STABILITY OF ERROR BOUNDS FOR CONVEX ... - Semantic Scholar

STABILITY OF ERROR BOUNDS FOR CONVEX CONSTRAINT SYSTEMS IN BANACH SPACES∗ § ´ ALEXANDER KRUGER† , HUYNH VAN NGAI‡ , AND MICHEL THERA

Abstract. This paper studies stability of error bounds for convex constraint systems in Banach spaces. We show that certain known sufficient conditions for local and global error bounds actually ensure error bounds for the family of functions being in a sense small perturbations of the given one. A single inequality as well as semi-infinite constraint systems are considered. Key words. error bounds, Hoffman constants, subdifferential AMS subject classifications. 49J52, 49J53, 90C30, 90C34 DOI. @@

1. Introduction. In this paper we continue our study started in [21] of stability of error bounds of convex constraint systems under data perturbations. For the summary of the theory of error bounds and its various applications to sensitivity analysis, convergence analysis of algorithms, penalty functions methods in mathematical programming the reader is referred to the survey papers by Az´e [2], Lewis & Pang [18], Pang [24], as well as the book by Auslender & Teboule [1]. For an extended real-valued function f : X → R∞ := R ∪ {+∞} on a Banach space X the error bound property is defined by the inequality (1)

d(x, Sf ) ≤ c[f (x)]+ ,

where Sf denotes the lower level set of f : (2)

Sf := {x ∈ X : f (x) ≤ 0},

c ≥ 0, and the notation α+ := max(α, 0) is used. Given an x ¯ ∈ X with f (¯ x) = 0 we say that f admits an (local) error bound at x ¯ if there exist reals c ≥ 0 and δ > 0 such that (1) holds for all x ∈ Bδ (¯ x). The best bound – the exact lower bound of all such c – coincides with [Er f (¯ x)]−1 , where (3)

Er f (¯ x) := lim inf x→¯ x f (x)>0

f (x) d(x, S(f ))

is the error bound modulus [9]) (also known as conditioning rate [25]) of f at x ¯. Thus, f admits an error bound at x ¯ if and only if Er f (¯ x) > 0. If (1) holds for some c ≥ 0, and all x ∈ X then we say that f admits a global error bound. In this case, the best bound – the exact lower bound of all such c – coincides ∗ Received by the editors @@; accepted for publication January @@@; published electronically @@. http://www.siam.org/journals/siopt/@@ † Centre for Informatics and Applied Optimization, School of Information Technology and Mathematical Sciences, University of Ballarat, Australia ([email protected]). ‡ Department of Mathematics, University of Quynhon, 170 An Duong Vuong, Qui Nhon, Vietnam ([email protected]). The research of Huynh Van Ngai was supported by XLIM (Department of Mathematics and Informatics), UMR 6172, University of Limoges and has been partially supported by NAFOSTED. § Laboratoire XLIM, UMR-CNRS 6172, Universit´ e de Limoges, France ([email protected]).

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´ ALEXANDER KRUGER, HUYNH VAN NGAI, AND MICHEL THERA

with [Er f ]−1 , where f (x) f (x)>0 d(x, S(f ))

(4)

Er f := inf

is the global error bound modulus. (In [3, 4] the last constant is denoted σ0 (f ).) A huge literature deals with criteria for the error bound property in terms of various derivative-like objects defined either in the primal space (directional derivatives, slopes, etc.) or in the dual space (different kinds of subdifferentials) [3–5,7,9–13,15,16, 18–20, 22–25, 27–29]. The convex case has attracted a special attention starting with the pioneering work by Hoffman [14] on error bounds for systems of affine functions, see [3, 7, 10, 18]. If f is a lower semicontinuous convex function the following conditions are known to provide sufficient criteria for the error bound property. • Local criteria: (L1) lim inf d(0, ∂f (x)) > 0; x→¯ x, f (x)>f (¯ x)

(L2) 0 ∈ / Bdry ∂f (¯ x). • Global criteria: (G1) inf d(0, ∂f (x)) > 0; f (x)>0

(G2)

inf d(0, Bdry ∂f (x)) > 0. f (x)=0

The following implications are true: (L2)

⇒

(L1)

(G2)

⇒

(G1),

while conditions (L1) and (G1) are actually necessary and sufficient for the corresponding error bound properties – see Theorems 1 and 22 below. We show in Theorems 8 and 25 that the stronger conditions (L2) and (G2) characterize stronger properties than just the existence of local or global error bounds for f , namely, they guarantee respectively the local or global error bound property for the family of functions being in a sense small perturbations of f . In this paper we consider also semi-infinite constraint systems of the form (5)

ft (x) ≤ 0

for all

t ∈ T,

where T is a compact, possibly infinite, Hausdorff space, ft : X → R, t ∈ T, are given continuous functions such that t 7→ ft (x) is continuous on T for each x ∈ X, and establish similar characterizations of stability of local and global error bounds with respect to perturbations of the functions ft – see Theorems 18 and 28. The organization of the paper is simple: besides the current introductory section it contains two more sections devoted to local and global error bounds respectively. If not specified otherwise, we consider extended real-valued functions on a Banach space X. The class of all lower semicontinuous proper convex functions on X will be denoted Γ0 (X). Bδ (¯ x) is the closed ball with center at x ¯ and radius δ. B ∗ denotes the dual unit ball. For a set Q, the notations int Q and Bdry Q mean the interior and the boundary of Q respectively. 2. Stability of local error bounds. In this section we discuss relationships between the local error bound criteria (L1) and (L2) and establish conditions for stability of local error bounds for the constraint systems (2) and (5). Theorem 1. Let f ∈ Γ0 (X), f (¯ x) = 0. Consider the following properties:

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STABILITY OF ERROR BOUNDS

(i). f admits an error bound at x ¯, that is, Er f (¯ x) > 0; (ii). τ (f, x ¯) := lim inf d(0, ∂f (x)) > 0; x→¯ x, f (x)>f (¯ x)

(iii). ς(f, x ¯) := d(0, Bdry ∂f (¯ x)) > 0; (iv). 0 6∈ ∂f (¯ x); (v). 0 ∈ int ∂f (¯ x). Each of the properties (ii)–(v) is sufficient for the error bound property (i). Moreover, (a). ς(f, x ¯) ≤ τ (f, x ¯) = Er f (¯ x); (b). [(iv) or (v)] ⇔ (iii) ⇒ (ii) ⇔ (i). Proof. (a). We first prove the inequality ς(f, x ¯) ≤ τ (f, x ¯). If 0 ∈ int ∂f (¯ x) then f (x) ≥ hv ∗ , x − x ¯i for all

v ∗ ∈ ς(f, x ¯)B ∗ , x ∈ X,

and consequently, f (x) ≥ ς(f, x ¯)kx − x ¯k

x ∈ X.

for all

On the other hand, −f (x) ≥ hx∗ , x ¯ − xi for all

x ∈ X, x∗ ∈ ∂f (x).

Adding the last two inequalities together we obtain hx∗ , x − x ¯i ≥ ς(f, x ¯)kx − x ¯k

x ∈ X, x∗ ∈ ∂f (x),

for all

and consequently, kx∗ k ≥ ς(f, x ¯) if x∗ ∈ ∂f (x) and x 6= x ¯. Hence, τ (f, x ¯) ≥ ς(f, x ¯). If 0 ∈ / int ∂f (¯ x) then ς(f, x ¯) = d(0, ∂f (¯ x)), and consequently, for any ε > 0 there exists a δ > 0 such that kx∗ k ≥ ς(f, x ¯) − ε for all x∗ ∈ ∂f (x) and x ∈ Bδ (¯ x). It follows that τ (f, x ¯) ≥ ς(f, x ¯). The next step is to show that Er f (¯ x) ≤ τ (f, x ¯). Consider any x ∈ X with f (x) > 0 and any x∗ ∈ ∂f (x). By definition of the subdifferential, f (u) − f (x) ≥ hx∗ , u − xi for all

u ∈ X.

In particular, −f (x) ≥ hx∗ , u − xi ≥ −kx∗ k ku − xk

for all

u ∈ Sf ,

and consequently, f (x) ≤ kx∗ k, d(x, S(f )) which immediately implies the inequality Er f (¯ x) ≤ τ (f, x ¯). The proof of the opposite inequality τ (f, x ¯) ≤ Er f (¯ x) is a typical example of the application of the Ekeland variational principle [8]. Suppose that Er f (¯ x) < α. We are going to show that τ (f, x ¯) ≤ α. Choose a β ∈ (τ (f, x ¯), α). By definition (3), for any δ > 0 there exists an x ∈ Bδ/2 (¯ x) such that 0 < f (x) < βd(x, S(f )). Consider a lower semicontinuous function g : X → R∞ given by g(u) = [f (u)]+ . It holds g(u) ≥ 0 for all u ∈ X and g(x) = f (x) < βd(x, S(f )). By Ekeland’s theorem, there exists an x ˆ ∈ X such that kˆ x − xk ≤ (β/α)d(x, S(f )) and g(u) − g(ˆ x) + αku − x ˆk ≥ 0

for all

u ∈ X.

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´ ALEXANDER KRUGER, HUYNH VAN NGAI, AND MICHEL THERA

Since kˆ x − xk < d(x, S(f )) and f is lower semicontinuous, we have g(u) = f (u) for all u near x, and it follows from the last inequality that kx∗ k ≤ α for some x∗ ∈ ∂f (ˆ x). Besides, kˆ x−x ¯k < 2kx − x ¯k ≤ δ. Hence, τ (f, x ¯) ≤ α. (b). The equivalence [(iv) or (v)] ⇔ (iii) is obvious. The chain (iii) ⇒ (ii) ⇔ (i) follows from (a). Remark 2. Constant τ (f, x ¯) providing a necessary and sufficient characterization of the local error bound property is also known as the strict outer subdifferential x) of f at x ¯ [9]. Criterion (ii) was used in [16, Theorem 2.1 (c)], [23, slope |∂f |> (¯ Corollary 2 (ii)], [25, Theorem 4.12], [28, Theorem 3.1]. Criterion (iii) was used in [10, Corollary 3.4], [12, Theorem 4.2]. The equality Er f (¯ x) = τ (f, x ¯) seems to be well known. See also characterizations of linear and nonlinear conditionings in [6, Theorem 5.2]. The inequality in (a) and the implication (iii) ⇒ (ii) in (b) in Theorem 1 can be strict. Example 3. f (x) ≡ 0, x ∈ R. Obviously 0 ∈ Bdry ∂f (¯ x), ς(f, x ¯) = 0, while τ (f, x ¯) = ∞ for any x ¯ ∈ R. Example 4. f (x) = 0 if x ≤ 0, and f (x) = x if x > 0. Then ∂f (0) = [0, 1] and 0 ∈ Bdry ∂f (0), ς(f, x ¯) = 0, while τ (f, 0) = 1. Thus, condition (iii) in Theorem 1 is in general stronger than each of the equivalent conditions (i) and (ii). It characterizes a stronger property than just the existence of a local error bound for f at x ¯, namely, it guaranties the local error bound property for the family of functions being small perturbations of f . Definition 5. Let f (¯ x) < ∞ and ε ≥ 0. We say that g : X → R∞ is an ε-perturbation of f near x ¯ and write g ∈ Ptb (f, x ¯, ε) if g(¯ x) = f (¯ x) and

(6)

lim sup x→¯ x

|g(x) − f (x)| ≤ ε. kx − x ¯k

Obviously, if g ∈ Ptb (f, x ¯, ε) then f ∈ Ptb (g, x ¯, ε). Remark 6. If the functions are continuous at x ¯ then condition (6) implies g(¯ x) = f (¯ x). The last requirement can be dropped from Definition 5 if condition (6) is replaced by a more general one: (7)

lim sup x→¯ x

|(g(x) − f (x)) − (g(¯ x) − f (¯ x))| ≤ ε. kx − x ¯k

In this case, a perturbation function does not have to coincide with the given one at the point of reference. In fact, the difference α := g(¯ x) − f (¯ x) can be arbitrarily large. However, this seemingly more general case can be easily reduced to the above one: if a function g satisfies (7) then the function x 7→ g(x) − α satisfies (6). Note also that neither g nor f in the above definition are assumed convex. The characterization below is (partially) in terms of Fr´echet subdifferentials which in the convex case reduce to subdifferentials in the sense of convex analysis. Proposition 7. Let f be convex, f (¯ x) < ∞, and ε ≥ 0. If g ∈ Ptb (f, x ¯, ε) then (i). ∂g(¯ x) ⊆ ∂f (¯ x) + εB ∗ ; (ii). d(0, ∂g(¯ x)) ≥ d(0, ∂f (¯ x)) − ε; (iii). d(0, Bdry ∂g(¯ x)) ≥ d(0, Bdry ∂f (¯ x)) − ε. Proof. (i). Let x∗ ∈ ∂g(¯ x). Then, by definition of the Fr´echet subdifferential and

STABILITY OF ERROR BOUNDS

5

by (6), for any ξ > 0 there exists a δ > 0 such that ξ g(x) − g(¯ x) − hx∗ , x − x ¯i ≥− , kx − x ¯k 2 g(x) − f (x) ξ ≤ε+ kx − x ¯k 2 for all x, 0 < kx − x ¯k ≤ δ. Subtracting the second inequality from the first one and recalling that g(¯ x) = f (¯ x) we obtain f (x) + εkx − x ¯k − f (¯ x) − hx∗ , x − x ¯i ≥ −ξ kx − x ¯k for all x, 0 < kx − x ¯k ≤ δ, and consequently x∗ ∈ ∂(f + εk · −¯ xk)(¯ x) = ∂f (¯ x) + εB ∗ . (ii) follows immediately from (i). (iii). If ε ≥ d(0, Bdry ∂f (¯ x)) the assertion is trivial. Let ε < d(0, Bdry ∂f (¯ x)). Then due to (i) zero is either outside both ∂f (¯ x) and ∂g(¯ x) or inside both of them. In the first case, the assertion coincides with (ii), while in the second one, it follows from (i). The next theorem shows that condition (iii) in Theorem 1 provides a characterization of the “combined” error bound property for the family of ε-perturbations of f near x ¯. Theorem 8. Let f ∈ Γ0 (X), f (¯ x) = 0, ε > 0. The following assertions hold true: (i). Er g(¯ x) ≥ ς(f, x ¯) − ε for any g ∈ Γ0 (X) ∩ Ptb (f, x ¯, ε); (ii). if 0 ∈ Bdry ∂f (¯ x) then the function g ∈ Γ0 (X) ∩ Ptb (f, x ¯, ε) defined by (8)

g(u) := f (u) + εku − x ¯k,

u ∈ X,

satisfies Er g(¯ x) ≤ ε; (iii). if dim X < ∞ and 0 ∈ Bdry ∂f (¯ x) then there exists an x∗ ∈ εB ∗ such that the function g ∈ Γ0 (X) ∩ Ptb (f, x ¯, ε) defined by (9)

g(u) := f (u) + hx∗ , u − x ¯i,

u ∈ X,

satisfies Er g(¯ x) ≤ ε. Proof. (i). If g ∈ Γ0 (X) ∩ Ptb (f, x ¯, ε) then, due to Proposition 7 (iii) and Theorem 1 (a), we have Er g(¯ x) ≥ ς(g, x ¯) ≥ ς(f, x ¯) − ε. (ii). Let 0 ∈ Bdry ∂f (¯ x), ξ > 0. Then f (u) ≥ 0

for all u ∈ X,

and there exists an u∗ ∈ (ξ/2)B ∗ such that u∗ 6∈ ∂f (¯ x), that is, there is a y ∈ B2ξ/3 (¯ x) such that f (y) < hu∗ , y − x ¯i ≤ (ξ/2)ky − x ¯k. Obviously, y 6= x ¯. By virtue of the Ekeland variational principle [8], we can select an x ∈ X satisfying kx−yk ≤ ky− x ¯k/2 ≤ ξ/3, such that the function u 7→ f (u)+ξku−xk attains its minimum at x. Hence x ∈ Bξ (¯ x) \ {¯ x} and 0 ∈ ∂f (x) + ξB ∗ , that is, there

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´ ALEXANDER KRUGER, HUYNH VAN NGAI, AND MICHEL THERA

exists a v ∗ ∈ ∂f (x), such that kv ∗ k ≤ ξ. Then the function g ∈ Γ0 (X) defined by (8) obviously satisfies g ∈ Ptb (f, x ¯, ε), g(x) ≥ εkx − x ¯k > 0, d(0, ∂g(x)) ≤ ε + ξ. As ξ > 0 can be chosen arbitrarily small, thanks to Theorem 1 (a), Er g(¯ x) = τ (g, x ¯) ≤ ε. (iii). Let dim X < ∞ and 0 ∈ Bdry ∂f (¯ x). Setting ξ = 1/k in the above proof of (ii) we obtain sequences {xk } ⊂ X and {vk∗ } ⊂ X ∗ such that f (xk ) ≥ 0, 0 < kxk − x ¯k ≤ 1/k, vk∗ ∈ ∂f (xk ), kvk∗ k ≤ 1/k. Without loss of generality (xk − x ¯)/kxk − x ¯k → z, kzk = 1. Choose an x∗ ∈ X ∗ such ∗ ∗ ∗ that kx k = hx , zi = ε. Then hx , xk − x ¯i > 0 for all sufficiently large k. It follows that for such k the function g ∈ Γ0 (X) defined by (9) satisfies g(xk ) > 0, d(0, ∂g(xk )) ≤ ε + 1/k. By virtue of Theorem 1 (a), Er g(¯ x) = τ (g, x ¯) ≤ ε. The last assertion of the theorem providing a statement in terms of a perturbation by a linear term is important when dealing with semi-infinite linear constraint systems [21]. Given a function f ∈ Γ0 (X) with f (¯ x) = 0 and a number ε ≥ 0 denote (10)

Er {Ptb (f, x ¯, ε)}(¯ x) :=

inf

Er g(¯ x).

g∈Γ0 (X)∩Ptb (f,¯ x,ε)

This number characterizes the error bound property for the whole family of convex ε-perturbations of f near x ¯. Obviously, Er {Ptb (f, x ¯, ε)}(¯ x) ≤ Er f (¯ x) for any ε ≥ 0. Corollary 9. Let f ∈ Γ0 (X), f (¯ x) = 0, ε > 0. The following assertions hold true: (i). Er {Ptb (f, x ¯, ε)}(¯ x) ≥ ς(f, x ¯) − ε; (ii). if 0 ∈ Bdry ∂f (¯ x) then Er {Ptb (f, x ¯, ε)}(¯ x) = 0. Due to Corollary 9 (i), condition 0 6∈ Bdry ∂f (¯ x) is sufficient for the error bound property of the family of ε-perturbations of f as long as ε < ς(f, x ¯). If 0 ∈ Bdry ∂f (¯ x) then, due to Corollary 9 (ii), none family of ε-perturbations possesses the error bound property. Corollary 10. Let f ∈ Γ0 (X), f (¯ x) = 0. The following properties are equivalent: (i). there exists an ε > 0 such that Er {Ptb (f, x ¯, ε)}(¯ x) > 0; (ii). 0 6∈ Bdry ∂f (¯ x). We consider now a semi-infinite constraint system (5), where T is a compact, possibly infinite, Hausdorff space, ft : X → R, t ∈ T, are given continuous functions such that t 7→ ft (x) is continuous on T for each x ∈ X. System (5) is equivalent to the single inequality f (x) ≤ 0 in terms of the continuous function f : X → R defined by (11)

f (x) := sup ft (x). t∈T

STABILITY OF ERROR BOUNDS

7

Stability of error bounds criterion for system (5) with respect to perturbations of the function (11) is given by Theorem 8. We are looking here for stability criteria with respect to perturbations of the original family of functions {ft }t∈T . Consider another family of continuous functions gt : X → R, t ∈ T, such that t 7→ gt (x) is continuous on T for each x ∈ X, and the corresponding function g : X → R defined by g(x) := sup gt (x). t∈T

Given x ¯ ∈ X and ε ≥ 0, the following conditions can qualify to be extensions to families of functions of the ε-perturbation property introduced in Definition 5: |gt (x) − ft (x)| ≤ ε; (C1) lim sup sup kx − x ¯k x→¯ x t∈T |(gt (x) − ft (x)) − (gt (¯ x) − ft (¯ x))| (C2) g(¯ x) = f (¯ x) and lim sup sup ≤ ε; kx − x ¯k x→¯ x t∈T |(gt (x) − ft (x)) − (gt (¯ x) − ft (¯ x))| ≤ ε, where (C3) g(¯ x) = f (¯ x) and sup kx − x ¯k x∈Bδ (¯ x)\{¯ x}, t∈T δ is a given positive number. All above conditions are symmetric, and consequently {gt }t∈T is an ε-perturbation of {ft }t∈T near x ¯ (with respect to any of these conditions) if and only if {ft }t∈T is an ε-perturbation of {gt }t∈T . Since functions ft and gt are continuous, condition (C1) implies equality gt (¯ x) = ft (¯ x) for all t ∈ T , and consequently equality g(¯ x) = f (¯ x) and the inequality in (C2). Hence (C1) ⇒ (C2). Obviously (C3) ⇒ (C2) and conversely, if {gt }t∈T is an ε-perturbation of {ft }t∈T near x ¯ in the sense of (C2) then for any ξ > ε there exists a δ > 0 such that {gt }t∈T is a ξ-perturbation of {ft }t∈T in the sense of (C3). Condition (C1) looks like a natural generalization of condition (6). Proposition 11. Let ε > 0. If {gt }t∈T satisfies (C1) then g ∈ Ptb (f, x ¯, ε) and ∂g(¯ x) ⊆ ∂f (¯ x) + εB ∗ . Proof. By (C1), we have g(¯ x) = f (¯ x), and for any ε0 > ε there exists a δ > 0 such that |gt (x) − ft (x)| ≤ ε0 kx − x ¯k

for all x ∈ Bδ (¯ x), t ∈ T,

and consequently, |g(x) − f (x)| ≤ ε0 kx − x ¯k

for all x ∈ Bδ (¯ x).

Since ε0 can be taken arbitrarily close to ε this condition implies (6). The inclusion follows from Proposition 7 (i). The following denotation is used in the sequel: (12)

Tf (x) := {t ∈ T : ft (x) = f (x)}.

Remark 12. Proposition 11 remains valid if instead of (C1) one employs the weaker set of conditions: |gt (x) − ft (x)| ≤ ε, kx − x ¯k t∈Tf (¯ x)

lim sup sup x→¯ x

Tg (¯ x) = Tf (¯ x).

´ ALEXANDER KRUGER, HUYNH VAN NGAI, AND MICHEL THERA

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The requirement that gt (¯ x) = ft (¯ x) for all t ∈ T (even for all t ∈ Tf (¯ x)) can be too restrictive for applications. That is why conditions (C2) and (C3) can be of interest. Unlike condition (C1), these weaker properties are not sufficient in general to guarantee the conclusions of Proposition 11. Some additional assumptions are needed. From now on in this section we limit ourselves to considering convex functions. We shall denote by G(X, T ) the class of all families {ft }t∈T of convex continuous functions ft : X → R such that t 7→ ft (x) is continuous on T for each x ∈ X. For {ft }t∈T ∈ G(X, T ) the convex continuous function f and the set Tf (x) are defined by (11) and (12) respectively. Under the assumptions made, if {ft }t∈T ∈ G(X, T ) then the directional derivative f 0 (¯ x; ·) of f at x ¯ is given by the formula (see, for instance, [17, Theorem 4.2.3], [26, Proposition 4.5.2]): f 0 (¯ x; x) = sup ft0 (¯ x; x),

(13)

x ∈ X.

t∈Tf (¯ x)

Given two families {ft }t∈T , {gt }t∈T ∈ G(X, T ) denote αf,g (x) := f (x) −

(14)

inf t∈Tg (x)

ft (x).

Obviously αf,g (x) ≥ 0, and αf,g (x) = 0 if and only if Tg (x) ⊆ Tf (x). Note that in general αg,f (x) 6= αf,g (x). Proposition 13. Let ε > 0 and families {ft }t∈T , {gt }t∈T ∈ G(X, T ) satisfy condition (C2). The following  assertions hold true:  [ \ [  (i). ∂g(¯ x) ⊆ ∂f (x) + (ε + αf,g (¯ x)/ρ)B ∗ ; δ>0 0 0. If αf,g (¯ x) = 0 then assertion (iv) coincides with (ii), otherwise it is a particular case of (iii) with ρ = . Remark 14. Analyzing the proof of Proposition 13 (iii) one can easily notice that it remains true if the inequality in condition (C3) is replaced by a weaker one: |(gt (x) − ft (x)) − (gt (¯ x) − ft (¯ x))| ≤ ε. kx − x ¯ k x∈Bδ (¯ x)\{¯ x}, t∈Tg (¯ x) sup

Furthermore, since the assertion establishes a “one-sided relation” (inclusion), it is sufficient to require a one-sided estimate: (gt (x) − ft (x)) − (gt (¯ x) − ft (¯ x)) ≤ ε. kx − x ¯ k x∈Bδ (¯ x)\{¯ x}, t∈Tg (¯ x) sup

Similarly, the inequality in condition (C2) can be replaced by the following one: (gt (x) − ft (x)) − (gt (¯ x) − ft (¯ x)) ≤ ε. kx − x ¯k t∈Tg (¯ x)

lim sup sup x→¯ x

Remark 15. The number αf,g (¯ x) in Proposition 13 is determined by the values f (¯ x) and ft (¯ x). Note that it also depends on the other family of functions {gt } since

´ ALEXANDER KRUGER, HUYNH VAN NGAI, AND MICHEL THERA

10

the infimum in its definition is taken over t ∈ Tg (¯ x). If g(¯ x) = f (¯ x) which is a part of any of the conditions (C1), (C2), and (C3), then αf,g (¯ x) can be rewritten equivalently as αf,g (¯ x) = sup (gt (¯ x) − ft (¯ x)). t∈Tg (¯ x)

Proposition 13 yields certain relations between distances from zero to subdifferentials of g and f and their boundaries. Proposition 16. Let the conditions of Proposition 13 be satisfied. The following assertions hold true: (i). if 0 ∈ ∂g(¯ x) and Tg (¯ x) ⊆ Tf (¯ x) then d(0, Bdry ∂g(¯ x)) ≤ d(0, Bdry ∂f (¯ x)) + ε;   (ii). d(0, ∂g(¯ x)) ≥ inf

sup

δ>0 0 0, and consequently 0 6∈ ∂g(¯ x) and d(0, ∂g(¯ x)) = d(0, Bdry ∂g(x)). In Proposition 16 (i) the subdifferentials of f and g are computed at x ¯. In all other assertions in Proposition 16 the subdifferentials of f are computed at nearby points, which is not exactly what is needed for establishing stability of error bounds estimates. Fortunately Proposition 16 (iv) allows us to establish the desired estimate in terms of ∂f (¯ x). Proposition 17. Let {ft }t∈T ∈ G(X, T ). Then for any ξ > 0 and δ > 0 there exists an ε > 0 such that for all {gt }t∈T ∈ G(X, T ) satisfying condition (C3) and αf,g (¯ x) ≤ ε

(15) it holds

d(0, ∂g(¯ x)) ≥ d(0, ∂f (¯ x)) − ξ. Proof. Let ξ > 0 be given. Due to the upper semicontinuity of the subdifferential mapping, there exists an η > 0 such that d(0, ∂f (x)) > d(0, ∂f (¯ x)) − ξ/3 for all x ∈ Bη (¯ x). Take a positive ε < min(ξ/3, ξ 2 /9, δ 2 , η 2 ). Ifpthe family {gt }t∈T ∈ G(X, T ) satisfies the assumptions of the proposition then  := αf,g (¯ x) < min(δ, η, ξ/3) and it follows from Proposition 16 (iv) that d(0, ∂g(¯ x)) ≥

inf x∈B (¯ x)

d(0, ∂f (x)) − ε −  ≥ d(0, ∂f (¯ x)) − ξ.

STABILITY OF ERROR BOUNDS

11

The following theorem gives a characterization of the stability of local error bounds for system (5). Theorem 18. Let {ft }t∈T ∈ G(X, T ) and f (¯ x) = 0. The following assertions hold true: (i). If 0 ≤ τ < ς(f, x ¯), δ > 0 then there exists an ε > 0 such that for any {gt }t∈T ∈ G(X, T ) satisfying (a) condition (C2) and Tf (¯ x) ⊆ Tg (¯ x) if 0 ∈ int ∂f (¯ x), or (b) conditions (C3) and (15), otherwise, one has Er g(¯ x) ≥ τ . (ii). If ς(f, x ¯) = 0 then for any ε > 0 there exists a family {gt }t∈T ∈ G(X, T ) satisfying condition (C1) such that Er g(¯ x) ≤ ε. Proof. (i). Let 0 ≤ τ < ς(f, x ¯), {gt }t∈T ∈ G(X, T ). By the definition of ς(f, x ¯), 0 6∈ Bdry ∂f (¯ x), that is, either 0 ∈ int ∂f (¯ x) or 0 6∈ ∂f (¯ x). If 0 ∈ int ∂f (¯ x), then it is sufficient to take ε = ς(f, x ¯) − τ . Indeed, by the definition of ς(f, x ¯) one has ς(f, x ¯)B ∗ ⊆ ∂f (¯ x). If condition (C2) is satisfied and Tf (¯ x) ⊆ Tg (¯ x) then, by Proposition 13 (ii), ς(f, x ¯)B ∗ ⊆ ∂g(¯ x) + εB ∗ . It follows that ∗ τ B ⊆ ∂g(¯ x), and consequently, by Theorem 1, Er g(¯ x) ≥ ς(g, x ¯) ≥ τ . Suppose now 0 ∈ / ∂f (¯ x). Then ς(f, x ¯) = d(0, ∂f (¯ x)). Take any ξ ∈ (0, ς(f, x ¯) − τ ). Proposition 17 implies the existence of an ε > 0 such that for any {gt }t∈T ∈ G(X, T ) satisfying conditions (C3) and (15) it holds d(0, ∂g(¯ x)) ≥ d(0, ∂f (¯ x)) − ξ > τ. Hence 0 ∈ / ∂g(¯ x) and, by Theorem 1, Er g(¯ x) ≥ d(0, ∂g(¯ x)) > τ . (ii). By Theorem 8 (ii), for any ε > 0 there exists a g ∈ Γ0 (X)∩Ptb (f, x ¯, ε), given by (9), such that Er g(¯ x) < ε. Since f is continuous, g is continuous too. For t ∈ T and x ∈ X, set gt (x) = ft (x) + g(x) − f (x). Then {gt }t∈T ∈ G(X, T ), g(x) = supt∈T gt (x), and condition (C1) is satisfied. Remark 19. Due to the equivalent representation of αf,g (¯ x) formulated in Remark 15, condition (15) in Theorem 18 is equivalent to the following one sup (gt (¯ x) − ft (¯ x)) ≤ ε. t∈Tg (¯ x)

The last inequality is obviously ensured by a stronger condition from [21, Theorem 3]: sup |gt (¯ x) − ft (¯ x)| ≤ ε. t∈T

The next corollary strengthens [21, Theorem 3]. Corollary 20. Let {ft }t∈T ∈ G(X, T ), and f (¯ x) = 0. The following properties are equivalent: (i). there exists an ε > 0 such that Er g(¯ x) ≥ ε for any {gt }t∈T ∈ G(X, T ) satisfying the conditions in Theorem 18 (i); (ii). 0 6∈ Bdry ∂f (¯ x). Remark 21. The inclusion Tf (¯ x) ⊆ Tg (¯ x) in Theorem 18 (b) cannot be dropped – see [21, Remark 5]. 3. Stability of global error bounds. In this section, we deal with the error bound property of the set Sf = {x ∈ X : f (x) ≤ 0} without relating it to a particular point x ¯ ∈ Sf= := {x ∈ X : f (x) = 0}. The next theorem represents a nonlocal analog of Theorem 1.

´ ALEXANDER KRUGER, HUYNH VAN NGAI, AND MICHEL THERA

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Theorem 22. Let f ∈ Γ0 (X), Sf 6= ∅. Consider the following properties: (i). f admits a global error bound, that is, Er f > 0; (ii). τ (f ) := inf d(0, ∂f (x)) > 0; f (x)>0

(iii). ς(f ) := inf d(0, Bdry ∂f (x)) > 0; f (x)=0

(iv).

inf d(0, ∂f (x)) > 0; f (x)=0

(v). 0 ∈ int ∂f (x) for some x such that f (x) = 0. Each of the properties (ii)–(v) is sufficient for the error bound property (i). Moreover, (a). ς(f ) ≤ τ (f ) = Er f ; (b). [(iv) or (v)] ⇔ (iii) ⇒ (ii) ⇔ (i). Proof. The equality in (a) is well known (see [4, Theorem 3.1]), [29, Theorem 7]. If 0 ∈ int ∂f (¯ x) for some x ¯ ∈ Sf= then Sf = {¯ x}, ς(f ) = ς(f, x ¯), and ς(f )B ∗ ⊆ ∂f (¯ x). It follows that f (x) ≥ ς(f )kx − x ¯k. On the other hand, if x∗ ∈ ∂f (x) for some x 6= x ¯ then −f (x) ≥ hx∗ , x ¯ − xi. Adding the last two inequalities together we obtain hx∗ , x − x ¯i ≥ ς(f )kx − x ¯k. Hence, kx∗ k ≥ ς(f ), and consequently, τ (f ) ≥ ς(f ). If 0 6∈ int ∂f (u) for all u ∈ Sf= then ς(f ) = inf d(0, ∂f (u)) and the inequality f (u)=0

in (a) follows from [4, Theorem 3.2]. The equivalence [(iv) or (v)] ⇔ (iii) in (b) is obvious, while the other one (ii) ⇔ (i) and the implication (iii) ⇒ (ii) follow from (a). Examples 3 and 4 in Section 2 are also applicable to global error bounds to show that inequality (a) and implication (iii) ⇒ (ii) in Theorem 22 can be strict. Theorem 22 (a) guarantees that kx∗ k ≥ ς(f ) for any x∗ ∈ ∂f (x) with f (x) > 0. The next Asymptotic qualification condition (AQC) ensures the limiting form of this estimate also for elements of ∂f (x) at some points x with f (x) < 0. (AQC) lim inf kx∗k k ≥ ς(f ) for any sequences xk ∈ X with f (xk ) < 0 and x∗k ∈ k→∞

∂f (xk ), k = 1, 2, . . . , satisfying (a) ether the sequence {xk } is bounded and lim f (xk ) = 0, k→∞

(b) or lim kxk k = ∞ and lim f (xk )/kxk k = 0. k→∞

k→∞

Note that if dim X < ∞ and Sf is closed then part (a) of (AQC) automatically implies lim inf kx∗k k ≥ ς(f ) and consequently (AQC) coincides with the asymptotic k→∞

qualification condition introduced in [21]. Proposition 23. Let f ∈ Γ0 (X), Sf 6= ∅, and 0 6∈ int ∂f (x) for all x ∈ Sf= . Suppose that (AQC) holds true. Then ς(f ) = sup

inf

ε>0 f (x)>−εkx−x0 k−ε

for any x0 ∈ X.

d(0, ∂f (x))

STABILITY OF ERROR BOUNDS

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Proof. Since 0 6∈ int ∂f (x) for all x ∈ Sf= , thanks to Theorem 22(a) we have ς(f ) = inf d(0, ∂f (x)) = inf d(0, ∂f (x)). f (x)=0

f (x)≥0

Hence, (16)

sup

inf

ε>0 f (x)>−εkx−x0 k−ε

d(0, ∂f (x)) ≤ ς(f ).

Consider sequences εk ↓ 0, xk ∈ X, and x∗k ∈ ∂f (xk ), k = 1, 2, . . . , such that −εk kxk − x0 k − εk < f (xk ) < 0.

(17)

If {xk } is bounded then it follows from (17) that f (xk ) → 0, and consequently part (a) of (AQC) is satisfied. If {xk } is unbounded we can assume that kxk k → ∞. Then (17) implies that limk→∞ f (xk )/kxk k = 0, that is, part (b) of (AQC) is satisfied. In both cases, thanks to (AQC), lim inf kx∗k k ≥ ς(f ). Together with (16) this proves the k→∞

assertion. Condition (iii) in Theorem 22 corresponds to the existence of a global error bound for a family of functions being small perturbations of f . Definition 24. Let Sf 6= ∅ and ε ≥ 0. We say that g : X → R∞ is an ε-perturbation of f relative to x0 ∈ dom f and write g ∈ Ptb x0 (f, ε) if Sg 6= ∅ and (18)

|g(x0 ) − f (x0 )| ≤ ε,

(19)

|(g(x) − f (x)) − (g(u) − f (u))| ≤ ε. kx − uk x6=u sup

Condition (19) constitutes the Lipschitz continuity (with modulus ε) property of the difference g − f . Unlike the case of Definition 5, this condition does not involve the reference point x0 which participates only in Condition (18). Clearly g ∈ Ptb x0 (f, ε) ⇒ g + f (x0 ) − g(x0 ) ∈ Ptb (f, x0 , ε). Basically an ε-perturbation is a result of a small shift and a small rotation of the original function. Note that if there is an x ∈ X such that f (x) < 0 (in particular, if 0 6∈ ∂f (¯ x) for some x ¯ ∈ Sf= ), then condition Sg 6= ∅ in Definition 24 is satisfied automatically for a sufficiently small ε. Theorem 25. Let f ∈ Γ0 (X), Sf 6= ∅, and x0 ∈ dom f . The following assertions hold true: (i). Suppose that either 0 ∈ int ∂f (¯ x) for some x ¯ ∈ Sf= or (AQC) holds. If 0 ≤ τ < ς(f ) then there exists an ε > 0 such that Er g ≥ τ for any g ∈ Γ0 (X) ∩ Ptb x0 (f, ε). (ii). Suppose that ς(f ) = 0. For ε > 0 and x ¯ ∈ Sf= set (20) (21)

ξ = ε min(1/2, kx0 − x ¯k−1 ), g(u) := f (u) + ξku − x ¯k,

u ∈ X.

Then g ∈ Γ0 (X) ∩ Ptb x0 (f, ε) and Er g ≤ ε. (iii). Suppose that dim X < ∞ and ς(f ) = 0. For ε > 0 and x ¯ ∈ Sf= let ξ > 0 be ∗ ∗ defined by (20). Then there exists an x ∈ ξB such that the function g in assertion (ii) can be replaced by the following one: (22)

g(u) := f (u) + hx∗ , u − x ¯i,

u ∈ X.

´ ALEXANDER KRUGER, HUYNH VAN NGAI, AND MICHEL THERA

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Proof. (i). Let 0 ≤ τ < ς(f ). If 0 ∈ int ∂f (¯ x) for some x ¯ ∈ Sf= , then Sf = {¯ x}, and ς(f ) = ς(f, x ¯). Take an ε ∈ (0, ς(f ) − τ ). If g ∈ Γ0 (X) ∩ Ptb x0 (f, ε) then g − g(¯ x) ∈ Ptb (f, x ¯, ε), and it follows from Proposition 7 (iii) that 0 ∈ int ∂g(¯ x) and ς(g −g(¯ x)) = ς(g, x ¯) ≥ ς(f )−ε > τ . If x 6= x ¯ and x∗ ∈ ∂g(x) then, due to Theorem 22, kx∗ k > τ . Since, by assumption, Sg 6= ∅, applying Theorem 22 again, we conclude that Er g ≥ τ . Let 0 6∈ int ∂f (x) for all x ∈ Sf= , (AQC) holds true, and τ 0 ∈ (τ, ς(f )). Then it follows from Proposition 23 that there exists an ε ∈ (0, τ 0 − τ ] such that kx∗ k ≥ τ 0 if x∗ ∈ ∂f (x) and f (x) > −εkx − x0 k − ε. Consider a function g ∈ Γ0 (X) ∩ Ptb x0 (f, ε). By definition, f (x) ≥ g(x) − εkx − x0 k − ε for all

x ∈ X.

Hence, if g(x) > 0 then f (x) > −εkx − x0 k − ε, and consequently d(0, ∂g(x)) ≥ d(0, ∂f (x)) − ε ≥ τ 0 − ε ≥ τ. The conclusion follows from Theorem 22. (ii). Let ς(f ) = 0, ε > 0, and x ¯ ∈ Sf= . If the function g ∈ Γ0 (X) is defined by (21) then g − f is obviously Lipschitz continuous with modulus ξ and |g(x0 ) − f (x0 )| ≤ ε. Hence g ∈ Ptb x0 (f, ε). We need to show that Er g ≤ ε. By definition of ς(f ), there exists a y ∈ Sf= and an u∗ ∈ Bdry ∂f (y) such that ku∗ k < ε/2. If it is possible to choose y 6= x ¯, then g(y) > 0 and τ (g) ≤ ku∗ k + ξ < ε; thanks to Theorem 22, Er g < ε. Otherwise, ku∗ k ≥ ε/2 for any u∗ ∈ Bdry ∂f (y) with y ∈ Sf= \ {¯ x}. Since ς(f ) = 0 this means that 0 ∈ Bdry ∂f (¯ x). Then, by Theorem 8 (ii), Er g ≤ Er g(¯ x) ≤ ξ ≤ ε. (iii). Let dim X < ∞, ς(f ) = 0, ε > 0, and x ¯ ∈ Sf= . If in the above proof of (ii) ∗ it is possible to choose y 6= x ¯, then take y ∈ X ∗ such that hy ∗ , y − x ¯i = ky − x ¯k, ∗ ∗ ∗ ky k = 1 and set x = ξy ; otherwise apply Theorem 8 (iii) instead of (ii). In both cases, if the function g ∈ Γ0 (X) is defined by (22), then g ∈ Ptb x0 (f, ε) and Er g ≤ ε. Given a function f ∈ Γ0 (X) with Sf 6= ∅, a point x0 ∈ dom f , and a number ε ≥ 0 denote Er {Ptb x0 (f, ε)} :=

(23)

inf

Er g.

g∈Γ0 (X)∩Ptb x0 (f,ε)

This number characterizes the error bound property for the whole family of convex ε-perturbations of f relative to x0 . Obviously, Er {Ptb x0 (f, ε)} ≤ Er f for any x0 ∈ dom f and ε ≥ 0. Corollary 26. Let f ∈ Γ0 (X), Sf 6= ∅, and x0 ∈ dom f . The following assertions hold true: (i). Suppose that either 0 ∈ int ∂f (¯ x) for some x ¯ ∈ Sf= or (AQC) holds. Then sup Er {Ptb x0 (f, ε)} ≥ ς(f ). ε>0

(ii). If ς(f ) = 0 then sup Er {Ptb x0 (f, ε)} = 0. ε>0

Due to Corollary 26 (i), under the assumption that either 0 ∈ int ∂f (¯ x) for some x ¯ ∈ Sf= or (AQC) holds, condition ς(f ) > 0 is sufficient for the error bound property of the family of ε-perturbations of f if ε > 0 is sufficiently small. If ς(f ) = 0 then, due to Corollary 26 (ii), the “uniform” error bound property does not hold. Corollary 27. Let f ∈ Γ0 (X) and Sf 6= ∅. Suppose that either 0 ∈ int ∂f (¯ x) for some x ¯ ∈ Sf= or (AQC) holds. The following properties are equivalent:

STABILITY OF ERROR BOUNDS

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(i). for any x0 ∈ dom f there exists an ε > 0 such that Er {Ptb x0 (f, ε)} > 0; (ii). for some x0 ∈ dom f there exists an ε > 0 such that Er {Ptb x0 (f, ε)} > 0; (iii). ς(f ) > 0. The next theorem gives a characterization of the stability of global error bounds for the infinite convex constraint system (5). Along with the family of continuous functions {ft }t∈T we consider the function f : X → R∞ and set valued mapping Tf : X ⇒ T defined by (11) and (12) respectively. To formulate stability criteria we need another family of continuous functions {gt }t∈T together with the corresponding mappings g and Tg . Recall that G(X, T ) denotes the class of all families {ft }t∈T of convex continuous functions ft : X → R such that t 7→ ft (x) is continuous on T for each x ∈ X. If {ft }t∈T ∈ G(X, T ) then f is obviously convex continuous too. If {ft }t∈T ∈ G(X, T ) and {gt }t∈T ∈ G(X, T ) then αf,g (x) is defined by (14). Theorem 28. Let {ft }t∈T ∈ G(X, T ), Sf 6= ∅, and x0 ∈ X. The following assertions hold true: (i). If 0 ≤ τ < ς(f ) then there exists an ε > 0 such that for any {gt }t∈T ∈ G(X, T ) satisfying Sg 6= ∅ and one of the following two groups of conditions: (a) if 0 ∈ int ∂f (¯ x) for some x ¯ ∈ Sf= then (24)

lim sup sup x→¯ x

t∈T

|(gt (x) − ft (x)) − (gt (¯ x) − ft (¯ x))| ≤ ε, kx − x ¯k Tf (¯ x) ⊆ Tg (¯ x)

(25) (b) if 0 6∈ ∂f (x) for all x ∈ (26)

sup x6=u, t∈T

Sf=

then (AQC) holds and

|(gt (x) − ft (x)) − (gt (u) − ft (u))| ≤ ε, kx − uk sup |gt (x0 ) − ft (x0 )| ≤ ε,

(27)

t∈T

one has Er g ≥ τ . (ii). If ς(f ) = 0 then for any ε > 0 there exists a family {gt }t∈T ∈ G(X, T ), satisfying (25)–(27), such that Er g < ε. Proof. (i). Let 0 ≤ τ < ς(f ), {gt }t∈T ∈ G(X, T ). By the definition of ς(f ), two cases are possible. (a) 0 ∈ int ∂f (¯ x) for some x ¯ ∈ Sf= . Then Sf = {¯ x}, ς(f ) = ς(f, x ¯), and ς(f )B ∗ ⊆ ∂f (¯ x). Let ε ∈ (0, ς(f ) − τ ). If condition (24) is satisfied then the family {gt − g(¯ x)}t∈T ∈ G(X, T ) satisfies condition (C2) formulated in Section 2. If additionally condition (25) is satisfied then, by Proposition 13 (ii), ς(f )B ∗ ⊆ ∂g(¯ x) + εB ∗ . It ∗ follows that (ς(f ) − ε)B ⊆ ∂g(¯ x), and consequently, Sg−g(¯x) = {¯ x} and ς(g − g(¯ x)) ≥ ς(f ) − ε > τ . By Theorem 22, τ (g − g(¯ x)) ≥ ς(g − g(¯ x)) > τ . Since Sg 6= ∅, we have g(¯ x) ≤ 0, and consequently, Er g = τ (g) = inf d(0, ∂g(x)) ≥

inf

g(x)>0

g(x)>g(¯ x)

d(0, ∂g(x)) = τ (g − g(¯ x)) > τ.

(b) 0 6∈ ∂f (u) for all u ∈ Sf= . Then ς(f ) = inf f (u)=0 d(0, ∂f (u)). For any u ∈ X and t ∈ Tg (u) we have g(u) − f (u) = gt (u) − sup fs (u) ≤ gt (u) − ft (u) ≤ |gt (u) − ft (u)|, s∈T

and consequently (28)

|g(u) − f (u)| ≤ sup |gt (u) − ft (u)|. t∈T

´ ALEXANDER KRUGER, HUYNH VAN NGAI, AND MICHEL THERA

16

By definition (14), (29) αf,g (u) = f (u) −

inf t∈Tg (u)

ft (u)

= f (u) − g(u) + sup (gt (u) − ft (u)) ≤ 2 sup |gt (u) − ft (u)|. t∈Tg (u)

t∈T

Let τ 0 ∈ (τ, ς(f )). If (AQC) holds true then it follows from Proposition 23 that, for a sufficiently small ξ > 0, one has√ku∗ k ≥ τ 0 if u∗ ∈ ∂f (u) √ and f (u) > −ξ(ku − x0 k + 1). Choose an ε > 0 satisfying ε(τ + ε) + 3ε ≤ ξ and ε + 2 ε ≤ τ 0 − τ . Let x∗ ∈ ∂g(x) for some x ∈ X with g(x) > 0. If condition (26) holds true then the family {gt − g(x)}t∈T ∈ G(X, T ) satisfies condition (C3) (with x ¯ = x) for any δ > 0. It follows from Proposition 13 (iii) that [ (30) x∗ ∈ ∂f (v) + (ε + αf,g (x)/ρ)B ∗ , v∈Bρ (x)

where ρ := (31)

√

ε(kx − x0 k + 1). Thanks to (30), for any v ∈ B (x), it holds

f (v) − f (x) ≥ −(kx∗ k + ε + αf,g (x)/ρ)kv − xk ≥ −ρ(kx∗ k + ε) + αf,g (x).

If, additionally, condition (27) holds true then it follows from (28) and (29) that (32)

f (x) ≥ g(x) − sup |gt (x0 ) − ft (x0 )| − εkx − x0 k > −ε(kx − x0 k + 1), t∈T

(33)

αf,g (x) ≤ 2 sup |gt (x0 ) − ft (x0 )| + 2εkx − x0 k ≤ 2ε(kx − x0 k + 1). t∈T

Suppose that kx∗ k < τ . Then (31), (32), and (33) yield √ f (v) > −[ ε(τ + ε) + 3ε](kx − x0 k + 1) ≥ −ξ(kx − x0 k + 1), √ ε + αf,g (x)/ρ ≤ ε + 2 ε ≤ τ 0 − τ. Hence, ku∗ k ≥ τ 0 for any u∗ ∈ ∂f (v) and, thanks to (30), kx∗ k ≥ τ . By Theorem 22, Er g = τ (g) ≥ τ . (ii). By Theorem 25 (ii), for any ε > 0 there exists a g ∈ Γ0 (X)∩Ptb x0 (f, ε), given by (22), such that Er g < ε. Since f is continuous, g is continuous too. For t ∈ T and x ∈ X, set gt (x) = ft (x) + g(x) − f (x). Then {gt }t∈T ∈ G(X, T ), g(x) = supt∈T gt (x), Tf (x) = Tg (x), and conditions (26), (27) are satisfied. The next corollary strengthens [21, Theorem 7]. Corollary 29. Let {ft }t∈T ∈ G(X, T ) and Sf 6= ∅. The following properties are equivalent: (i). for any x0 ∈ X there exists an ε > 0 such that Er g ≥ ε for all {gt }t∈T ∈ G(X, T ) satisfying the conditions in Theorem 28 (i); (ii). for some x0 ∈ X there exists an ε > 0 such that Er g ≥ ε for all {gt }t∈T ∈ G(X, T ) satisfying the conditions in Theorem 28 (i); (iii). ς(f ) > 0. Acknowledgement. The authors wish to thank the anonymous referees for careful reading of the paper and valuable comments and suggestions, which helped us improve the presentation. REFERENCES

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