Calmness for Closed Multifunctions over Constraint Sets in Banach Spaces

In this paper, we mainly study calmness and strong calmness of closed multifunctions over constraint sets in Banach spaces. In terms of tangent cones, normal cones and coderivatives, we provide some dual necessary/sufficient conditions ensuring calmness over constraint sets. In particular we proved a dual characterization for strong calmness of a closed multifunction over constraint closed sets with mild assumptions.

Calmness is essentially equivalent to metric subregularity which is another well-known and important concept in mathematical programming and optimization.If we take F(x) := {y ∈ Y : x ∈ M(y)} for all x ∈ X, then (2) is equivalent to d(x, F −1 (ȳ) ∩ A) ≤ τ(d(ȳ, F(x)) + d(x, A)) for all x close to x, (3) and (3) means that the generalized equation with constraint: ȳ ∈ F(x) subject to x ∈ A is metrically subregular at x ∈ F −1 (ȳ) ∩ A. The metric subregularity can be used to estimate the distance of a candidate x to the solution set of generalized equation.
Calmness is known to be a weakened version of the Aubin pseudo-Lipschitz property and closely relates to the upper Lipschitz property of multifunctions.Several subdifferential conditions ensuring calmness for multifunctions in finite-dimensional spaces have been developed.Reader are invited to consult (Henrion, 2001;Henrion & Jourani, 2002;Herion, Jourani & Outrata, 2002;Herion & Outrata, 2005;Mordukhovich, 1995) and references therein for more details.It is noted that Zheng and Ng studied calmness of convex closed multifunctions in Banach spaces and provided its dual characterizations in terms of normal cones and coderivative (see Zheng & Ng, 2007).Subsequently, in (Zheng & Ng 2009;Zheng & Ng, 2010), they further consider calmness of closed (not convex necessarily) multifunctions and gave several necessary and/or sufficient dual conditions for calmness.Motivated by (Zheng & Ng 2007;Zheng & Ng 2009;Zheng & Ng, 2010), we mainly discuss calmness of closed multifunctions over constraint subsets in this paper and aim to establish several subdifferential conditions ensuring calmness over constraint subsets via normal cones and coderivatives.
This paper is organized as follows.Several preliminaries and known results will be given in Section 2. Section 3 is devoted to main results on sufficient and/or necessary conditions for calmness and strong calmness over constraint subsets which are established by using some results in Section 2 and in terms of normal cone and coderivative.Applications of main results to calmness of one special multifunction are also given therein.The conclusion of this paper is presented in Section 4.

Preliminaries
Let X, Y be Banach spaces with the closed unit balls denoted by B X and B Y , and let X * , Y * denote the dual spaces of X and Y respectively.Let A be a closed subset of X and a ∈ A. Denote T c (A, a) the Clarke tangent cone of A at a which is defined as We denote by N c (A, a) the Clarke normal cone of A at a which is defined by Let N(A, a) denote the Fréchet normal cone of A at a which is defined by where Nε (A, x) is the set of ε-normal to A at x and defined as This means that x * ∈ N(A, a) if and only if there exist For the case when X is an Asplund space (see Phelps, 1989 for definitions and their equivalences), it has been proved in (Mordukhovich & Shao, 1996) that where co w * denotes the weak * closed convex hull.Thus, x * ∈ N(A, a) if and only if there exist It is known from (Mordukhovich, 2006) that If A is convex, all normal cones coincide and reduce to the normal cone in the sense of convex analysis; that is It is known that prox-regularity is an important extension of convexity, and prox-regularity of a set express a variational behavior of "order two".Recall that A is said to be prox-regular at a Clarke, Stern & Wolenski, 1995;Poliquin & Rockafellar, 1996;Rockafelar & Wets, 1998).
In 2005, Aussel, Daniilidis and Thibault introduced the concept of subsmoothness which is the extension of proxregularity and smoothness (see Aussel, Daniilidis & Thibault, 2005).This concept expresses a variational behavior of "order one".
Let A be a closed subset of X and a ∈ A. Recall that (i) A is said to be subsmooth at a if for any ε > 0 there exists δ > 0 such that (ii) A is said to satisfy Condition (S) at a if for any ε > 0 there exists δ > 0 such that It is easy to verify that A is subsmooth at a if and only if for any ε > 0 there exists δ > 0 such that In 2008, Zheng and Ng further studied the concept of subsmooth and provide a characterization for this concept; that is, A is subsmooth at a if and only if for any ε > 0 there exists δ > 0 such that holds for all u ∈ B(a, δ) ∩ A and u * ∈ N c (A, u) ∩ B X * (see Zheng & Ng, 2008).Further, they considered a weakened notion which is called L-subsmooth, and studied calmness for closed multifunctions with L-subsmooth assumptions (see Zheng & Ng, 2009).
Let M : Y ⇒ X be a closed multifunction and (ȳ, x) ∈ gph(M).Recall that (i) M is said to subsmooth (resp.satisfy Condition (S)) at (ȳ, x) if gph(M) is subsmooth (resp.satisfies Condition (S)) at (ȳ, x); (ii) M is said to be L-subsmooth at (ȳ, x) if for any ε > 0 there exists δ > 0 such that for any u It is easy to verify that L-smoothness is weaker than subsmoothness but stronger than Condition (S).We refer readers to (Zheng & Ng, 2009) for more properties and examples with respect to L-subsmoothness.
For a proper lower semicontinuous convex function ψ : X → R ∪ {+∞}, recall that the subdifferential of ψ at x ∈ dom(ψ) := {x ∈ X : ψ(x) < +∞} is defined as We close this section with the following result which is a cornerstone in convex analysis and convex optimization.Readers could consulte Theorem 3.16 in (Phelps, 1989).

Main Results
In this section, we main study calmness and strong calmness of closed multifunctions over constraint subsets, and aim to provide sufficient conditions for calmness and strong calmness.We begin with the definition of calmness over constraint subset.
Let M : Y ⇒ X be a closed multifunction and A be a closed subset of X.Let ȳ ∈ Y and x ∈ M(ȳ) ∩ A. Recall that M is said to be calm at (ȳ, x) over A if there exist τ, δ ∈ (0, +∞) such that Noting that d(x, ∅) = +∞ and d(x, M(ȳ)) ≤ ∥x − x∥, it follows that M is calm at (ȳ, x) if and only if there exist τ, δ ∈ (0, +∞) such that The following proposition is on necessary conditions for calmness of closed multifunctions over constraint subsets.
The proof can be obtained by using Proposition 4.1 and Theorem 4.2 in (Huang, He & Wei, 2014) and Theorem 4.2 in (Zheng & Ng, 2009).
Proposition 3.1.Let M : Y ⇒ X be a closed multifunction, A be a closed subset of X and let ȳ ∈ Y and x ∈ M(ȳ)∩A.
If M is calm at (ȳ, x) over A, then there exist τ, δ ∈ (0, +∞) such that Assume further that X is finite-dimensional and Y is an Asplund space.If M is calm at (ȳ, x) over A, then there exist τ, δ ∈ (0, +∞) such that Let F := M −1 and consider the following generalized equation with constraint: By using ( 7), one has that M is calm at (ȳ, x) over A if and only if the generalized equation with constraint of ( 10) is metrically subregular at x ∈ F −1 (ȳ) ∩ A. Thus, the proof of Proposition 3.1 can be obtained by Proposition 4.1 and Theorem 4.2 in (Huang, He & Wei, 2014).
The following proposition provides a sufficient condition for calmness of closed multifunctions over constraint subset with the help of subsmooth assumptions.We give its proof for the sake of completeness.
Lemma 3.1 Let M : Y ⇒ X be a closed multifunction and A be a closed subset of X. Suppose that (ȳ, x) ∈ gph(M) with x ∈ A and τ ∈ (0, +∞).
holds for all y ∈ Y and x ∈ D c M(ȳ, x)(y).
The necessity part.Let y ∈ Y and x ∈ D c M(ȳ, x)(y)\T c (M(ȳ) ∩ A, x).Take any γ ∈ (0, 1).Applying Lemma 2.1, there exist z ∈ T c (M(ȳ) ∩ A, x) and Since T c (M(ȳ) ∩ A, x) is a closed and convex cone, then one has that ⟨z * , z⟩ = 0 by using z * ∈ N(T c (M(ȳ) ∩ A, x), z) and consequently By ( 23), there exist This with ( 25) and ( 26) implies that The following theorem presents one sufficient condition for calmness of closed multifunctions over constraint closed subset under subsmooth assumptions.The proof is immediate from Proposition 3.2 and Lemma 3.1.
Theorem 3.1.Let M : Y ⇒ X be a closed multifunction, A be a closed subset of X and let ȳ ∈ Y and x ∈ M(ȳ) ∩ A. Suppose that M is L-subsmooth at (ȳ, x), A is subsmooth at x and there exist τ, δ ∈ (0, +∞) such that for any u ∈ M(ȳ) ∩ A ∩ B( x, δ), one has holds for all y ∈ Y and x ∈ D c M(ȳ, u)(y).Then for any ε ∈ (0, 1 2τ+1 ), M is calm at (ȳ, x) over A with constant Next, we study the strong calmness for close multifunctions over constraint subsets.Recall that M is said to be strongly calm at (ȳ, x) over A if there exist τ, δ > 0 such that It is easy to verify that M is strongly calm at (ȳ, x) over A if and only if M is calm at (ȳ, x) over A and M(ȳ) ∩ A ∩ B( x, δ 0 ) = { x} for some δ 0 > 0.
Under the assumption of Condition (S), we provide a characterization for strong calmness of closed multifunctions over constraint closed subsets through the following theorem.
Theorem 3.2.Let M : Y ⇒ X be a closed multifunction, A be a closed subset of X and let ȳ ∈ Y and x ∈ M(ȳ) ∩ A. Suppose that M satisfies Condition (S) at (ȳ, x) and A satisfies Condition (S) at x. Then M is strongly calm at (ȳ, x) over A if and only if there exists η ∈ (0, +∞) such that Proof.The necessity part.Suppose that M is strongly calm at (ȳ, x) over A. Then there exist τ, δ > 0 such that M is calm at (ȳ, x) over A with constant τ > 0 and M(ȳ) ∩ A ∩ B( x, δ) = { x}.By virtue of Proposition 3.1, one has follows that (28) holds with η := 1 τ > 0. The sufficiency part.Let ε ∈ (0, η η+2 ).Since M satisfies Condition (S) at (ȳ, x) and A satisfies Condition (S) at x, there exists δ ∈ (0, +∞) such that whenever and We claim that Granting this, it follows from ( 28) that ( 11) holds with τ := 1 η for all u ∈ M(ȳ) ∩ A ∩ B( x, δ), and thus M is calm at (ȳ, x) over A by Proposition 3.2.
Let x ∈ M(ȳ) ∩ A ∩ B( x, δ).By the Hahn-Banach Theorem, there exists u * ∈ B X * such that ⟨u * , x − x⟩ = ∥x − x∥.By virtue of (28), there exist x * 1 ∈ D * c M −1 ( x, ȳ)(y * ) for some y * ∈ B Y * and x * 2 ∈ N c (A, x) ∩ B X * such that Noting that ( Under the assumptions that M satisfies Condition (S) at (ȳ, x) and A satisfies Condition (S) at x, by using the proof of Theorem 3.2, it is easy to verify that 1 τ(M, A; ȳ, x) = η(M, A; ȳ, x), here we use the convention that the infimum over the empty set is +∞ and the supremum over the empty set is 0.
As applications of main results in this paper, we are now in a position to consider calmness and strong calmness of the following multifunction (see Herion, 2001;Herion, Jourani & Outrata, 2002): where A is a closed subset of X, g : X → Y and D is a closed subset of Y.
Let ȳ ∈ Y and x ∈ G(ȳ).We set M(y) := g −1 (D − y) for any y ∈ Y. Then it is easy to verify that M is calm (resp.strongly calm) at (ȳ, x) over A implies the calmness (resp.strong calmness) of G at (ȳ, x).
The following proposition provides one sufficient condition for calmness of G in (33).
Therefore, v ∈ T c (A, a) if and only if for any a n A → a and any t n → 0 + , there exists v n → v such that a n + t n v n ∈ A for all n.
A→ a means that x → a with x ∈ A.