# Metric Subregularity for Subsmooth Generalized Constraint Equations in Banach Spaces

Authors: He Qinghai; Yang Ji; Zhang Binbin
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

This paper is devoted to metric subregularity of a kind of generalized constraint equations. In particular, in terms of coderivatives and normal cones, we provide some necessary and sufficient conditions for subsmooth generalized constraint equations to be metrically subregular and strongly metrically subregular in general Banach spaces and Asplund spaces, respectively.


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## Body

## 1. Introduction

LetX be a Banach space and f:X→R¯ be a function. Consider the following inequality:
(1.1)f(x)≤0.
Let S:={x∈X:f(x)≤0}. Recall that (1.1) has a local error bound at a∈S if there exist τ,δ∈(0,+∞) such that
(1.2)d(x,S)≤τ[f(x)]+∀x∈B(a,δ),
where [f(x)]+:=max{f(x),0} and B(a,δ) denotes the open ball of center a and radius δ. The error bound has been studied by many authors (see [1–3] and the references therein).LetY be another Banach space, b∈Y, and let F:X⇉Y be a closed multifunction. The following generalized equation:
(GE)b∈F(x)
concludes most of systems in optimization and was investigated by many researchers (see [4–9] and the references therein). Let x∈X and b∈F(a). Recall that (GE) is metrically subregular at (a,b) if there exist τ,δ∈(0,∞) such that
(1.3)d(x,F-1(b))≤τd(b,F(x))∀x∈B(a,δ)
(see [4–6] and the references therein). This property provides an estimate how far for an element x near a can be from the solution set of (GE). A stronger notion is the metric regularity: a multifunction F is metrically regular at (a,b) if there exist τ,δ∈(0,+∞) such that
(1.4)d(x,F-1(y))≤τd(y,F(x))∀(x,y)∈B((a,b),δ).
There exists a wide literature on this topic. We refer the interested readers to [3, 7–11] and to the references contained therein. Let A be a closed subset of X. Consider the generalized constraint equation as follows:
(GCE)b∈F(x)subjecttox∈A.LetS denote the solution set of (GCE), that is, S={x∈A:b∈F(x)}. We say that (GCE) is metrically subregular at a∈S if there exist τ,δ∈(0,∞) such that
(1.5)d(x,S)≤τ(d(b,F(x))+d(x,A))∀x∈B(a,δ).WhenA=X, (GCE) reduces (GE) and (1.5) means that (GE) is metrically subregular at (a,b). When F(x)=[f(x),+∞),b=0 and A=X, (GCE) reduces the inequality (1.1) and (1.5) means that this inequality has a local error bound at a. Error bounds, metric subregularity and regularity have important applications in mathematical programming and have been extensively studied (see [1–12] and the references therein). The Authors [13] introduced the notions of primal smoothness and investigated the properties of primal smooth functions. Under proper conditions, the distance function is primal smooth. Differentiability of the distance function was discussed in [14]. As extension of primal smoothness and convexity, the notion of subsmoothness was introduced and some functional characterizations were provided in [15]. Recently, by variational analysis techniques (for more details, see [16–19]), Zheng and Ng [6] investigated metric subregularity of (GE) under the subsmooth assumption. In this paper, in terms of normal cones and coderivatives, we devote to metric subregularity of generalized constraint equation (GCE) under the subsmooth assumption. We will build some new necessary and sufficient conditions for (GCE) to be metrically subregular and strongly metrically subregular.


## 2. Preliminaries

LetX be a Banach space. We denote by BX and X* the closed unit ball and the dual space of X, respectively. Let A be a nonempty subset of X, int(A) and bd(A), respectively, denote the interior and the boundary of A. For a∈X and δ>0, let B(a,δ) denote the open ball with center a and radius δ.We introduce some notions of variations and derivatives needed to state our results.For a closed subsetA of X and a∈A, let Tc(A,a) and T(A,a), respectively, denote the Clarke tangent cone and contingent (Bouligand) cone of A at a defined by
(2.1)Tc(A,a):=liminfx→Aa,t→0+A-xt,T(A,a):=limsupt→0+A-at,
where x→Aa means that x→a with x∈A. It is easy to verify that v∈Tc(A,a) if and only if for each sequence {an} in A converging to a and each sequence {tn} in (0, ∞) decreasing to 0, there exists a sequence {vn} in X converging to v such that an+tnvn∈A for each natural number n; while v∈T(A,a) if and only if there exists a sequence {vn} in X converging to v and a sequence {tn} in (0,∞) decreasing to 0 such that a+tnvn∈A for all n.We denote byNc(A,a) the Clarke normal cone of A at a, that is,
(2.2)Nc(A,a):={x*∈X*:〈x*,h〉≤0∀h∈Tc(A,a)}.
For ε≥0 and a∈A, the nonempty set
(2.3)N^ε(A,a):={x*∈X*:limsupx→Aa〈x*,x-a〉∥x-a∥≤ε}
is called the set of Fréchetε-normals of A at a. When ε=0, N^ε(A,a) is a convex cone which is called the Fréchet normal cone of A at a and is denoted by N^(A,a). Let N(A,a) denote the Mordukhovich limiting or basic normal cone of A at a, that is,
(2.4)N(A,a):=limsupx→Aa,ε→0+N^ε(A,x),
that is, x*∈N(A,a) if and only if there exist sequences {(xn,εn,xn*)} in A×R+×X* such that (xn,εn)→(a,0),xn*→w*x* and xn*∈N^εn(A,xn) for each natural number n. It is known that
(2.5)N^(A,a)⊆N(A,a)⊆Nc(A,a),
(see [4, 9, 16, 18, 19] and the references contained therein). If A is convex, then
(2.6)T(A,a)=Tc(A,a),N^(A,a)=N(A,a)=Nc(A,a)={x*∈X*:〈x*,x-a〉≤0∀x∈A}.
Recall that a Banach space X is called an Asplund space if every continuous convex function on X is Fréchet differentiable at each point of a dense subset of X (for other definitions and their equivalents, see [19]). It is well known that X is an Asplund space if and only if every separable subspace of X has a separable dual space. In particular, every reflexive Banach space is an Asplund space. When X is an Asplund space, it is well known that
(2.7)Nc(A,a)=cl*(co(N(A,a))),N(A,a)=limsupx→AaN^(A,x),
where cl*(·) denotes the closure with respect to the weak* topology, see [9, 19]. Recently, Zheng and Ng [5] established an approximate projection result for a closed subset of X, which will play a key role in the proofs of our main results.Lemma 2.1.
LetA be a closed nonempty subset of a Banach space X and let β∈(0,1). Then for any x∉A there exist a∈bd(A) and a*∈Nc(A,a) with ∥a*∥=1 such that
(2.8)β∥x-a∥<min{d(x,A),〈a*,x-a〉}.
If X is an Asplund space, then Nc(A,a) can be replaced by N^(A,a).LetF:X⇉Y be a multifunction and let Gr(F) denote the graph of F, that is,
(2.9)Gr(F):={(x,y)∈X×Y:y∈F(x)}.
As usual, F is said to be closed (resp., convex) if Gr(F) is a closed (resp., convex) subset of X×Y. Let (x,y)∈Gr(F). The Clarke tangent and contingent derivatives DcF(x,y),DF(x,y) of F at (x,y) are defined by
(2.10)DcF(x,y)(u):={v∈Y:(u,v)∈Tc(Gr(F),(x,y))}∀u∈X,DF(x,y)(u):={v∈Y:(u,v)∈T(Gr(F),(x,y))}∀u∈X,
respectively. Let D^*F(x,y),D*F(x,y), and Dc*F(x,y) denote the coderivatives of F at (x,y) associated with the Fréchet, Mordukhovich, and Clarke normal structures, respectively. They are defined by the following:
(2.11)D^*F(x,y)(y*):={x*∈X*:(x*,-y*)∈N^(Gr(F),(x,y))}∀y*∈Y*,D*F(x,y)(y*):={x*∈X*:(x*,-y*)∈N(Gr(F),(x,y))}∀y*∈Y*,Dc*F(x,y)(y*):={x*∈X*:(x*,-y*)∈Nc(Gr(F),(x,y))}∀y*∈Y*.
The more details of the coderivatives can be found in [9, 18, 19] and the references therein.


## 3. Subsmooth Generalized Constraint Equation

LetA be a closed subset of X. Recall (see [13, 14]) that A is said to be prox-regular at a∈A if there exist τ,δ>0 such that
(3.1)〈x*-u*,x-u〉≥-τ∥x-u∥2
whenever x,u∈A∩B(a,δ),x*∈Nc(A,x)∩BX*, and u*∈Nc(A,u)∩BX*.As a generalization of the prox-regularity, Aussel et al. [15] introduced and studied the subsmoothness. A is said to be subsmooth at a∈A if for any ε>0 there exist τ,δ>0 such that
(3.2)〈x*-u*,x-u〉≥-ε∥x-u∥,
whenever x,u∈A∩B(a,δ),x*∈Nc(A,x)∩BX*, and u*∈Nc(A,u)∩BX*.It is easy to verify thatA is subsmooth at a∈A if and only if for any ε>0, there exists δ>0 such that
(3.3)〈u*,x-u〉≤ε∥x-u∥
whenever x,u∈A∩B(a,δ) and u*∈Nc(A,u)∩BX*.LetF:X⇉Y be a closed multifunction, b∈Y and a∈F-1(b). Zheng and Ng [6] introduce the concept of the L-subsmoothness of F at a for b: F is called to be L-subsmooth at a for b if for any ε>0 there exists δ>0 such that
(3.4)〈u*,x-a〉+〈v*,y-v〉≤ε(∥x-a∥+∥y-v∥),
whenever v∈F(a)∩B(b,δ),(u*,v*)∈Nc(Gr(F),(a,v))∩BX*×Y*and(x,y)∈Gr(F)with∥x-a∥+∥y-b∥<δ. Next, we introduce the concept of the subsmoothness of generalized constraint equation (GCE) which will be useful in our discussion.Definition 3.1.
Generalized equation(GCE) is subsmooth at a∈S if for any ε>0, there exists δ>0 such that
(3.5)〈u*,x-u〉-〈v*,y-b〉≤ε(∥x-u∥+∥y-b∥),〈w*,x′-u〉≤ε∥x′-u∥,
whenever x∈B(a,δ),x′∈A∩B(a,δ),u∈S∩B(a,δ),y∈F(x)∩B(b,δ),v*∈BY*,u*∈Dc*F(u,b)(v*)∩BX*,andw*∈Nc(A,u)∩BX*.Remark 3.2.
The subsmoothness of(GCE) at a means the subsmoothness of F at a for b when A=X, while the subsmoothness of (GCE) at a means the subsmoothness of A at a when F(x)=b for all x∈X. If A=X and Gr(F) is prox-regular at (a,b), then generalized equation (GCE) is subsmooth at a. If A and Gr(F) are convex, then F is also subsmooth at a. Finally when A is prox-regular and F is single-valued and smooth, (GCE) is subsmooth at a, too. Hence, Definition 3.1 extends notions of smoothness, convexity and prox-regularity.Proposition 3.3.
Suppose that Generalized equation(GCE) is subsmooth at a∈S. Then for any ε>0 there exists δ>0 such that
(3.6)〈u*,x-u〉≤(2+ε)d(b,F(x))+ε∥x-u∥,(3.7)〈w*,x-u〉≤d(x,A)+ε∥x-u∥,
whenever x∈B(a,δ),u∈S∩B(a,δ), u*∈Dc*F(u,b)(BY*)∩BX*, and w*∈Nc(A,u)∩BX*.Proof.
Suppose that(GCE) is subsmooth at a∈S. Then for any ε>0, there exists δ>0 such that
(3.8)〈u*,x-u〉-〈v*,y-b〉≤ε2(∥x-u∥+∥y-b∥),〈w*,x′-u〉≤ε2∥x′-u∥,
whenever x∈B(a,2δ),x′∈A∩B(a,2δ),u∈S∩B(a,2δ),y∈F(x)∩B(b,2δ),v*∈BY*,u*∈Dc*F(u,b)(v*)∩BX*,andw*∈Nc(A,u)∩BX*.
Letx∈B(a,δ),u∈S∩B(a,δ),v*∈BY*,u*∈Dc*F(u,b)(v*)∩BX*,andw*∈Nc(A,u)∩BX*. If F(x)∩B(b,δ)=∅, then
(3.9)〈u*,x-u〉≤∥x-u∥≤∥x-a∥+∥a-u∥≤2δ,d(b,F(x))≥δ.
Thus, (3.6) holds. Otherwise, one has
(3.10)〈u*,x-u〉≤〈v*,y-b〉+ε2(∥x-u∥+∥y-b∥)≤(1+ε)∥y-b∥+ε∥x-u∥,
whenever y∈F(x)∩B(b,δ). Noting that d(b,F(x))=d(b,F(x)∩B(b,δ)) (sine F(x)∩B(b,δ)≠∅), it follows that
(3.11)〈u*,x-u〉≤(1+ε)d(b,F(x))+ε∥x-u∥.
It remains to show that (3.7) holds. Since
(3.12)〈w*,x-u〉=〈w*,x-x′〉+〈w*,x′-u〉≤∥x-x′∥+ε2∥x′-u∥≤(1+ε2)∥x-x′∥+ε2∥x-u∥,
whenever x′∈A∩B(a,δ). One has
(3.13)〈w*,x-u〉≤d(x,A)+ε2(d(x,A)+∥x-u∥).
Noting that u∈A, it follows that d(x,A)≤∥x-u∥ which implies that (3.7) holds and completes the proof.


## 4. Main Results

This section is devoted to metric subregularity of generalized equation(GCE). We divide our discussion into two subsections addressing the necessary conditions and the sufficient conditions for metric subregularity.


### 4.1. Necessary Conditions for Metric Subregularity

There are two results in this subsection: one is on the Banach space setting and the other on the Asplund space setting.Theorem 4.1.
Suppose thatX,Y are Banach spaces and that generalized equation (GCE) is metrically subregular at a∈S. Then there exist τ,δ∈(0,+∞) such that
(4.1)N^(S,u)∩BX*⊆τ(Dc*F(u,b)(BY*)+Nc(A,u)∩BX*)∀u∈S∩B(a,δ).Proof.
LetδGr(F) denote the indicator function of Gr(F) and δ>0 such that (1.5) holds. Then (1.5) can be rewritten as
(4.2)d(x,S)≤δGr(F)(x,y)+τ(∥y-b∥+d(x,A))∀(x,y)∈B(a,δ)×Y.
Let u∈S∩B(a,δ) and u*∈N^(S,u)∩BX*. Noting (cf. [9, Corollary 1.96]) that N^(S,u)∩BX*=∂^d(·,S)(u), one gets that for any natural number n, there exists r∈(0,δ) such that B(u,r)⊆B(a,δ) and
(4.3)〈u*,x-u〉≤d(x,S)+1n∥x-u∥∀x∈B(u,r).
Hence, by (4.2), it follows that
(4.4)〈u*,x-u〉≤δGr(F)(x,y)+τ∥y-b∥+τd(x,A)+1n∥x-u∥∀(x,y)∈B(u,r)×Y,
that is, (u,b) is a local minimizer of ϕ defined by
(4.5)ϕ(x,y):=-〈u*,x-u〉+δGr(F)(x,y)+τ∥y-b∥+τd(x,A)+1n∥x-u∥∀(x,y)∈X×Y.
Hence, (0,0)∈∂cϕ(u,b). It follows from [16] that
(4.6)(0,0)∈(-u*,0)+Nc(Gr(F),(u,b))+{0}×τBY*+τ∂cd(·,A)(u)×{0}+1nBX*×{0},
that is,
(4.7)(1τu*+1τnxn*,-yn*)∈Nc(Gr(F),(u,b))+∂cd(·,A)(u)×{0},
for some xn*∈BX* and yn*∈BY*. Since BX* and BY* are weak* compact, without loss of generality (otherwise take a generalized subsequence), we can assume xn*→w*x*,yn*→w*y* for some x*∈BX* and y*∈BY* as n→∞. Noting that
(4.8)Nc(Gr(F),(u,b))+∂cd(·,A)(u)×{0}
is weak* closed (since Nc(Gr(F),(u,b)) is weak* closed and ∂cd(·,A)(u)×{0} is weak* compact), one has
(4.9)(u*τ,-y*)∈Nc(Gr(F),(u,b))+∂cd(·,A)(u)×{0}.
This implies that
(4.10)u*∈τ(Dc*F(u,b)(BY*)+Nc(A,u)∩BX*).
This shows that (4.1) holds true. The proof is completed.WhenX and Y are Asplund spaces, the conclusion in Theorem 4.1 can be strengthened with Dc*F(u,b)(BY*) and Nc(A,u)∩BX*) replaced by D*F(u,b)(BY*) and N(A,u)∩BX*), respectively. Its proof is similar to that of Theorem 4.1.Theorem 4.2.
Suppose thatX and Y are Asplund spaces and that generalized equation (GCE) is metrically subregular at a∈S. Then there exist τ,δ∈(0,+∞) such that
(4.11)N^(S,u)∩BX*⊆τ(D*F(u,b)(BY*)+N(A,u)∩BX*)∀u∈S∩B(a,δ).


### 4.2. Sufficient Conditions for Metric Subregularity

Under the subsmooth assumption, we will show in the next result that some conditions similar to (4.1) turns out to be sufficient conditions for metric subregularity.Theorem 4.3.
LetX and Y be Banach spaces. Suppose that generalized constraint equation (GCE) is subsmooth at a and that there exist τ,δ∈(0,+∞) such that
(4.12)Nc(S,u)∩BX*⊆τ(Dc*F(u,b)(BY*)+Nc(A,u)∩BX*),
whenever u∈bd(S)∩B(a,δ). Then (GCE) is metrically subregular at a and, more precisely, for any ε∈(0,1/(2(1+τ))) there exists δε∈(0,δ/2) such that
(4.13)d(x,S)≤(1+τ)(2+ε)1-2(1+τ)ε(d(b,F(x))+d(x,A))∀x∈B(a,δε).Proof.
Letε∈(0,1/(2(1+τ))). Then, by subsmooth assmption of (GCE) at a and Proposition 3.3, there exists δ′∈(0,δ/2) such that
(4.14)〈u1*,x-u〉≤(2+ε)d(b,F(x))+ε∥x-u∥,〈u2*,x-u〉≤d(x,A)+ε∥x-u∥,
whenever x∈B(a,δ′),u∈S∩B(a,δ′),u1*∈Dc*F(u,b)(BY*)∩BX*andu2*∈Nc(A,u)∩BX*.
Letδε∈(0,δ′/2) and x∈B(a,δε)∖S. Now we need only show (4.13).(i)
IfF(x)∩B(b,δ′)=∅, then d(x,S)≤∥x-a∥<δε, d(b,F(x))≥δ′. Hence (4.13) holds.(ii)
SupposeF(x)∩B(b,δ′)≠∅ and let
(4.15)β∈(max{d(x,S)δε,2(1+τ)ε,12},1).By Lemma 2.1 there exist u0∈bd(S) and u*∈Nc(S,u0) with ∥u*∥=1 such that
(4.16)β∥x-u0∥≤min{〈u*,x-u0〉,d(x,S)}.
Thus, ∥x-u0∥≤(d(x,S)/β)<δε. Hence,
(4.17)∥u0-a∥≤∥u0-x∥+∥x-a∥<2δε<δ′<δ.
By (4.12) there exist y1*∈BY*,x1*∈Dc*F(u0,b)(y1*), and x2*∈Nc(A,u0)∩BX* such that u*=τ(x1*+x2*). Applying (4.14) with ((τ/(1+τ))x1*,(τ/(1+τ))x2*,u0) in place of (u1*,u2*,u), it follows that
(4.18)〈u*,x-u0〉=τ(〈x1*,x-u0〉+〈x2*,x-u0〉)≤(1+τ)(2+ε)(d(b,F(x))+d(x,A))+2(1+τ)ε∥x-u0∥.
This and (4.16) imply that
(4.19)d(x,S)≤∥x-u0∥≤(1+τ)(2+ε)β-2(1+τ)ε(d(b,F(x))+d(x,A)).
Letting β→1, it follows that (4.13) holds. The proof is completed.WhenX and Y are Asplund spaces, the assumption in Theorem 4.3 can be weakened with Nc(S,u) replaced by N^(S,u).Theorem 4.4.
SupposeX and Y are Asplund spaces. Suppose that generalized constraint equation (GCE) is subsmooth at a and that there exist τ,δ∈(0,+∞) such that
(4.20)N^(S,u)∩BX*⊆τ(Dc*F(u,b)(BY*)+Nc(A,u)∩BX*),
whenever u∈bd(S)∩B(a,δ). Then for any ε>0 there exists δε>0 such that (4.13) holds.With the Asplund space version of Lemma2.1 applied in place of the Banach space version, similar to the proof of Theorem 4.3, it is easy to verify Theorem 4.4.In general,(GCE) is not necessarily metrically subregular at a if (GCE) only has that Nc(S,a)∩BX*⊆τ(Dc*F(a,b)(BY*)+Nc(A,a)∩BX*).Finally, we end this subsection with a sufficient and necessary condition for the Clarke tangent derivative mappingDcF(a,b) to be metrically subregular at 0 for 0 over the Clerke tangent cone Tc(A,a).Let(4.21)τ(F,a,b;A):=inf{τ>0:thereexistsδ>0suchthat(1.5)holds}.
For u∈S, let
(4.22)γ(F,u,b;A):=inf{τ>0:Nc(S,u)∩BX*⊆τ(Dc*F(a,b)(BY*)+Nc(A,u)∩BX*)}.
The following lemma is known ([5, Theorem 3.2]) and useful for us in the sequel.Lemma 4.5.
Assume thatF:X⇉Y is a closed convex multifunction, A is a closed convex subset of X, and a∈S. And suppose that there exist a cone C and a neighborhood V of a such that S∩V=(a+C)∩V. Then,
(4.23)τ(F,a,b;A)=γ(F,a,b;A).
Consequently, (GCE) is metrically subregular at a if and only if γ(F,a,b;A)<+∞.Theorem 4.6.
Leta∈S and
(4.24)τ:=inf{τ>0:d(h,Tc(S,a))≤τ(d(0,DcF(a,b)(h))+d(h,Tc(A,a)))∀h∈X}.
Suppose that
(4.25)Tc(S,a)⊆Tc(A,a)∩DcF(a,b)-1(0).
Then,
(4.26)τ=γ(F,a,b;A).
If, in addition, τ<+∞, then
(4.27)Tc(S,a)=Tc(A,a)∩DcF(a,b)-1(0).
Consequently, DcF(a,b) is metrically subregular at (0,0) over Tc(A,a) if and only if γ(F,a,b;A)<+∞.Proof.
First, we assume thatτ<+∞. By the definition of τ, we have
(4.28)d(x,Tc(S,a))≤τ(d(0,DcF(a,b)(x))+d(x,Tc(A,a)))∀x∈X.
This implies that
(4.29)Tc(A,a)∩DcF(a,b)-1(0)⊆Tc(S,a).
This and (4.25) imply that (4.27) holds.
We consider the following constraint equation:(GCE′)0∈DcF(a,b)(x)subjecttox∈Tc(A,a).
Let S′ denote the solution set of (GCE′). Then,
(4.30)S′=Tc(A,a)∩DcF(a,b)-1(0).
Noting that
(4.31)Nc(S′,0)=Nc(Tc(S,a),0)=Nc(S,a),Nc(Gr(F),(a,b))=Nc(Gr(DcF(a,b)),(0,0)),
it is straightforward to verify that
(4.32)τ=τ(DcF(a,b),0,0;Tc(A,a)),γ(F,a,b;A)=γ(DcF(a,b),0,0;Tc(A,a)).
On the other hand, since DcF(a,b) is a closed convex multifunction from X to Y and Tc(A,a) is a closed convex cone, Lemma 4.5 implies that
(4.33)τ(DcF(a,b),0,0;Tc(A,a))=γ(DcF(a,b),0,0;Tc(A,a)).
This gives us τ=γ(F,a,b;A).
It remains to show thatγ(F,a,b;A)=+∞ when τ=+∞. Suppose that
(4.34)γ(F,a,b;A)<+∞.
We need only show that τ<+∞. Let x∈X∖Tc(S,a) and β∈(0,1). By Lemma 2.1 there exist u∈Tc(S,a) and x*∈Nc(Tc(S,a),u) with ∥x*∥=1 such that
(4.35)β∥x-u∥≤〈x*,x-u〉.
Noting that Tc(S,a) is a convex cone, it is easy to verify that
(4.36)x*∈Nc(Tc(S,a),0)=Nc(S,a),〈x*,u〉=0.
Take a fixed r in (γ(F,a,b;A),+∞). Then there exist y*∈rBY*,x1*∈Dc*F(a,b)(y*) and x2*∈Nc(A,a)∩rBX* such that
(4.37)x*=x1*+x2*.
We equip the product space X×Y with norm
(4.38)∥(x,y)∥r:=r1+r∥x∥+∥y∥∀(x,y)∈X×Y.
Noting that the unit ball of the dual space of (X×Y,∥·∥r) is (((1+r)/r)BX*)×BY*, it follows from the convexity of DcF(x,y) and Tc(A,a) that
(4.39)1r(x1*,-y*)∈Nc(Gr(F),(a,b))∩((1+rrBX*)×BY*)=Nc(Gr(DcF(a,b)),(0,0))∩((1+rrBX*)×BY*)=∂cd∥·∥r(·,Gr(DcF(a,b)))(0,0),1rx2*∈Nc(A,a)∩BX*=Nc(Tc(A,a),0)∩BX*=∂cd(·,Tc(A,a))(0).
Hence,
(4.40)1r〈x1*,x〉≤d∥·∥r((x,0),Gr(DcF(a,b)))≤d(0,DcF(a,b)(x)),1r〈x1*,x〉1r〈x2*,x〉≤d(x,Tc(A,a)),
whenever x∈X. Noting that 〈x*,u〉=0, it follows from (4.35) that
(4.41)β∥x-u∥r≤d(0,DcF(a,b)(x))+d(x,Tc(A,a)).
Therefore,
(4.42)βd(x,Tc(S,a))r≤d(0,DcF(a,b)(x))+d(x,Tc(A,a)).
Letting β→1, one has
(4.43)d(x,Tc(S,a))≤r(d(0,DcF(a,b)(x))+d(x,Tc(A,a))).
This contradicts with τ=+∞. The proof is completed.


### 4.3. Strongly Metric Subregularity

LetF:X⇉Y be a multifunction and b∈F(a). Recall that F is strongly subregular at a if there exist τ∈(0,+∞), neighborhoods U of a, and V of b such that
(4.44)∥x-a∥≤τd(b,F(x)∩V)∀x∈U.
It is clear that this definition is equivalent to the next one when A=X.Definition 4.7.
One says that generalized constraint equation(GCE) is strongly metrically subregular at a if there exists τ,δ∈(0,∞) such that
(4.45)∥x-a∥≤τ(d(b,F(x))+d(x,A))∀x∈B(a,δ).
It is clear that (GCE) is strongly metrically subregular at a if and only if a is an isolated point of S (i.e., S∩B(a,r)={a} for some r>0) and it is metrically subregular at a. Thus, if (GCE) is strongly metrically subregular at a, Then Nc(S,a)=X*. We immediately have the following Corollary 4.8 from Theorem 4.1.Corollary 4.8.
Suppose that there existsτ,δ∈(0,∞) such that (4.45) holds. Then,
(4.46)BX*⊆τ(Dc*F(a,b)(BY*)+Nc(A,a)∩BX*).Applying Theorem4.3, one obtains a sufficient condition for (GCE) to be strongly metrically subregular at a.Corollary 4.9.
LetX,Y be Banach spaces. Suppose that generalized constraint equation (GCE) is subsmooth at a and that there exists τ∈(0,+∞) such that
(4.47)BX*⊆τ(Dc*F(a,b)(BY*)+Nc(A,a)∩BX*).
Then (GCE) is strongly metrically subregular at a and, more precisely, for any ε∈(0,1/(1+2τ)) there exists δε>0 such that
(4.48)∥x-a∥≤(1+τ)(2+ε)1-2(1+τ)ε(d(b,F(x))+d(x,A))∀x∈B(a,δε).Proof.
From Theorem4.3, we need only show that S∩B(a,δ)={a} for some δ>0. Since the assumption that (GCE) is subsmooth at a, by Proposition 3.3, for any ε∈(0,1/2(1+τ)), there exists δ>0 such that
(4.49)〈a1*,x-a〉≤(2+ε)d(b,F(x))+ε∥x-u∥,(4.50)〈a2*,x-a〉≤d(x,A)+ε∥x-u∥,
whenever x∈B(a,δ),a1*∈Dc*F(a,b)(BY*)∩BX*anda2*∈Nc(A,a)∩BX*.
Take an arbitraryx*∈BX*. By (4.47), there exist x1*∈Dc*F(a,b)(BY*), x2*∈Nc(A,a)∩BX* such that x*=τ(x1*+x2*). Let x∈S∩B(a,δ). Applying (4.49) with (τ/(1+τ))x1* in place of a1*, it follows from this and (4.50) that we have
(4.51)τ〈x1*,x-a〉≤(1+τ)ε∥x-a∥,〈x2*,x-a〉≤ε∥x-a∥.
Then,
(4.52)〈x*,x-a〉=τ(〈x1*,x-a〉+〈x2*,x-a〉)≤(1+2τ)ε∥x-a∥.
And so,
(4.53)∥x-a∥=supx*∈BX*〈x*,x-a〉≤(1+2τ)ε∥x-a∥.
This shows that S∩B(a,δ)={a}. The proof is completed.From Corollaries4.8 and 4.9, we also have the following equivalent results.Corollary 4.10.
Suppose that generalized constraint equation(GCE) is subsmooth at a. Then the following statements are equivalent:(i)
(GCE) is strongly metrically subregular at a;(ii)
there existsτ∈(0,∞) such that BX*⊆τ(Dc*F(a,b)(BY*)+Nc(A,a)∩BX*));(iii)
0∈int(Dc*F(a,b)(Y*)+Nc(A,a));(iv)
X*=Dc*F(a,b)(Y*)+Nc(A,a);(v)
DcF(a,b) is strongly metrically subregular at 0 for 0 over Tc(A,a).Proof.
First, by Corollaries4.8 and 4.9, it is clear that (i)⇔(ii). Noting that Dc*(DcF(a,b))(0,0)(BY*)=Dc*F(a,b)(BY*), (ii)⇔(v) is immediate from (i)⇔(ii).
It is clear that (ii)⇔(iii). Noting that Nc(A,a) and Dc*F(a,b)(Y*) are cones, hence,
(4.54)Dc*F(a,b)(Y*)+Nc(A,a)=⋃n=1∞‍(Dc*F(a,b)(nBY*)+Nc(A,a)∩nBX*).
This shows that (ii)⇒(iv).
It remains to show that (iv)⇒(ii). Suppose that (iv) holds, by the Alaogu theorem, for each n, the set Dc*F(a,b)(nBY*)+Nc(A,a)∩nBX* is weakly star-closed, it follows from the well-known Baire category theorem and (iv) that
(4.55)0∈int(Dc*F(a,b)(BY*)+Nc(A,a)∩BX*).
Hence, (ii) holds. The proof is completed.


## 4.1. Necessary Conditions for Metric Subregularity

There are two results in this subsection: one is on the Banach space setting and the other on the Asplund space setting.Theorem 4.1.
Suppose thatX,Y are Banach spaces and that generalized equation (GCE) is metrically subregular at a∈S. Then there exist τ,δ∈(0,+∞) such that
(4.1)N^(S,u)∩BX*⊆τ(Dc*F(u,b)(BY*)+Nc(A,u)∩BX*)∀u∈S∩B(a,δ).Proof.
LetδGr(F) denote the indicator function of Gr(F) and δ>0 such that (1.5) holds. Then (1.5) can be rewritten as
(4.2)d(x,S)≤δGr(F)(x,y)+τ(∥y-b∥+d(x,A))∀(x,y)∈B(a,δ)×Y.
Let u∈S∩B(a,δ) and u*∈N^(S,u)∩BX*. Noting (cf. [9, Corollary 1.96]) that N^(S,u)∩BX*=∂^d(·,S)(u), one gets that for any natural number n, there exists r∈(0,δ) such that B(u,r)⊆B(a,δ) and
(4.3)〈u*,x-u〉≤d(x,S)+1n∥x-u∥∀x∈B(u,r).
Hence, by (4.2), it follows that
(4.4)〈u*,x-u〉≤δGr(F)(x,y)+τ∥y-b∥+τd(x,A)+1n∥x-u∥∀(x,y)∈B(u,r)×Y,
that is, (u,b) is a local minimizer of ϕ defined by
(4.5)ϕ(x,y):=-〈u*,x-u〉+δGr(F)(x,y)+τ∥y-b∥+τd(x,A)+1n∥x-u∥∀(x,y)∈X×Y.
Hence, (0,0)∈∂cϕ(u,b). It follows from [16] that
(4.6)(0,0)∈(-u*,0)+Nc(Gr(F),(u,b))+{0}×τBY*+τ∂cd(·,A)(u)×{0}+1nBX*×{0},
that is,
(4.7)(1τu*+1τnxn*,-yn*)∈Nc(Gr(F),(u,b))+∂cd(·,A)(u)×{0},
for some xn*∈BX* and yn*∈BY*. Since BX* and BY* are weak* compact, without loss of generality (otherwise take a generalized subsequence), we can assume xn*→w*x*,yn*→w*y* for some x*∈BX* and y*∈BY* as n→∞. Noting that
(4.8)Nc(Gr(F),(u,b))+∂cd(·,A)(u)×{0}
is weak* closed (since Nc(Gr(F),(u,b)) is weak* closed and ∂cd(·,A)(u)×{0} is weak* compact), one has
(4.9)(u*τ,-y*)∈Nc(Gr(F),(u,b))+∂cd(·,A)(u)×{0}.
This implies that
(4.10)u*∈τ(Dc*F(u,b)(BY*)+Nc(A,u)∩BX*).
This shows that (4.1) holds true. The proof is completed.WhenX and Y are Asplund spaces, the conclusion in Theorem 4.1 can be strengthened with Dc*F(u,b)(BY*) and Nc(A,u)∩BX*) replaced by D*F(u,b)(BY*) and N(A,u)∩BX*), respectively. Its proof is similar to that of Theorem 4.1.Theorem 4.2.
Suppose thatX and Y are Asplund spaces and that generalized equation (GCE) is metrically subregular at a∈S. Then there exist τ,δ∈(0,+∞) such that
(4.11)N^(S,u)∩BX*⊆τ(D*F(u,b)(BY*)+N(A,u)∩BX*)∀u∈S∩B(a,δ).


## 4.2. Sufficient Conditions for Metric Subregularity

Under the subsmooth assumption, we will show in the next result that some conditions similar to (4.1) turns out to be sufficient conditions for metric subregularity.Theorem 4.3.
LetX and Y be Banach spaces. Suppose that generalized constraint equation (GCE) is subsmooth at a and that there exist τ,δ∈(0,+∞) such that
(4.12)Nc(S,u)∩BX*⊆τ(Dc*F(u,b)(BY*)+Nc(A,u)∩BX*),
whenever u∈bd(S)∩B(a,δ). Then (GCE) is metrically subregular at a and, more precisely, for any ε∈(0,1/(2(1+τ))) there exists δε∈(0,δ/2) such that
(4.13)d(x,S)≤(1+τ)(2+ε)1-2(1+τ)ε(d(b,F(x))+d(x,A))∀x∈B(a,δε).Proof.
Letε∈(0,1/(2(1+τ))). Then, by subsmooth assmption of (GCE) at a and Proposition 3.3, there exists δ′∈(0,δ/2) such that
(4.14)〈u1*,x-u〉≤(2+ε)d(b,F(x))+ε∥x-u∥,〈u2*,x-u〉≤d(x,A)+ε∥x-u∥,
whenever x∈B(a,δ′),u∈S∩B(a,δ′),u1*∈Dc*F(u,b)(BY*)∩BX*andu2*∈Nc(A,u)∩BX*.
Letδε∈(0,δ′/2) and x∈B(a,δε)∖S. Now we need only show (4.13).(i)
IfF(x)∩B(b,δ′)=∅, then d(x,S)≤∥x-a∥<δε, d(b,F(x))≥δ′. Hence (4.13) holds.(ii)
SupposeF(x)∩B(b,δ′)≠∅ and let
(4.15)β∈(max{d(x,S)δε,2(1+τ)ε,12},1).By Lemma 2.1 there exist u0∈bd(S) and u*∈Nc(S,u0) with ∥u*∥=1 such that
(4.16)β∥x-u0∥≤min{〈u*,x-u0〉,d(x,S)}.
Thus, ∥x-u0∥≤(d(x,S)/β)<δε. Hence,
(4.17)∥u0-a∥≤∥u0-x∥+∥x-a∥<2δε<δ′<δ.
By (4.12) there exist y1*∈BY*,x1*∈Dc*F(u0,b)(y1*), and x2*∈Nc(A,u0)∩BX* such that u*=τ(x1*+x2*). Applying (4.14) with ((τ/(1+τ))x1*,(τ/(1+τ))x2*,u0) in place of (u1*,u2*,u), it follows that
(4.18)〈u*,x-u0〉=τ(〈x1*,x-u0〉+〈x2*,x-u0〉)≤(1+τ)(2+ε)(d(b,F(x))+d(x,A))+2(1+τ)ε∥x-u0∥.
This and (4.16) imply that
(4.19)d(x,S)≤∥x-u0∥≤(1+τ)(2+ε)β-2(1+τ)ε(d(b,F(x))+d(x,A)).
Letting β→1, it follows that (4.13) holds. The proof is completed.WhenX and Y are Asplund spaces, the assumption in Theorem 4.3 can be weakened with Nc(S,u) replaced by N^(S,u).Theorem 4.4.
SupposeX and Y are Asplund spaces. Suppose that generalized constraint equation (GCE) is subsmooth at a and that there exist τ,δ∈(0,+∞) such that
(4.20)N^(S,u)∩BX*⊆τ(Dc*F(u,b)(BY*)+Nc(A,u)∩BX*),
whenever u∈bd(S)∩B(a,δ). Then for any ε>0 there exists δε>0 such that (4.13) holds.With the Asplund space version of Lemma2.1 applied in place of the Banach space version, similar to the proof of Theorem 4.3, it is easy to verify Theorem 4.4.In general,(GCE) is not necessarily metrically subregular at a if (GCE) only has that Nc(S,a)∩BX*⊆τ(Dc*F(a,b)(BY*)+Nc(A,a)∩BX*).Finally, we end this subsection with a sufficient and necessary condition for the Clarke tangent derivative mappingDcF(a,b) to be metrically subregular at 0 for 0 over the Clerke tangent cone Tc(A,a).Let(4.21)τ(F,a,b;A):=inf{τ>0:thereexistsδ>0suchthat(1.5)holds}.
For u∈S, let
(4.22)γ(F,u,b;A):=inf{τ>0:Nc(S,u)∩BX*⊆τ(Dc*F(a,b)(BY*)+Nc(A,u)∩BX*)}.
The following lemma is known ([5, Theorem 3.2]) and useful for us in the sequel.Lemma 4.5.
Assume thatF:X⇉Y is a closed convex multifunction, A is a closed convex subset of X, and a∈S. And suppose that there exist a cone C and a neighborhood V of a such that S∩V=(a+C)∩V. Then,
(4.23)τ(F,a,b;A)=γ(F,a,b;A).
Consequently, (GCE) is metrically subregular at a if and only if γ(F,a,b;A)<+∞.Theorem 4.6.
Leta∈S and
(4.24)τ:=inf{τ>0:d(h,Tc(S,a))≤τ(d(0,DcF(a,b)(h))+d(h,Tc(A,a)))∀h∈X}.
Suppose that
(4.25)Tc(S,a)⊆Tc(A,a)∩DcF(a,b)-1(0).
Then,
(4.26)τ=γ(F,a,b;A).
If, in addition, τ<+∞, then
(4.27)Tc(S,a)=Tc(A,a)∩DcF(a,b)-1(0).
Consequently, DcF(a,b) is metrically subregular at (0,0) over Tc(A,a) if and only if γ(F,a,b;A)<+∞.Proof.
First, we assume thatτ<+∞. By the definition of τ, we have
(4.28)d(x,Tc(S,a))≤τ(d(0,DcF(a,b)(x))+d(x,Tc(A,a)))∀x∈X.
This implies that
(4.29)Tc(A,a)∩DcF(a,b)-1(0)⊆Tc(S,a).
This and (4.25) imply that (4.27) holds.
We consider the following constraint equation:(GCE′)0∈DcF(a,b)(x)subjecttox∈Tc(A,a).
Let S′ denote the solution set of (GCE′). Then,
(4.30)S′=Tc(A,a)∩DcF(a,b)-1(0).
Noting that
(4.31)Nc(S′,0)=Nc(Tc(S,a),0)=Nc(S,a),Nc(Gr(F),(a,b))=Nc(Gr(DcF(a,b)),(0,0)),
it is straightforward to verify that
(4.32)τ=τ(DcF(a,b),0,0;Tc(A,a)),γ(F,a,b;A)=γ(DcF(a,b),0,0;Tc(A,a)).
On the other hand, since DcF(a,b) is a closed convex multifunction from X to Y and Tc(A,a) is a closed convex cone, Lemma 4.5 implies that
(4.33)τ(DcF(a,b),0,0;Tc(A,a))=γ(DcF(a,b),0,0;Tc(A,a)).
This gives us τ=γ(F,a,b;A).
It remains to show thatγ(F,a,b;A)=+∞ when τ=+∞. Suppose that
(4.34)γ(F,a,b;A)<+∞.
We need only show that τ<+∞. Let x∈X∖Tc(S,a) and β∈(0,1). By Lemma 2.1 there exist u∈Tc(S,a) and x*∈Nc(Tc(S,a),u) with ∥x*∥=1 such that
(4.35)β∥x-u∥≤〈x*,x-u〉.
Noting that Tc(S,a) is a convex cone, it is easy to verify that
(4.36)x*∈Nc(Tc(S,a),0)=Nc(S,a),〈x*,u〉=0.
Take a fixed r in (γ(F,a,b;A),+∞). Then there exist y*∈rBY*,x1*∈Dc*F(a,b)(y*) and x2*∈Nc(A,a)∩rBX* such that
(4.37)x*=x1*+x2*.
We equip the product space X×Y with norm
(4.38)∥(x,y)∥r:=r1+r∥x∥+∥y∥∀(x,y)∈X×Y.
Noting that the unit ball of the dual space of (X×Y,∥·∥r) is (((1+r)/r)BX*)×BY*, it follows from the convexity of DcF(x,y) and Tc(A,a) that
(4.39)1r(x1*,-y*)∈Nc(Gr(F),(a,b))∩((1+rrBX*)×BY*)=Nc(Gr(DcF(a,b)),(0,0))∩((1+rrBX*)×BY*)=∂cd∥·∥r(·,Gr(DcF(a,b)))(0,0),1rx2*∈Nc(A,a)∩BX*=Nc(Tc(A,a),0)∩BX*=∂cd(·,Tc(A,a))(0).
Hence,
(4.40)1r〈x1*,x〉≤d∥·∥r((x,0),Gr(DcF(a,b)))≤d(0,DcF(a,b)(x)),1r〈x1*,x〉1r〈x2*,x〉≤d(x,Tc(A,a)),
whenever x∈X. Noting that 〈x*,u〉=0, it follows from (4.35) that
(4.41)β∥x-u∥r≤d(0,DcF(a,b)(x))+d(x,Tc(A,a)).
Therefore,
(4.42)βd(x,Tc(S,a))r≤d(0,DcF(a,b)(x))+d(x,Tc(A,a)).
Letting β→1, one has
(4.43)d(x,Tc(S,a))≤r(d(0,DcF(a,b)(x))+d(x,Tc(A,a))).
This contradicts with τ=+∞. The proof is completed.


## 4.3. Strongly Metric Subregularity

LetF:X⇉Y be a multifunction and b∈F(a). Recall that F is strongly subregular at a if there exist τ∈(0,+∞), neighborhoods U of a, and V of b such that
(4.44)∥x-a∥≤τd(b,F(x)∩V)∀x∈U.
It is clear that this definition is equivalent to the next one when A=X.Definition 4.7.
One says that generalized constraint equation(GCE) is strongly metrically subregular at a if there exists τ,δ∈(0,∞) such that
(4.45)∥x-a∥≤τ(d(b,F(x))+d(x,A))∀x∈B(a,δ).
It is clear that (GCE) is strongly metrically subregular at a if and only if a is an isolated point of S (i.e., S∩B(a,r)={a} for some r>0) and it is metrically subregular at a. Thus, if (GCE) is strongly metrically subregular at a, Then Nc(S,a)=X*. We immediately have the following Corollary 4.8 from Theorem 4.1.Corollary 4.8.
Suppose that there existsτ,δ∈(0,∞) such that (4.45) holds. Then,
(4.46)BX*⊆τ(Dc*F(a,b)(BY*)+Nc(A,a)∩BX*).Applying Theorem4.3, one obtains a sufficient condition for (GCE) to be strongly metrically subregular at a.Corollary 4.9.
LetX,Y be Banach spaces. Suppose that generalized constraint equation (GCE) is subsmooth at a and that there exists τ∈(0,+∞) such that
(4.47)BX*⊆τ(Dc*F(a,b)(BY*)+Nc(A,a)∩BX*).
Then (GCE) is strongly metrically subregular at a and, more precisely, for any ε∈(0,1/(1+2τ)) there exists δε>0 such that
(4.48)∥x-a∥≤(1+τ)(2+ε)1-2(1+τ)ε(d(b,F(x))+d(x,A))∀x∈B(a,δε).Proof.
From Theorem4.3, we need only show that S∩B(a,δ)={a} for some δ>0. Since the assumption that (GCE) is subsmooth at a, by Proposition 3.3, for any ε∈(0,1/2(1+τ)), there exists δ>0 such that
(4.49)〈a1*,x-a〉≤(2+ε)d(b,F(x))+ε∥x-u∥,(4.50)〈a2*,x-a〉≤d(x,A)+ε∥x-u∥,
whenever x∈B(a,δ),a1*∈Dc*F(a,b)(BY*)∩BX*anda2*∈Nc(A,a)∩BX*.
Take an arbitraryx*∈BX*. By (4.47), there exist x1*∈Dc*F(a,b)(BY*), x2*∈Nc(A,a)∩BX* such that x*=τ(x1*+x2*). Let x∈S∩B(a,δ). Applying (4.49) with (τ/(1+τ))x1* in place of a1*, it follows from this and (4.50) that we have
(4.51)τ〈x1*,x-a〉≤(1+τ)ε∥x-a∥,〈x2*,x-a〉≤ε∥x-a∥.
Then,
(4.52)〈x*,x-a〉=τ(〈x1*,x-a〉+〈x2*,x-a〉)≤(1+2τ)ε∥x-a∥.
And so,
(4.53)∥x-a∥=supx*∈BX*〈x*,x-a〉≤(1+2τ)ε∥x-a∥.
This shows that S∩B(a,δ)={a}. The proof is completed.From Corollaries4.8 and 4.9, we also have the following equivalent results.Corollary 4.10.
Suppose that generalized constraint equation(GCE) is subsmooth at a. Then the following statements are equivalent:(i)
(GCE) is strongly metrically subregular at a;(ii)
there existsτ∈(0,∞) such that BX*⊆τ(Dc*F(a,b)(BY*)+Nc(A,a)∩BX*));(iii)
0∈int(Dc*F(a,b)(Y*)+Nc(A,a));(iv)
X*=Dc*F(a,b)(Y*)+Nc(A,a);(v)
DcF(a,b) is strongly metrically subregular at 0 for 0 over Tc(A,a).Proof.
First, by Corollaries4.8 and 4.9, it is clear that (i)⇔(ii). Noting that Dc*(DcF(a,b))(0,0)(BY*)=Dc*F(a,b)(BY*), (ii)⇔(v) is immediate from (i)⇔(ii).
It is clear that (ii)⇔(iii). Noting that Nc(A,a) and Dc*F(a,b)(Y*) are cones, hence,
(4.54)Dc*F(a,b)(Y*)+Nc(A,a)=⋃n=1∞‍(Dc*F(a,b)(nBY*)+Nc(A,a)∩nBX*).
This shows that (ii)⇒(iv).
It remains to show that (iv)⇒(ii). Suppose that (iv) holds, by the Alaogu theorem, for each n, the set Dc*F(a,b)(nBY*)+Nc(A,a)∩nBX* is weakly star-closed, it follows from the well-known Baire category theorem and (iv) that
(4.55)0∈int(Dc*F(a,b)(BY*)+Nc(A,a)∩BX*).
Hence, (ii) holds. The proof is completed.



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*Source: 185249-2012-12-26.xml*