1 Introduction

DC APPROACH TO REGULARITY OF CONVEX MULTIFUNCTIONS WITH APPLICATIONS TO INFINITE SYSTEMS1

B. S. MORDUKHOVICH 2 and T. T. A. NGHIA3

Abstract. The paper develops a new approach to the study of metric regularity and related well-posedness properties of convex set-valued mappings between general Banach spaces by reducing them to unconstrained minimization problems with objectives given as the diﬀerence of convex (DC) functions. In this way we establish new formulas for calculating the exact regularity bound of closed and convex multifunctions and apply them to deriving explicit conditions ensuring well-posedness of inﬁnite convex systems described by inequality and equality constraints.

Key Words: variational analysis, optimization, convex multifunctions, metric regularity

AMS Classiﬁcation: 49J53, 49J52, 90C25

1 Introduction

Let X and Y be Banach spaces with norms generically denoted by k · k. For any x ∈ X and

r > 0 the symbol IBr(x) stands for the closed ball centered at x with radius r, while the unit closed

ball and the unit sphere in X are denoted by IBX and SX , respectively. Recall that a set-valued mapping/multifunction F : X → Y is metrically regular around (¯x, y¯) ∈ gph F with modulus K > 0 if there exist neighborhoods U ofx ¯ and V ofy ¯ such that

dx; F −1(y) ≤ Kdy; F (x) for any x ∈ U, y ∈ V, (1.1)

where d(x; Ω) stands for the distance from x to Ω in X. The inﬁmum of all the moduli K over (K, U, V ) in (1.1), denoted by reg F (¯x, y¯), is called the exact regularity bound. It has been well recognized that the concept of metric regularity is fundamental in nonlinear analysis and optimization and is used not only in theoretical studies but also in numerical methods. For closed and convex multifunctions (i.e., those with the closed and convex graph gph F := {(x, y) ∈ X × Y | y ∈ F (x)}), this notion is characterized by the Robinson-Ursescu Theorem, which says that F is metrically regular around (¯x, y¯) ∈ gph F if and only ify ¯ belongs to the interior of the range rge F := {y ∈ Y | y ∈ F (x), x ∈ X}; see, e.g., [1, 2]. This is a natural and far-going extension of the classical Open Mapping Theorem for linear operators to convex set-valued mappings. It is well known that metric regularity can be equivalently described in other forms; in particular, as the local covering property of F : X → Y around (¯x, y¯) ∈ gph F (also known as linear openness as well as local surjection), which means that there is C > 0 such that F (x) ∩ V + CrIBY ⊂ F IBr(x) whenever IBr(x) ⊂ U as r > 0

1This research was supported by the USA National Science Foundation under grant DMS-1007132. 2Distinguished University Professor, Department of Mathematics, Wayne State University, Detroit, MI 48202, USA; email: [email protected]. Research of this author was also partially supported by the Australian Research Council under grant DP-12092508, by the European Regional Development Fund (FEDER), and by the following Portuguese agencies: Foundation for Science and Technology, Operational Program for Competitiveness Factors, and Strategic Reference Framework under grant PTDC/MAT/111809/2009. 3Doctoral Student, Department of Mathematics, Wayne State University, Detroit, MI 48202, USA.

1 for some neighborhoods U ofx ¯ and V ofy ¯. The supremum of all such C is denoted by cov F (¯x, y¯), the exact covering bound. Furthermore, the reciprocal of cov F (¯x, y¯) is the exact regularity bound reg F (¯x, y¯); see, e.g., [3] and the bibliographies therein. A highly important issue for many aspects of nonlinear analysis, especially for its variational frameworks, is the calculation of the exact regularity bound and related quantities; see [3] for results, discussions, and references. When X is an Asplund space (i.e., each of its separable subspaces has a separable dual) and Y is ﬁnite-dimensional, this constant is calculated by Mordukhovich [3, Theorem 4.21] via the coderivative norm (see Section 2):

∗ −1 ∗ ∗ ∗ −1 ∗ ∗ reg F (¯x, y¯) = kD F (¯y, x¯)k := sup ky k y ∈ D F (¯y, x¯)(x ), x ∈ IBX∗ , (1.2) extending his previous coderivative criterion for mappings between ﬁnite-dimensional spaces. A natural question arises about the equality in (1.2) when Y is inﬁnite-dimensional. This has been intensively studied in the recent years; see, e.g., [4, 5, 6, 7, 8]. A simple counterexample to the equality in (1.2) with dim Y = ∞ can be found in [8], while there are a number of positive results in this direction discussed in what follows. In particular, the answer is positive in the framework of F : IRn → IRS × C(T ) given by F (x) := (z, p) ha(t), xi − p(t) ≤ 0, t ∈ T ; hb(s), xi − z(s) = 0, s ∈ S}, (1.3)

where C(T ) is the space of continuous functions on a Hausdorff compact T , IRS is a product space with |S| < n for the cardinality of S, a : T → IRn is continuous, and b : S → IRn is a function over discrete domain. The equality in (1.2) in the setting of (1.3) can be derived from [4, Theorem 3.1] and the explicit calculations provided in Corollary 3.2 therein. A different setting of the equality in (1.2) is identified in [5] for the multifunction F : X → Y

from an arbitrary Banach space X to Y = l∞(T ) with the inverse

−1 ∗ F (y) = x ∈ X hat , xi − bt ≤ yt, t ∈ T for all y ∈ l∞(T ), (1.4)

∗ ∗ where T is an arbitrary index set and (at , bt) ∈ X ×IR for all t ∈ T . Besides diﬀerent spaces X and Y in (1.3) and (1.4), the latter system addresses general inﬁnite constraints (in particular, countable ones) in contrast to the compact index set T in (1.3). Another approach to the equality in (1.2) for closed and convex multifunctions F : X → Y is developed in [8] under the condition

∗ ∗ ∗ ∗ ∗ ∗ cl y ∈ SY ∗ σΩ(x , y ) < ∞, x ∈ X ⊂ SY ∗ (1.5)

imposed on the Banach spaces, where σΩ is the support function of the set Ω := gph F −(¯x, y¯). Con- dition (1.5) holds automatically when Y is finite-dimensional; otherwise, it seems to be restrictive, since the dual unit sphere is never weak∗ closed in infinite dimensions. However, a careful analysis verifies the validity of (1.5) in both cases (1.3) and (1.4); see below along with the discussion of the recent developments in [6, 7]. The primary goal of this paper is to derive precise formulas of type (1.2) for calculating the exact regularity bound for convex multifunctions between general Banach spaces with no imposing condition (1.5) while involving ε-coderivatives. Our approach is based on a simple observation that F is metrically regular around (¯x, y¯) if and only if (¯x, y¯) is a local solution to the following unconstrained DC (difference of convex functions) program:

minimize Kdy; F (x) − dx; F −1(y) subject to (x, y) ∈ X × Y

2 for some K > 0. To the best of our knowledge, this approach is new in literature. The rest of our paper is organized as follows. In Section 2 we present some basic definitions and preliminaries, which are widely used in the sequel. Section 3 contains the main results of this paper that provide relationships between the exact regularity and the coderivative norms for closed and convex multifunctions. In Section 4 we apply the results obtained in the preceding section to the convex infinite systems containing linear equality and infinitely many convex inequality constraints indexed by an arbitrary set T . A significant part in this section is to show that when adding linear equality constraints to system (1.4), the problem becomes essentially more complicated. Moreover, the results in this section also extend some known ones established recently in [6, 7]. Several instructive examples are constructed to make comparisons of our results with those in [4, 5, 6, 7]. Our notation and terminology are basically standard and conventional in the area of variational analysis and semi-infinite/infinite programming; see, e.g., [3, 9]. As usual, h·, ·i signifies the canonical ∗ pairing between X and its dual X∗ with the symbol w→ indicating the convergence in the weak∗ topology of X∗ and the cl∗ standing for the weak∗ topological closure of a set. Given an arbitrary T index set T , we consider the product space of multipliers IR := {λ = (λt)| t ∈ T } with λt ∈ IR for T T t ∈ T and denote by IRe the collection of λ ∈ IR such that λt 6= 0 for only finitely many t ∈ T . The positive cone in IRe T is defined by

T T IRe + := λ ∈ IRe λt ≥ 0 for all t ∈ T .

2 Preliminaries

Let X and Y be Banach spaces, and let ϕ: X → IR := IR∪{∞} be an extended-real-valued function with the domain dom ϕ := {x ∈ X| ϕ(x) < ∞}. We always assume that ϕ is proper (i.e., dom ϕ 6= ∅) and convex. Its conjugate function ϕ∗ : X∗ → IR is deﬁned by

∗ ∗ ∗ ∗ ϕ (x ) := sup hx , xi − ϕ(x) x ∈ X = sup hx , xi − ϕ(x) x ∈ dom ϕ . (2.1)

The ε-subdiﬀerential of ϕ: X → IR atx ¯ ∈ dom ϕ is given by

∗ ∗ ∗ ∂εϕ(¯x) := x ∈ X hx , x − x¯i ≤ ϕ(x) − ϕ(¯x) + ε for all x ∈ X , ε ≥ 0. (2.2)

When ε = 0 in (2.2), the set ∂ϕ(¯x) := ∂0ϕ(¯x) is the classical subdiﬀerential of convex analysis. Taking (2.1) into account, the ε-subdiﬀerential (2.2) can be written as the form

∗ ∗ ∗ ∗ ∗ ∂εϕ(¯x) = x ∈ X ϕ (x ) ≤ hx , x¯i − ϕ(¯x) + ε . (2.3)

The following sum rule is well-known in convex analysis (see, e.g., [2, Theorem 2.8.7]): [ h i ∂ε(ϕ1 + ϕ2)(¯x) = ∂ε1 ϕ1(¯x) + ∂ε2 ϕ2(¯x) (2.4)

ε1+ε2=ε ε1,ε2≥0

provided that one of the functions ϕi is continuous atx ¯ ∈ dom ϕ1 ∩ dom ϕ2. Since the above subdiﬀerential constructions and results are given for extended-real-valued functions, they encompass the case of sets Ω ⊂ X by considering their indicator functions δ(x; Ω) equal to 0 when x ∈ Ω and ∞ otherwise. In this way, the collection of ε-normals to a convex set Ω at x¯ ∈ Ω is deﬁned by

∗ ∗ ∗ Nε(¯x; Ω) := ∂εδ(¯x; Ω) = x ∈ X hx , x − x¯i ≤ ε for all x ∈ Ω , ε ≥ 0. (2.5)

3 Then it follows from (2.3) that

∗ ∗ ∗ ∗ Nε(¯x; Ω) = x ∈ X σΩ(x ) ≤ hx , x¯i + ε , (2.6)

∗ ∗ ∗ ∗ where σΩ stands for the support function of Ω deﬁned by σΩ(x ) := sup{hx , xi| x ∈ Ω} for x ∈ X . Furthermore, we use the notation co Ω and cone Ω to signify the convex hull and the conic convex hull of the set Ω, respectively. → ∗ ∗ → ∗ Given a set-valued mapping F : X → Y , deﬁne its ε-coderivative Dε F (¯x, y¯): Y → X at (¯x, y¯) ∈ gph F in the direction y∗ ∈ Y ∗ by

∗ ∗ ∗ ∗ ∗ ∗ Dε F (¯x, y¯)(y ) := x ∈ X (x , −y ) ∈ Nε (¯x, y¯); gph F , ε ≥ 0, (2.7)

∗ ∗ with D F (¯x, y¯) := D0F (¯x, y¯) for the case of coderivative. Consider the ε-coderivative norm

∗ ∗ ∗ ∗ ∗ ∗ kDε F (¯x, y¯)k := sup kx k x ∈ Dε F (¯x, y¯)(y ), y ∈ IBY ∗ . (2.8)

It follows from [3, Theorem 1.44] that D∗F −1(¯y, x¯)(0) = {0} if F is metrically regular around (¯x, y¯) ∈ gph F , and hence we have in this case that

∗ −1 ∗ ∗ ∗ −1 ∗ ∗ kD F (¯y, x¯)k = sup ky k y ∈ D F (¯y, x¯)(x ), x ∈ SX∗ . (2.9)

We conclude this section with two results for DC programs used in this paper.

Lemma 2.1 [10] (necessary and suﬃcient conditions for global DC minimizers). Let

ϕ1, ϕ2 : X → IR be proper and convex functions on a Banach space X. Then x¯ is a global minimizer of the unconstrained DC program:

minimize ϕ1(x) − ϕ2(x) over x ∈ X (2.10)

if and only ∂εϕ2(¯x) ⊂ ∂εϕ1(¯x) for all ε ≥ 0.

For the case of local minimizers of (2.10), the following suﬃcient condition is established in [11] with a counterexample showing that this condition is not necessary.

Lemma 2.2 [11] (suﬃcient conditions for local DC minimizers). Let ϕ1, ϕ2 : X → IR be

proper and convex functions on a Banach space X, and let ϕ2 be locally Lipschitzian around the point x¯ ∈ dom ϕ1 ∩ dom ϕ2. Then x¯ is a local minimizer of (2.10) if there is ε0 > 0 such that

∂εϕ2(¯x) ⊂ ∂εϕ1(¯x) for all ε ∈ [0, ε0].

3 Metric Regularity of Closed and Convex Multifunctions

Throughout this section we assume that F : X → Y is a closed and convex multifunction between two Banach spaces with (¯x, y¯) ∈ gph F . Let us present the main result of the paper that establishes relationships between metric regularity and ε-coderivatives.

Theorem 3.1 (calculating the exact regularity bound via ε-coderivatives). Given (¯x, y¯) ∈ gph F , assume that y¯ ∈ int (rge F ). Then we have

∗ −1 reg F (¯x, y¯) = lim kDε F (¯y, x¯)k, (3.1) ε↓0

h n ∗ −1 ∗ ∗ ∗ ∗ oi reg F (¯x, y¯) = lim sup kx k x ∈ Dε F (¯x, y¯)(y ), y ∈ SY ∗ . (3.2) ε↓0

4 Proof. Sincey ¯ ∈ int (rge F ), it follows from the Robinson-Ursescu theorem that F is metrically regular around (¯x, y¯). We can rewrite (1.1) as: there are η, K > 0 such that

−1 d x; F (y) ≤ Kd y; F (x) whenever (x, y) ∈ IBη(¯x, y¯) ⊂ X × Y. (3.3)

Now deﬁne proper and convex functions ϕ1, ϕ2 : X × Y → IR by

−1 ϕ1(x, y) := d y; F (x) and ϕ2(x, y) := d x; F (y) . (3.4)

It follows from the covering property in [1, Theorem 1] that there is r > 0 such that IB2r(¯y) ⊂

F (¯x + IBX ). Combining this with the construction of ϕ2 in (3.4) gives us

ϕ2(x, y) ≤ kx − x¯k + 1 for all y ∈ IB2r(¯y),

which implies that ϕ2 is bounded above around (¯x, y¯), and thus it is locally Lipschitzian around this point due to [2, Corollary 2.2.13]. We have from metric regularity (3.3) that (¯x, y¯) is a local minimizer of the DC program:

minimize Kϕ1(x, y) − ϕ2(x, y) subject to (x, y) ∈ X × Y. (3.5)

Consequently, (¯x, y¯) is a global minimizer of the following DC program: minimize Kϕ1 + δ(·; IBη(¯x, y¯)) (x, y) − ϕ2(x, y) subject to (x, y) ∈ X × Y. (3.6)

Applying Lemma 2.1 to problem (3.6), we get the inclusion ∂εϕ2(¯x, y¯) ⊂ ∂ε Kϕ1 + δ(·; IBη(¯x, y¯)) (¯x, y¯) for all ε ≥ 0.

Since δ((¯x, y¯); IBη(¯x, y¯)) is continuous at (¯x, y¯), it follows from the ε-subdiﬀerential sum rule (2.4) that the latter inclusion reduces to [ h i ∂εϕ2(¯x, y¯) ⊂ ∂ε1 (Kϕ1)(¯x, y¯) + ∂ε2 δ(·; IBη(¯x, y¯))(¯x, y¯)

ε1+ε2=ε ε1,ε2≥0 [ h ε2 i (3.7) = ∂ (Kϕ )(¯x, y¯) + IB ∗ ∗ ε1 1 η X ×Y ε1+ε2=ε ε1,ε2≥0

ε due to the fact that ∂εδ(·; IBr(x))(x) = r IBX∗ for all ε ≥ 0 and r > 0. Next we compute the ε-subdiﬀerentials of the functions Kϕ1 and ϕ2 at (¯x, y¯) by using the ∗ ∗ conjugate functions. It follows from (2.3) that (x , y ) ∈ ∂ε1 (Kϕ1)(¯x, y¯) if and only if

∗ ∗ ∗ ∗ ∗ (Kϕ1) (x , y ) ≤ hx , x¯i + hy , y¯i + ε1. (3.8)

Furthermore, elementary transformations ensure the equalities:

∗ ∗ ∗ ∗ ∗ (Kϕ1) (x , y ) = sup hx , xi + hy , yi − Kd(y; F (x)) x,y = sup hx∗, xi + hy∗, yi − inf(Kky − uk + δ(u; F (x))) x,y u = sup hx∗, xi + hy∗, y − ui + hy∗, ui − Kky − uk − δ(u; F (x)) u,x,y = sup hx∗, xi + hy∗, ui − δ(u; F (x)) + hy∗, yi − Kkyk u,x,y ∗ ∗ ∗ = σgph F (x , y ) + δ(y ; KIBY ∗ ).

5 By using (2.6) and (3.8), the latter yields that

∗ ∗ ∂ε1 (Kϕ1)(¯x, y¯) = Nε1 (¯x, y¯); gph F ∩ X × KIBY . (3.9)

Similarly, by taking into account the form of ϕ2 in (3.4), we arrive at the representation

∗ ∂εϕ2(¯x, y¯) = Nε (¯x, y¯); gph F ∩ IBX∗ × Y . (3.10)

Thus the inclusion in (3.7) reduces to

∗ [ ∗ ∗ ∗ Nε (¯x, y¯); gph F ∩ IBX × Y ⊂ Nε1 (¯x, y¯); gph F ∩ X × KIBY

ε1+ε2=ε ε1,ε2≥0 (3.11) ε2 + IB ∗ ∗ . η X ×Y

∗ ∗ ∗ ∗ To justify the equality in (3.1), ﬁx ε > 0 and pick any (x , y ) ∈ IBX∗ × Y satisfying y ∈ ∗ −1 ∗ ∗ ∗ Dε F (¯y, x¯)(x ), which means that (−x , y ) ∈ Nε((¯x, y¯); gph F ). It follows from (3.11) that there ∗ ∗ ∗ ∗ ∗ −1 are ε1 ∈ [0, ε] and (u , v ) ∈ Nε1 ((¯x, y¯); gph F ) such that kv k ≤ K and ky − v k ≤ (ε − ε1)η . Hence we get the estimate

∗ ∗ −1 −1 ky k ≤ kv k + (ε − ε1)η ≤ K + εη .

∗ −1 Observe from the deﬁnition in (2.8) that the function kDε F (¯y, x¯)k is nondecreasing with respect to ε ≥ 0, which implies therefore that

∗ −1 ∗ −1 −1 lim kDε F (¯y, x¯)k = inf kDε F (¯y, x¯)k ≤ inf K + εη = K. ε↓0 ε>0 ε>0

Letting K ↓ reg F (¯x;y ¯) above gives us the estimate

∗ −1 lim kDε F (¯y, x¯)k ≤ reg F (¯x, y¯). (3.12) ε↓0

It follows from (3.12) that the equality in (3.1) is trivial if reg F (¯x, y¯) = 0. Considering further the case of reg F (¯x, y¯) > 0, we conclude from the deﬁnition of the exact regularity bound that (¯x, y¯) is not a local minimizer of the DC problem (3.5) when 0 < K < reg F (¯x, y¯). Then Lemma 2.2 allows us ∗ ∗ ∗ ∗ to ﬁnd sequences εn ↓ 0 and (xn, yn) ∈ ∂εn ϕ2(¯x, y¯) such that (xn, yn) ∈/ ∂εn (Kϕ1)(¯x, y¯). Combining this with (3.9) and (3.10) implies that

∗ ∗ kxnk ≤ 1 and kynk > K. (3.13)

Since IB2r(¯y) ⊂ F (¯x + IBX ) as mentioned above, we get from (3.10) and (3.13) that

∗ ∗ ∗ ∗ εn ≥ sup hxn, x − x¯i + hyn, y − y¯i ≥ sup hyn, y − y¯i − kxnk (x,y)∈gph F y∈IB2r (¯y) (3.14) ∗ ∗ ∗ ≥ 2rkynk − kxnk ≥ 2rK − kxnk.

∗ Suppose without loss of generality that kxnk ≥ rK for all n ∈ IN and deﬁne

∗ ∗ ∗ −1 ∗ ∗ ∗ −1 ∗ −1 yen := ynkxnk , xen := −xnkxnk , and εen := εnkxnk .

∗ ∗ ∗ −1 ∗ We have kx k = 1, εn ↓ 0, and y ∈ D F (¯y, x¯)(x ). It follows from (3.13) that en e en εen en ∗ ∗ ∗ −1 ∗ ∗ ∗ ∗ ∗ −1 sup ky k y ∈ D F (¯y, x¯)(y ), x ∈ S ∗ ≥ ky k = ky kkx k > K. εn X en n n

6 Letting n → ∞ and K ↑ reg F (¯x, y¯) gives us

∗ ∗ ∗ −1 ∗ ∗ lim sup ky k y ∈ Dε F (¯y, x¯)(x ), x ∈ SX∗ } ≥ reg F (¯x;y ¯), ε↓0 which implies the equality in (3.1) by taking in to account the estimate in (3.12). It remains to prove formula (3.2). By arguments similar to those following (3.14) we get that

∗ ∗ ∗ Dε F (¯x, y¯)(y ) ∩ rIBX∗ = ∅ for all 0 < ε < r and y ∈ SY ∗ . (3.15)

∗ ∗ ∗ ∗ ∗ ∗ Pick any (x , y ) ∈ X × SY ∗ such that x ∈ Dε F (¯x, y¯)(y ) for some 0 < ε < r. Define further ∗ ∗ ∗ −1 ∗ ∗ ∗ −1 ∗ −1 ∗ ∗ ∗ −1 xb := −x kx k , yb := −y kx k , and εb := εkx k . This ensures that xb ∈ SX∗ , kyb k = kx k , and y∗ ∈ D∗F −1(¯y, x¯)(x∗). Observe from (3.15) that ε ≤ εr−1, and thus we have b εb b b ∗ −1 ∗ ∗ −1 ∗ −1 kx k = ky k ≤ kD F (¯y, x¯)k ≤ kD −1 F (¯y, x¯)k. b εb εr This together with (3.1) yields the inequality “≥” in (3.2) by letting ε ↓ 0. To justify the converse inequality in (3.2), note first that it is obvious when reg F (¯x, y¯) = 0. If reg F (¯x, y¯) > 0, we get from the equality in (3.1) that there exists a sufficiently small number 0 < s < reg F (¯x, y¯) ensuring the condition

∗ −1 ∗ ∗ Dε F (¯y, x¯)(x ) ∩ sIBX∗ = ∅ for all 0 < ε < s and x ∈ SX∗ .

By arguments similar to the above proof we also derive that

∗ −1 n ∗ −1 ∗ ∗ ∗ ∗ o ∗ kDε F (¯y, x¯)k ≤ sup kx k x ∈ Dεs−1 F (¯x, y¯)(y ), y ∈ SY . This together with (3.1) justiﬁes the inequality “≤” in (3.2) and completes the proof. 4 The following consequence of Theorem 3.1 and the classical Brøndsted-Rockafellar theorem establishes a precise formula for the exact regularity bound of a closed and convex multifunction F between Banach spaces by using the coderivative of F −1 instead of its ε-counterparts while involving points close to the reference one.

Corollary 3.2 (calculating the exact regularity bound via the coderivative norm at nearby points). In the setting of Theorem 3.1 we have

h n ∗ −1 oi reg F (¯x, y¯) = lim sup kD F (y, x)k (x, y) ∈ gph F ∩ IBε(¯x, y¯) . (3.16) ε↓0

Proof. To verify the inequality “≥” in (3.16), observe from (3.3) that for any K > reg F (¯x, y¯) and any suﬃciently small ε > 0 we get

−1 d x; F (y) ≤ Kd y; F (x) for all (x, y) ∈ IBε(x,e ye) whenever (x,e ye) ∈ IBε(¯x, y¯). It follows from (3.1) that

∗ −1 ∗ −1 K ≥ lim kDηF (y, x)k ≥ kD F (y, x)k for all (x, y) ∈ IBε(¯x, y¯). η↓0 e e e e e e

This clearly implies the estimate

h n ∗ −1 oi K ≥ lim sup kD F (y, x)k (x, y) ∈ gph F ∩ IBε(¯x, y¯) . ε↓0

Letting there K ↓ reg F (¯x, y¯), we arrive at the inequality “≥” in (3.16).

7 To prove the converse inequality in (3.16), take an arbitrary ε > 0 and observe from Theorem 3.1 ∗ −1 ∗ ∗ ∗ ∗ that reg F (¯x, y¯) ≤ kDε2 F (¯y, x¯)k. This allows us to ﬁnd (x , y ) ∈ X ×Y satisfying the condition ∗ ∗ −1 ∗ ∗ ∗ y ∈ Dε2 F (¯y, x¯)(x ), i.e., (−x , y ) ∈ Nε2 ((¯x, y¯); gph F ). We have furthermore that

kx∗k ≤ 1 and ky∗k + ε ≥ reg F (¯x, y¯). (3.17)

Due to the Brøndsted-Rockafellar theorem (see, e.g., [2]) there are (xε, yε) ∈ gph F ∩ IBε(¯x, y¯) ∗ ∗ ∗ ∗ ∗ ∗ and (−xε, yε ) ∈ N((xε, yε); gph F ) satisfying kxε − x k ≤ ε and kyε − y k ≤ ε. Thus we get ∗ ∗ ∗ ∗ kxεk ≤ kx k + ε ≤ 1 + ε and ky k ≤ kyε k + ε arriving in the way at

∗ ∗ −1 ky k ≤ (1 + ε)kD F (yε, xε)k + ε.

Combining this with (3.17) implies the estimate n o ∗ −1 reg F (¯x, y¯) ≤ (1 + ε) sup kD F (y, x)k (x, y) ∈ gph F ∩ IBε(¯x, y¯) + 2ε, which ensures the inequality “≤” in (3.16) by passing to the limit as ε ↓ 0. 4 It is worth mentioning that the exact bound formulas similar to (3.16) are established in [3, Theorem 4.5] and [12, Theorem 5.6] for general closed-graph multifunctions between Asplund spaces via the so-called Fr´echet/regular coderivative. Though Corollary 3.2 concerns convex multifunctions, both spaces X and Y in it are arbitrary Banach. The next consequence of Theorem 3.1 gives the main result of [8] proved in a diﬀerent way.

Corollary 3.3 (calculating the exact covering bound). Given a point (¯x, y¯) ∈ gph F with y¯ ∈ int (rge F ), the exact covering bound of F at (¯x, y¯) is calculated by

∗ ∗ h σgph F −(¯x,y¯)(x , y )i cov F (¯x, y¯) = lim inf inf kx∗k + . ∗ ∗ ∗ ε↓0 x ∈X y ∈SY ∗ ε Proof. Deﬁne Ω := gph F − (¯x, y¯). Since the number cov F (¯x, y¯) is the reciprocal of reg F (¯x, y¯), it suﬃces to show that h σ (x∗, y∗)−1i reg F (¯x, y¯) = lim sup sup kx∗k + Ω =: α. (3.18) ε↓0 ∗ ∗ ∗ x ∈X y ∈SY ∗ ε

∗ ∗ ∗ ∗ ∗ ∗ By (3.2) we ﬁnd sequences ε ↓ 0 and (x , y ) ∈ X × S ∗ such that x ∈ D F (¯x, y¯)(y ), which n n n Y n εn n ∗ ∗ ∗ −1 means σΩ(xn, −yn) ≤ εn due to (2.6), and that kxnk → reg F (¯x, y¯) as n → ∞. Hence

∗ ∗ −1 ∗ ∗ −1 −1 ∗ σΩ(x , y ) ∗ σΩ(xn, −yn) ∗ √ sup sup kx k + √ ≥ kxnk + √ ≥ kxnk + εn , ∗ ∗ ∗ x ∈X y ∈SY ∗ εn εn which implies the inequality “≤” in (3.18) by passing to the limit as n → ∞. Conversely, if the right-hand side of (3.18) equals to 0, the equality in (3.18) is obvious. Otherwise, ∗ ∗ ∗ we ﬁnd sequences εen ↓ 0 and (xen, yen) ∈ X × SY ∗ such that ∗ ∗ −1 ∗ σΩ(xen, yen) β < kxenk + → α as n → ∞ (3.19) εen ∗ ∗ −1 ∗ ∗ ∗ for some β > 0. It follows that σΩ(x , y ) ≤ εnβ for all n ∈ IN, which yields x ∈ D F (¯x, y¯)(−y ) en en e en εbn en −1 with εbn := εenβ → 0 by (2.6). Hence we have ∗ ∗ −1 ∗ σΩ(xen, yen) ∗ −1 n ∗ −1 ∗ ∗ ∗ ∗ o kx k + ≤ kx k ≤ sup kx k x ∈ D F (¯x, y¯)(y ), y ∈ SY ∗ . (3.20) en en εn εen b

8 Employing now the regularity formula (3.2) in (3.20) and taking (3.19) into account, we get the estimate α ≤ reg F (¯x, y¯) and thus complete the proof of the corollary. 4 Finally in this section, we introduce a condition improving the one in (1.5), which helps us to remove ε > 0 in the exact bound formula (3.1) and get the result in form (1.2) for convex multifunctions between general Banach spaces.

Theorem 3.4 (pointwise calculating the exact regularity bound via the coderivative norm). In the setting of Theorem 3.1, assume in addition that

Λ(SY ∗ ) ⊂ SY ∗ , (3.21) where the set Λ(SY ∗ ) is deﬁned sequentially by

n ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ o Λ(SY ∗ ) := y ∈ Y ∃ εn ↓ 0, y ∈ SY ∗ s.t. D F (¯x, y¯)(y ) 6= ∅ and y is a weak cluster point of y . n εn n n Then we have the equality

reg F (¯x, y¯) = kD∗F −1(¯y, x¯)k. (3.22)

If furthermore reg F (¯x, y¯) > 0, we get the improved formula

n ∗ −1 ∗ ∗ ∗ ∗ o ∗ reg F (¯x, y¯) = max kx k x ∈ D F (¯x, y¯)(y ), y ∈ SY . (3.23) Proof. Note that the equality in (3.22) is trivial when reg F (¯x, y¯) = 0. Otherwise, it follows from (3.2) that there are sequences ε ↓ 0 and x∗ ∈ D∗ F (¯x, y¯)(y∗ ) such that kx∗ k > 0, ky∗ k = 1, and n n εn n n n

∗ −1 reg F (¯x, y¯) = lim kxnk . (3.24) n→∞

∗ Since the sequence {xn} is bounded by (3.24), we get from (3.21) and the classical Alaoglu-Bourbaki ∗ ∗ ∗ ∗ ∗ ∗ ∗ theorem that there is a subnet (xα, yα, εα) of (xn, yn, εn) weak converging to some (¯x , y¯ , 0) ∈ ∗ X × SY ∗ × IR. Note further that

∗ ∗ ∗ ∗ hx¯ , x − x¯i − hy¯ , y − y¯i = limhxα, x − x¯i − hyα, y − y¯i ≤ lim sup εα = 0 α α for all (x, y) ∈ gph F , which yieldsx ¯∗ ∈ D∗F (¯x, y¯)(¯y∗). Moreover, the classical uniform boundedness ∗ ∗ principle tells us that kx¯ k ≤ lim inf kxαk. This together with (3.24) ensures that α n o ∗ −1 ∗ −1 ∗ ∗ ∗ ∗ reg F (¯x, y¯) ≤ kx¯ k ≤ sup kx k x ∈ D F (¯x, y¯)(y ), ky k = 1 . Combining the latter with (3.2) yields (3.22), which easily implies (3.23) under the additional assumption made. This completes the proof of the theorem. 4

It is obvious that assumption (3.21) automatically holds when the space Y is ﬁnite-dimensional. In this case Theorem 3.4 agrees with [3, Theorem 4.21] obtained under the Asplund property of X while for nonconvex set-valued mappings. Furthermore, the equality in (3.22) is equivalent to the perfect regularity property of F introduced and studied in [8] under assumption (1.5). Observe that the latter assumption with Ω = gph F − (¯x, y¯) is stronger than (3.21) due to the proper inclusion

∗h [ n ∗ ∗ ∗ oi ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ Λ(SY ) ⊂ cl y ∈ SY Dε F (¯x, y¯)(y ) 6= ∅ = cl y ∈ SY σΩ(x , y ) < ∞, x ∈ X . ε≥0

9 4 Applications to Inﬁnite Convex Systems

In this section we develop applications of the results obtained in Section 3 to the special class of

set-valued mappings F : X → Y := Z × l∞(T ) given by ( (z, p) ∈ Z × l∞(T ) Ax = z, ft(x) ≤ pt, t ∈ T if x ∈ C F (x) := (4.1) ∅ otherwise

and describing, in particular, sets of feasible solutions in parameterized semi-inﬁnite/inﬁnite programs with inequality, equality constraints, and geometric constraints. The data of (4.1) are as follows: A : X → Z is a bounded linear operator between two Banach

spaces; the functions ft : X → IR are proper, lower semicontinuous (l.s.c.), and convex for all t from the arbitrary index set T ; C is a closed and convex subset of X with nonempty interior. These assumptions clearly imply that F in (4.1) is closed and convex multifunction, and so we can implement the results on metric regularity at (x, (z, p)) ∈ gph F obtained above to the inﬁnite constraint system (4.1) provided the validity of the underlying condition

(z, p) ∈ int (rge F ). (4.2)

Note that this condition clearly implies that z ∈ int(AX), which ensures that A is an open mapping, and hence it must be surjective.

Throughout this section we denote f := supt∈T ft and suppose that the space Z × l∞(T ) is equipped with the maximum product norm

k(z, p)k = max kzk, kpk for all z ∈ Z, p ∈ l∞(T ).

As mentioned in Section 1, F is metrically regular around (x, (z, p)) ∈ gph F if and only if condition (4.2) holds (Robinson-Ursescu). This motivates us to introduce a qualification condition via the initial data of (4.1), which ensures (4.2) and extend the well-known strong Slater constraint qualification for infinite convex inequality systems to the more general constraint case of (4.1).

Definition 4.1 (strong Slater condition). We say the infinite system (4.1) satisfies the strong Slater condition (SSC) at (z, p) ∈ Z × l∞(T ) if there is xb ∈ int C such that the function f is bounded from above around xb, that Axb = z, and that

sup[ft(xb) − pt] < 0. (4.3) t∈T

The following proposition shows that the SSC is a suﬃcient condition for the validity of (4.2), being in fact “almost” necessary for this, up to the upper boundedness of the supremum function f.

Proposition 4.2 (strong Slater condition and metric regularity). Let (z, p) ∈ rge F for the inﬁnite system (4.1). Then the SSC for F at (z, p) implies the validity of (4.2). Conversely, if (4.2) holds, then there is xb ∈ int C such that Axb = z and that (4.3) is satisﬁed.

Proof. To justify the ﬁrst part, assume that F satisﬁes the SSC at (z, p). Then there are xb ∈ int C and ε > 0 such that the supremum function f is bounded from above around xb, A(xb) = z, and that f(xb) < −ε for some ε > 0. Note that f is also a proper, l.s.c., and convex. It follows from [2, Theorem 2.2.9] that it is actually continuous at xb. Since A is surjective and xb ∈ int C, the Open

10 ε Mapping Theorem allows us to ﬁnd 0 < s ≤ 2 such that IBs(z) ⊂ A(IBr(xb) ∩ C) for r > 0. Picking 0 0 0 any (z , p ) ∈ IBs(z, p), there is x ∈ IBr(xb) ∩ C such that Ax = z , and for each t ∈ T we have ε f (x) − p ≤ f(x) + s ≤ f(x) − f(x) + s + f(x) ≤ f(x) − f(x) + s − ε ≤ f(x) − f(x) − ≤ 0 t t b b b b 2 0 0 when r is suﬃciently small. This yields (z , p ) ∈ rge F , which implies that IBs(z, p) ⊂ rge F .

To verify the converse part, observe that (z, (pt − ε)t∈T ) ∈ rge F for some ε > 0 provided that (z, p) ∈ int (rge F ). Hence there is xb ∈ X such that Axb = z and ft(xb) − pt ≤ −ε for all t ∈ T , which completes the proof of the proposition. 4 Now we proceed with calculating the exact regularity bound for the constraint system (4.1) at (¯x, (¯z, 0)) ∈ gph F based on the results of Section 3. It follows from Theorem 3.1 that reg F (¯x, (¯z, 0)) can be calculated via the norms of ε-coderivatives. The next result, which is certainly of its own interest, accomplishes an important step in this direction.

Theorem 4.3 (explicit form of ε-coderivatives for inﬁnite convex systems). Let F be the inﬁnite constraint system (4.1), and let (¯x, (¯z, 0)) ∈ gph F . Then we have the following ε-coderivative representation on the unit sphere:

∗ ∗ ∗ ∗ ∗ ∗ Dε F x,¯ (¯z, 0) S(Z×l∞(T )) = x ∈ X (x , hx , x¯i + ε) ∈ M for each ε ≥ 0 (4.4)

[ ∗h ∗ [ ∗ ∗ i ∗ ∗ ∗ with M := cl (1−kz k) co epi ft +epi δ (·; C0) +(A z , hz , z¯i) and C0 := C∩dom f. ∗ z ∈IBZ∗ t∈T

∗ ∗ ∗ ∗ ∗ ∗ ∗ Proof. To prove the inclusion“⊂” in (4.4), pick (z , p ) ∈ S(Z×l∞(T )) and x ∈ Dε F (¯x, (¯z, 0))(z , p ). Then we have kz∗k + kp∗k = 1 and

hx∗, x − x¯i − hz∗, z − z¯i − hp∗, pi ≤ ε for all (x, z, p) ∈ gph F,

which can be equivalently written as

∗ ∗ ∗ ∗ hx − A z , x − x¯i − hp , pi ≤ ε whenever (x, p) ∈ C0 × l∞(T ), ft(x) − hδt, pi ≤ 0, t ∈ T, (4.5)

∗ where δt ∈ l∞(T ) stands for the classical Dirac function at each t ∈ T satisfying

hδt, pi = pt as t ∈ T for p = (pt)t∈T ∈ l∞(T ).

It follows from the extended Farkas lemma in [13, Theorem 4.1] that (4.5) is equivalent to

∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗h n [ ∗o ∗ i p , x − A z , hx − A z , x¯i + ε ∈ cl cone {δt} × epi ft + {0} × epi δ (·; C0) . (4.6) t∈T

T ∗ ∗ ∗ ∗ Hence there exist nets {λν }ν∈N ⊂ IRe +, {(wν , sν )}ν∈N ⊂ epi δ (·; C0), and {(utν , rtν )}ν∈N ⊂ epi ft for each t ∈ T such that

∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ X ∗ ∗ p , x − A z , hx − A z , x¯i + ε = w − lim λtν (δt, utν , rtν ) + (0, wν , sν ). (4.7) ν t∈T

∗ ∗ X Observe from the latter equality that p = w − lim λtν δt. Thus we have ν t∈T

X X ∗ ∗ ∗ X lim sup λtν ≥ sup lim λtν pt = sup hp , pi = kp k ≥ hp , ei = lim λtν (4.8) ν ν ν t∈T kpk≤1 t∈T kpk≤1 t∈T

11 with e ∈ l∞(T ) satisfying et = 1 for all t ∈ T . This yields that

∗ ∗ X 1 − kz k = kp k = lim λtν . (4.9) ν t∈T If kz∗k = 1, we get from (4.7) and (4.9) the relationships

hx∗ − A∗z∗, x − x¯i − ε = hx∗ − A∗z∗, xi − (hx∗ − A∗z∗, x¯i + ε) X ∗ ∗ X = lim λtν hutν , xi + hwν , xi − lim λtν rtν − sν ν ν t∈T t∈T X ∗ ∗ ≤ lim sup λtν hutν , xi − ft(x) − rtν + f(x) + hwν , xi − sν ν t∈T X ≤ lim sup λ [f ∗(u∗ ) − r ] + f(x) + [δ∗(·; C )(w∗) − s ] tν t tν tν 0 ν ν ν t∈T X ≤ lim sup λtν f(x) = 0 ν t∈T

∗ ∗ ∗ ∗ ∗ ∗ ∗ for any x ∈ C0. It follows from (2.1) that (x − A z , hx − A z , x¯i + ε) ∈ epi δ (·; C0), which yields

∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ (x , hx , x¯i + ε) ∈ epi δ (·; C0) + (A z , hA z , x¯i) = epi δ (·; C0) + (A z , hz , z¯i) ⊂ M.

∗ P If kz k < 1, by (4.9) we assume without loss of generality that t∈T λtν > 0 for all ν ∈ N and λtν deﬁne λetν := P for each t ∈ T and ν ∈ N . It gives by (4.7) that t0∈T λt0ν

∗ ∗ ∗ X ∗ ∗ ∗ ∗ ∗ (x , hx , x¯i + ε) = w − lim λtν (utν , rtν ) + (wν , sν ) + (A z , hz , z¯i) ν t∈T ∗ ∗ X ∗ ∗ ∗ ∗ ∗ = w − (1 − kz k) lim λetν (utν , rtν ) + (wν , sν ) + (A z , hz , z¯i) ⊂ M. ν t∈T In both cases above we have (x∗, hx∗, x¯i + ε) ∈ M, which justiﬁes the inclusion “⊂” in (4.4). ∗ ∗ ∗ ∗ ∗ Conversely, take an arbitrary element x ∈ X satisfying (x , hx , x¯i+ε) ∈ M. We ﬁnd z ∈ IBZ∗ T ∗ ∗ ∗ ∗ and nets {λν }ν∈N ⊂ IRe +, {(wν , sν )}ν∈N ⊂ epi δ (·; C0), and {(utν , rtν )}ν∈N ⊂ epi ft for each t ∈ T P such that t∈T λtν = 1 and

∗ ∗ ∗ ∗ X ∗ ∗ ∗ ∗ ∗ (x , hx , x¯i + ε) = w − (1 − kz k) λtν (utν , rtν ) + (wν , sν ) + (A z , hz , z¯i). t∈T ∗ ∗ P ∗ ∗ Defining pν := (1 − kz k) t∈T λtν δt, note that kpν k = 1 − kz k by similar arguments as in the ∗ proof of (4.8). It follows from the classical Alaoglu-Bourbaki theorem that there is a subnet of pν ∗ ∗ (without relabeling) weak converging to some p ∈ IB ∗ . By using the arguments as in (4.8) l∞(T ) again, we get kp∗k = 1 − kz∗k and then obtain (4.6). Due to the equivalence between (4.5) and (4.6), this justifies the inclusion “⊃” in (4.4) and thus completes the proof of the theorem. 4 In the coderivative case of Theorem 4.3 (i.e., when ε = 0) we can equivalently modify the representation in (4.4) and provide its further specification.

Proposition 4.4 (explicit forms of the coderivative for inﬁnite convex systems). Let (¯x, (¯z, 0)) ∈ gph F for the constraint system (4.1). Then we have the coderivative representation

∗ ∗ ∗ ∗ ∗ ∗ D F x,¯ (¯z, 0) S(Z×l∞(T )) = x ∈ X (x , hx , x¯i) ∈ L (4.10)

[ ∗h ∗ [ ∗ ∗ i ∗ ∗ ∗ with L := cl (1−kz k) co gph ft +gph δ (·; C0) +(A z , hz , z¯i) and C0 = C∩dom f. ∗ z ∈IBZ∗ t∈T ∗ Furthermore, the term gph δ (·; C0) can be removed in the expression for L if x¯ ∈ int C0.

12 Proof. To verify the inclusion “⊂”, for any x∗ ∈ D∗F (¯x, (¯z, 0))(z∗, p∗) with kz∗k + kp∗k = 1 we get from the proof of Theorem 4.3 the validity of inclusion (4.5) with ε = 0. This allows us to ﬁnd nets T ∗ ∗ ∗ ∗ {λν }ν∈N ⊂ IRe +, {ρν }ν∈N ⊂ IR+, {(wν , sν )}ν∈N ⊂ gph δ (·; C0), and {(utν , rtν )}ν∈N ⊂ gph ft for each t ∈ T providing the limiting representation

∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ X ∗ ∗ p , x − A z , hx − A z , x¯i = w − lim λtν (δt, utν , rtν ) + (0, wν , sν ) + (0, 0, ρν ). (4.11) ν t∈T

X ∗ Similarly to the proof of Theorem 4.3, suppose with no loss of generality that λtν = 1 − kz k for t∈T all ν ∈ N and then get

∗ ∗ ∗ ∗ ∗ ∗ ∗ rtν = ft (utν ) ≥ hutν , x¯i − ft(¯x) ≥ hutν , x¯i and sν = δ (·; C0)(wν ) ≥ hwν , x¯i.

This implies together with (0.2) the relationships

∗ ∗ ∗ X X ∗ ∗ hx − A z , x¯i = lim λtν rtν + sν + ρν ≥ lim sup λtν hutν , x¯i + hwν , x¯i + ρν ν ν t∈T t∈T ∗ ∗ ∗ ≥ hx − A z , x¯i + lim sup ρν , ν which ensure that lim sup ρν = 0. Then it follows from (4.11) that ν

∗ ∗ ∗ X ∗ ∗ ∗ ∗ ∗ (x , hx , x¯i) = w − lim λtν (utν , rtν ) + (wν , sν ) + (A z , hz , z¯i) ∈ L, ν t∈T and thus we arrive at the inclusion “⊂” in (4.10). The veriﬁcation of the opposite inclusion in (4.10) follows the lines on the proof of Theorem 4.3. ∗ ∗ ∗ ∗ ∗ ∗ ∗ Finally, letx ¯ ∈ int C0 and x ∈ D F (¯x, (¯z, 0))(z , p ) with (z , p ) ∈ S(Z×l∞(T )) . Using the same notation as in the proof of (4.10) above, we have

∗ ∗ ∗ ∗ ∗ ∗ X ∗ ∗ 0 = hx − A z , x¯i − hx − A z , x¯i = lim λtν (hutν , x¯i − rtν ) + hwν , x¯i − stν ν t∈T ∗ ∗ ∗ ≤ − lim sup sup hwν , xi − hwν , x¯i ≤ − lim sup ηkwν k, ν x∈C0 ν

∗ where η > 0 is such that IBη(¯x) ⊂ C0. This implies that lim sup kwν k = 0, and thus we can remove ν ∗ gph δ (·; C0) in the representation of L in (4.10) and complete the proof of the proposition. 4 The next major result provides a precise calculation of the exact regularity bound of the inﬁnite constraint system (4.1) entirely via its initial data.

Theorem 4.5 (exact regularity bound of infinite constraint systems). Let (¯x, (¯z, 0)) ∈ gph F for the infinite system in (4.1), which is assumed to satisfy the SSC from Definition 4.1 at (¯z, 0). Then the exact regularity bound of F at (¯x, (¯z, 0)) is calculated by

∗ −1 ∗ ∗ reg F (¯x, (¯z, 0)) = lim sup kx k (x , hx , x¯i + ε) ∈ M , (4.12) ε↓0 where the set M is deﬁned in Theorem 4.3. If in addition 0 < dim Z < ∞, then n o ∗ −1 ∗ −1 ∗ ∗ reg F (¯x, (¯z, 0)) = kD F ((¯z, 0), x¯)k = max kx k (x , hx , x¯i) ∈ L , (4.13) where the set L is deﬁned in Proposition 4.4.

13 Proof. It follows from Proposition 4.2 that (¯z, 0) ∈ int (rge F ), i.e., the mapping F is metrically regular around (¯x, (¯z, 0)). Substituting the ε-coderivative expression from Theorem 4.3 into the exact bound formula (3.2) of Theorem 3.1, we arrive at the limiting representation (4.12). Let us next justify the equalities in (4.13) under the additional assumption made. Employing The- orem 3.4 and Proposition 4.4, we need to check that condition (3.21) holds and that reg F (¯x, (¯z, 0)) > ∗ ∗ ∗ ∗ ∗ 0. To proceed, take any ε > 0 and (z , p ) ∈ S(Z×l∞(T )) satisfying DεF (¯x, (¯z, 0))(z , p ) 6= ∅. By the same arguments as in the proofs of relationships (4.6) and (4.9) we get the inclusion

∗ ∗ ∗ p ∈ 1 − kz k cl co δt t ∈ T .

∗ ∗ ∗ ∗ ∗ ∗ ∗ Thus the set cl {(z , p ) ∈ S(Z×l∞(T )) | Dε F (¯x, (¯z, 0))(z , p ) 6= ∅} is contained in that of

∗ [ h ∗ ∗ ∗ i cl z × 1 − kz k cl co δt t ∈ T . (4.14) ∗ z ∈IBZ∗

∗ ∗ Further, it follows from the proof in (4.9) that cl co {δt| t ∈ T } ⊂ Sl∞(T ) . Since dim Z < ∞, the

∗ latter implies that the set in (4.14) is a subset of S(Z×l∞(T )) , which ensures the validity of (3.21). It remains to verify that reg F (¯x, (¯z, 0)) > 0. Indeed, it is easy to see that

∗ −1 ∗ ∗ ∗ ∗ ∗ ∗ ∗ D F (¯z, 0), x¯ (x ) ⊃ (z , 0) ∈ Z × l∞(T ) A z = x .

Since the operator A is surjective, it follows from [3, Lemma 1.18] that k(A∗)−1k > 0. Therefore we conclude that kD∗F −1((¯z, 0), x¯)k > 0, which yields reg F (¯x, (¯z, 0)) > 0 by Theorem 3.1 and thus completes the proof of this theorem. 4 Observe from Theorem 3.4 that the exact bound formula

∗ −1 reg F x,¯ (¯z, 0) = D F (¯z, 0), x¯ (4.15)

∗ ∗ ∗ holds also in the case of dim Z = 0. For the linear functions ft(x) = hat , xi−bt with (at , bt) ∈ X ×IR as t ∈ T this equality was obtained in [5] when C = X under the additional assumption that ∗ ∗ the coeﬃcient set {at | t ∈ T } is bounded in X . It is easy to check that the latter assumption implies that the supremum function f is bounded from above around the Slater point xb and that x¯ ∈ int C0 = int (dom f). The following example demonstrates that the converse implication is not true even in a simple one-dimensional setting of semi-inﬁnite programming.

Example 4.6 (bounded from above constraint linear functions with unbounded coeﬃ- 1 cients). Let X = IR, Z = {0}, T = (0, 1), and ft(x) = − t x + t in (4.1). Note that 1 1 √ f (x) = − x + t = − x − t + 2t ≤ −2 x + 2t for all x > 0 and t ∈ T. t t t

Taking xb = 4 andx ¯ = 1, we observe that ft(xb) < −2, ft(¯x) ≤ 0, and the supremum function f is 1 bounded from above around xb. However, the coeﬃcient set − t t ∈ T is obviously unbounded. A natural question arising from Theorem 4.5 is whether formula (4.15) holds true for inﬁnite- dimensional spaces Z. The following counterexample is constructed for the case of the classical

Asplund space Z = c0, which is the space of sequences of real numbers converging to zero and endowed with the supremum norm.

Example 4.7 (failure of the coderivative exact bound formula for countable inequality

and inﬁnite-dimensional equality constraints.) Let X = Z = c0, and let T = IN. Deﬁne a

14 linear operator A : X → Z by Ax := (x2, x3,...) for all x = (x1, x2,...) ∈ X. It is easy to see that

A is bounded and surjective. We form a set-valued mapping F : c0 → c0 × l∞ of type (4.1) by F (x) := (z, p) ∈ Z × l∞ Ax = z, x1 + xn + 1 ≤ pn, n ∈ IN for any x ∈ X. (4.16)

Takex ¯ := − 1 ,z ¯ := Ax¯, and x := (−2, − 1 , − 1 ,...) ∈ X. Observe that the strong Slater n n∈IN b 2 3 −1 condition of Definition 4.1 is satisfied at xb for (4.16) and thatx ¯ ∈ F (¯z, 0). Defining further 1 1 1 1 1 1 1 2 1 1 xk := −1, − ,..., − , , − , − ,... , zk := − ,..., − , , − , − ,... , 2 k − 1 k k + 1 k + 2 2 k − 1 k k + 1 k + 2

k k shows that x → x¯ and z → z¯ in c0. Moreover, we have the equalities

k k n k k k k o n 1 1 o 1 d (z , 0); F (x ) = max sup(x1 + xn + 1)+, sup | xn+1 − zn| = max , = (4.17) n n k k k where (α)+ when α ∈ IR stands for max{0, α} as usual. It is easy to calculate the inverse mapping −1 k k k 2 value F (z , 0) = {(a, z1 , z2 ,...) ∈ c0| a ≤ − k − 1}, which gives us

k −1 k n k 2 k k o n 2 1 o 2 d x ; F (z , 0) = max x1 + + 1 , sup |xn+1 − zn| = max , = . (4.18) k + n k k k It follows from (4.17) and (4.18) that reg F (¯x, (0, z¯)) ≥ 2. Thus formula (4.15) fails if we show that

∗ ∗ ∗ ∗ kx k ≥ 1 for all x ∈ D F x,¯ (0, z¯) S(Z×l∞) . (4.19)

∗ To proceed, we employ Proposition 4.4 that gives us some z ∈ IBZ∗ with

∗ ∗ ∗h ∗ i ∗ ∗ ∗ (x , hx , x¯i) ∈ cl (1 − kz k) co (δ1 + δn, −1) n ∈ IN + (A z , hz , z¯i),

∗ IN where δn ∈ c0 and hδn, xi = xn for all x ∈ c0 and n ∈ IN. Hence there is a net (λν )ν∈N ⊂ IRe such P ∗ that n∈IN λnν = 1 − kz k for all ν ∈ N and that

∗ ∗ ∗ X ∗ ∗ ∗ (x , hx , x¯i) = w − lim λn (δ1 + δn, −1) + (A z , hz , z¯i), ν ν n∈IN which readily implies the limiting relationships

X X X λnν 0 = lim λn (−hδ1 + δn, x¯i − 1) = lim λn (−x¯1 − x¯n − 1) = lim . (4.20) ν ν ν ν ν n n∈IN n∈IN n∈IN

∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ Since c0 = l1, we write z in the form (z1 , z2 ,...) ∈ l1 and observe that A z = (0, z1 , z2 ,...) ∈ l1. P∞ ∗ Thus for any ε > 0 there is k ∈ IN suﬃciently large and such that n=k+1 |zn| ≤ ε, which ensures ∗ ∗ ∗ ∗ ∗ ∗ ∗ that kA z − zbkk ≤ ε with zbk := (0, z1 , . . . , zk, 0, 0,...) ∈ l1. Deﬁne further xbk by

∗ ∗ X ∗ xk := w − lim λnν (δ1 + δn) + zk b ν b n∈IN

k ∗ ∗ k and take e := (1, sign(z1 ),..., sign(zk), 0, 0,...) ∈ c0. We observe that ke k = 1 and that

k ∗ ∗ k X k k X ∗ k kx k ≥ hxk, e i = lim λnν (e1 + en) + znen+1 bk b ν n∈IN n=1 k k+1 (4.21) X X ∗ X ≥ lim λnν + lim inf |zn| − λnν . ν ν n∈IN n=1 n=1

15 It follows from the equations (4.20) that

k+1 X X λnν 0 ≤ lim sup λnν ≤ (k + 1) lim sup = 0. ν ν n n=1 n∈IN

Combining this with (4.21) gives us the estimates

k k ∗ ∗ X ∗ ∗ X ∗ ∗ ∗ kxbkk ≥ 1 − kz k + |zn| = 1 − kz k + |zn| ≥ 1 − kz k + kz k − ε = 1 − ε. n=1 n=1

∗ ∗ ∗ ∗ ∗ It is clear furthermore that kx − xbkk = kA z − zbkk ≤ ε. Thus we arrive at the inequalities

∗ ∗ ∗ ∗ kx k ≥ kxbkk − kx − xbkk ≥ 1 − ε − ε = 1 − 2ε for all ε > 0,

which show that kx∗k ≥ 1, i.e., we get (4.19). This veriﬁes the failure of (4.15).

The next example, which is a modiﬁcation of Example 4.7 shows that the coderivative formula (4.15) for calculating the exact regularity bound fails when dim Z = ∞ even for constraint systems (4.1) with a single convex inequality constraint while both Asplund spaces X and Z.

Example 4.8 ((failure of the coderivative exact bound formula for single inequality and

inﬁnite-dimensional equality constraints). Let X = Z = c0, and let T = {1}. Deﬁne the linear operator A : X → Z as in Example 4.7 and consider F : X → Z × IR given by F (x) := (z, p) ∈ Z × IR Ax = z, f(x) ≤ p for any ∈ X,

where f(x) := sup{x1 + xn + 1| n ∈ IN}. With the same notation as in Example 4.7 we get

d(zk, 0); F (xk) = k−1 and dxk; F −1(zk, 0) = 2k−1,

which shows that reg F (¯x, (¯z, 0)) ≥ 2. Note further that dom f = X and that

∗ ∗ epi f = cl co (δ1 + δn, −1) n ∈ IN + {0} × IR+;

the latter follows from the well-known formula for general supremum functions:

∗ ∗ [ ∗ epi f = cl co epi ft . t∈T

∗ ∗ −1 Taking now any x ∈ D F ((¯z, 0), x¯)(S(Z×IR)∗ ) and using Theorem 4.3 with the above representation of epi f ∗ to, we arrive at

∗ ∗ ∗h ∗ i ∗ ∗ ∗ (x , hx , x¯i) ∈ cl (1 − kz k) co (δ1 + δn, −1) n ∈ IN + (A z , hz , z¯i).

Then repeating the arguments of Example 4.7 gives us kx∗k ≥ 1, and thus (4.15) fails.

The next result provides efficient conditions, which ensure the validity of the major regularity formula (4.15) in the case of infinite-dimensional spaces Z. Note that its proof is different from that of (4.13) in Theorem 4.5 when Z in finite-dimensional. In particular, it does not rely on condition (3.21), or its predecessor (1.5), that may not hold. Indeed, even in the simplest setting of T = ∅ the ∗ left-hand side of (3.21) is cl SZ∗ , which is obviously not a subset of SZ∗ when dim Z = ∞.

16 Theorem 4.9 (validity of the coderivative exact bound formula for finite inequality and infinite-dimensional equality constraints). In the case of arbitrary Banach spaces X and Z in (4.1), assume that the index set T is finite, that

∗ ∗ ∗ ft(x) = hat , xi − bt for all x ∈ X, t ∈ T with (at , bt) ∈ X × IR,

and, given (¯x, (¯z, 0)) ∈ gph F , the constraint mapping F satisﬁes the SSC condition at (¯z, 0) with x¯ ∈ int C. Then we have the exact bound formula (4.15).

Proof. T = {1, . . . , k} and observe that dom f = X and C0 = C in our case. Since

∗ ∗ ∗ ∗ epi fn = (an, bn) + {0} × IR+ and {0} × IR+ + epi δ (·; C) ⊂ epi δ (·; C),

∗ for any z ∈ IBZ∗ we get the equality

∗ ∗ ∗ ∗ ∗ ∗ 1 − kz k co epi ft t ∈ T + epi δ (·; C0) = 1 − kz k co (an, bn) 1 ≤ n ≤ k + epi δ (·; C).

It is easy to see that the above set is weak∗ closed in X∗ × IR. Thus the set M in Theorem 4.3 is represented as follows:

[ n ∗ ∗ ∗ ∗ ∗ ∗ o M = (1 − kz k) co (an, bn) 1 ≤ n ≤ k + epi δ (·; C) + (A z , hz , z¯i) ∗ z ∈IBZ∗

∗ ∗ m k Invoking now the result in the ﬁrst part of Theorem 4.5, we ﬁnd sequences of xm ∈ X , λ ∈ IR+, ∗ ∗ ∗ Pk m ∗ (wm, sm) ∈ epi δ (·; C), and zm ∈ IBZ∗ such that n=1 λn = 1 − kzmk and

k ∗ ∗ −1 X m ∗ ∗ ∗ ∗ ∗ xm, hxm, x¯i + m = λn (an, bn) + (wm, sm) + (A zm, hzm, z¯i) (4.22) n=1 with the upper estimate of the regularity bound

k −1 ∗ −1 X m ∗ ∗ ∗ ∗ reg F x,¯ (¯z, 0 ) ≤ kxmk + O(m) = λn an + wm + A zm + O(m), (4.23) n=1

m m k where O(m) → 0 as m → ∞. Since kλ k ≤ 1, we suppose that λ → λ ∈ IR+ as m → ∞. Furthermore, observe from (4.22) that

k k 1 X X m−1 = hx∗ , x¯i + − hx∗ , x¯i = λmb + s − λmha∗ , x¯i − hw∗ , x¯i m m m n n m n n m n=1 n=1 k k X m ∗ ∗ X m ∗ ≥ λn (bn − han, x¯i) + sm − hwm, x¯i ≥ λn (bn − han, x¯i) ≥ 0, n=1 n=1

Pk ∗ which implies that n=1 λn(bn − han, x¯i) = 0 by passing to the limit as m → ∞. Deﬁne

k k k X m X ∗ ∗ X ∗ ∗ ∗ εm := |λn − λn|, ηm := λn + kzmk, and xbm := λnan + A zm n=1 n=1 n=1

and note that εm = O(m) and ηm = 1 − O(m). Applying Proposition 4.4, we have that

−1 ∗ ∗ ηm xbm ∈ D F x,¯ (¯z, 0) S(Z×IRn)∗ ,

17 Moreover, the same arguments as in the proof of the second part of Proposition 4.4 ensure that ∗ kwmk → 0. It follows therefore that

k ∗ ∗ X m ∗ ∗ ∗ ∗ kxm − xbmk = (λn − λn)an + wm ≤ εm sup kank + kwmk = O(m), n=1 1≤n≤k which implies together with (4.23) the estimates

∗ −1 −1 ∗ −1 reg F x,¯ (¯z, 0) ≤ kxbmk + O(m) + O(m) ≤ ηmkηm xbmk + O(m) + O(m) ∗ ∗ ∗ −1 ≤ (1 − O(m)) inf kx k (x , hx , x¯i) ∈ L + O(m) + O(m).

Letting m → ∞ therein, we arrive at

∗ −1 ∗ ∗ reg F x,¯ (¯z, 0) ≤ sup kx k (x , hx , x¯i) ∈ L , which implies (4.15) and thus completes the proof of the theorem. 4 Concluding Remarks. The reader can observe from the proofs of the results in this section that the imposed Slater condition can be actually replaced by the weaker condition (4.2) at (¯z, 0), which is necessary and sufficient for the metric regularity reg F (¯x, (¯z, 0)) < ∞ of F in (4.1). When dim Z = 0 therein, an equivalent equality to (4.15) is established in [6, Theorem 15] under the ∗ ∗ additional assumptions that either the set ∪t∈T dom ft is bounded in X , or the space X is reflexive. A result similar to (4.15) as dim Z < ∞ is presented in [7, Theorem 16] in a different form with a different proof (not completely clear to us) under a certain uniform boundedness condition on the functions ft.

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