM Journal on Control 4 (1966) 139-152.
27) Louis L. Pennisi: An Indirect Sufficiency Proof For the Problem . of Lagrange with Differential Inequalities as Added Side Conditions. Transactions of the American Math. Society 74 (1953) 177-198.
28) M.M. Vainberg: Variational Methods for the Study of Nonlinear Operations. Moscow 1956, English Translation Holden-Day, San Francisco, California 1964.
29) Fredric Valentine: The Problem of Lagrange with Differential Inequalities as Added Side Condition. In "Contributions to the Calculus of Variations 1933-1937. University of Chicago Press, Chicago 1937.