Variational Inequalities and Vector OptimizationView this Special Issue
Metric Subregularity for Subsmooth Generalized Constraint Equations in Banach Spaces
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.
Let be a Banach space and be a function. Consider the following inequality: Let . Recall that (1.1) has a local error bound at if there exist such that where and denotes the open ball of center and radius . The error bound has been studied by many authors (see [1–3] and the references therein).
Let be another Banach space, , and let be a closed multifunction. The following generalized equation: concludes most of systems in optimization and was investigated by many researchers (see [4–9] and the references therein). Let and . Recall that is metrically subregular at if there exist such that (see [4–6] and the references therein). This property provides an estimate how far for an element near can be from the solution set of . A stronger notion is the metric regularity: a multifunction is metrically regular at if there exist such that There exists a wide literature on this topic. We refer the interested readers to [3, 7–11] and to the references contained therein. Let be a closed subset of . Consider the generalized constraint equation as follows:
When , reduces and (1.5) means that is metrically subregular at . When and , reduces the inequality (1.1) and (1.5) means that this inequality has a local error bound at . 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  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 . As extension of primal smoothness and convexity, the notion of subsmoothness was introduced and some functional characterizations were provided in . Recently, by variational analysis techniques (for more details, see [16–19]), Zheng and Ng  investigated metric subregularity of under the subsmooth assumption. In this paper, in terms of normal cones and coderivatives, we devote to metric subregularity of generalized constraint equation under the subsmooth assumption. We will build some new necessary and sufficient conditions for to be metrically subregular and strongly metrically subregular.
Let be a Banach space. We denote by and the closed unit ball and the dual space of , respectively. Let be a nonempty subset of , and , respectively, denote the interior and the boundary of . For and , let denote the open ball with center and radius .
We introduce some notions of variations and derivatives needed to state our results.
For a closed subset of and , let and , respectively, denote the Clarke tangent cone and contingent (Bouligand) cone of at defined by where means that with . It is easy to verify that if and only if for each sequence in converging to and each sequence in (0, ) decreasing to , there exists a sequence in converging to such that for each natural number ; while if and only if there exists a sequence in converging to and a sequence in decreasing to such that for all .
We denote by the Clarke normal cone of at , that is, For and , the nonempty set is called the set of Fréchet -normals of at . When , is a convex cone which is called the Fréchet normal cone of at and is denoted by . Let denote the Mordukhovich limiting or basic normal cone of at , that is, that is, if and only if there exist sequences in such that and for each natural number . It is known that (see [4, 9, 16, 18, 19] and the references contained therein). If is convex, then Recall that a Banach space is called an Asplund space if every continuous convex function on is Fréchet differentiable at each point of a dense subset of (for other definitions and their equivalents, see ). It is well known that is an Asplund space if and only if every separable subspace of 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 where denotes the closure with respect to the topology, see [9, 19]. Recently, Zheng and Ng  established an approximate projection result for a closed subset of , which will play a key role in the proofs of our main results.
Lemma 2.1. Let be a closed nonempty subset of a Banach space and let . Then for any there exist and with such that If is an Asplund space, then can be replaced by .
Let be a multifunction and let denote the graph of , that is, As usual, is said to be closed (resp., convex) if is a closed (resp., convex) subset of . Let . The Clarke tangent and contingent derivatives of at are defined by respectively. Let , and denote the coderivatives of at associated with the Fréchet, Mordukhovich, and Clarke normal structures, respectively. They are defined by the following: The more details of the coderivatives can be found in [9, 18, 19] and the references therein.
3. Subsmooth Generalized Constraint Equation
As a generalization of the prox-regularity, Aussel et al.  introduced and studied the subsmoothness. is said to be subsmooth at if for any there exist such that whenever , and .
It is easy to verify that is subsmooth at if and only if for any , there exists such that whenever and .
Let be a closed multifunction, and . Zheng and Ng  introduce the concept of the L-subsmoothness of at for : is called to be L-subsmooth at for if for any there exists such that whenever . Next, we introduce the concept of the subsmoothness of generalized constraint equation which will be useful in our discussion.
Remark 3.2. The subsmoothness of at means the subsmoothness of at for when , while the subsmoothness of at means the subsmoothness of at when for all . If and Gr is prox-regular at , then generalized equation is subsmooth at . If and Gr are convex, then is also subsmooth at . Finally when is prox-regular and is single-valued and smooth, is subsmooth at , too. Hence, Definition 3.1 extends notions of smoothness, convexity and prox-regularity.
Proof. Suppose that is subsmooth at . Then for any , there exists such that
Let . If , then Thus, (3.6) holds. Otherwise, one has whenever . Noting that (sine ), it follows that
It remains to show that (3.7) holds. Since whenever . One has Noting that , it follows that which implies that (3.7) holds and completes the proof.
4. Main Results
This section is devoted to metric subregularity of generalized equation . 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.
Proof. Let denote the indicator function of and such that (1.5) holds. Then (1.5) can be rewritten as Let and . Noting (cf. [9, Corollary 1.96]) that , one gets that for any natural number , there exists such that and Hence, by (4.2), it follows that that is, is a local minimizer of defined by Hence, . It follows from  that that is, for some and . Since and are compact, without loss of generality (otherwise take a generalized subsequence), we can assume for some and as . Noting that is closed (since is closed and is compact), one has This implies that This shows that (4.1) holds true. The proof is completed.
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. Let and be Banach spaces. Suppose that generalized constraint equation is subsmooth at and that there exist such that whenever . Then is metrically subregular at and, more precisely, for any there exists such that
Proof. Let . Then, by subsmooth assmption of at and Proposition 3.3, there exists such that
Let and . Now we need only show (4.13).(i)If , then , . Hence (4.13) holds.(ii)Suppose and let By Lemma 2.1 there exist and with such that Thus, . Hence, By (4.12) there exist , and such that . Applying (4.14) with in place of , it follows that This and (4.16) imply that Letting , it follows that (4.13) holds. The proof is completed.
When and are Asplund spaces, the assumption in Theorem 4.3 can be weakened with replaced by .
Theorem 4.4. Suppose and are Asplund spaces. Suppose that generalized constraint equation is subsmooth at and that there exist such that whenever . Then for any there exists such that (4.13) holds.
Finally, we end this subsection with a sufficient and necessary condition for the Clarke tangent derivative mapping to be metrically subregular at for over the Clerke tangent cone .
Let For , let The following lemma is known ([5, Theorem 3.2]) and useful for us in the sequel.
Lemma 4.5. Assume that is a closed convex multifunction, is a closed convex subset of , and . And suppose that there exist a cone and a neighborhood of such that . Then, Consequently, is metrically subregular at if and only if .
Theorem 4.6. Let and Suppose that Then, If, in addition, , then Consequently, is metrically subregular at over if and only if .
Proof. First, we assume that . By the definition of , we have
This implies that
This and (4.25) imply that (4.27) holds.
We consider the following constraint equation: Let denote the solution set of . Then, Noting that it is straightforward to verify that On the other hand, since is a closed convex multifunction from to and is a closed convex cone, Lemma 4.5 implies that This gives us .
It remains to show that when . Suppose that We need only show that . Let and . By Lemma 2.1 there exist and with such that Noting that is a convex cone, it is easy to verify that Take a fixed in . Then there exist and such that We equip the product space with norm Noting that the unit ball of the dual space of is , it follows from the convexity of and that Hence, whenever . Noting that , it follows from (4.35) that Therefore, Letting , one has This contradicts with . The proof is completed.
4.3. Strongly Metric Subregularity
Let be a multifunction and . Recall that is strongly subregular at if there exist , neighborhoods of , and of such that It is clear that this definition is equivalent to the next one when .
Definition 4.7. One says that generalized constraint equation is strongly metrically subregular at if there exists such that It is clear that is strongly metrically subregular at if and only if is an isolated point of (i.e., for some ) and it is metrically subregular at . Thus, if is strongly metrically subregular at , Then . We immediately have the following Corollary 4.8 from Theorem 4.1.
Corollary 4.8. Suppose that there exists such that (4.45) holds. Then,
Applying Theorem 4.3, one obtains a sufficient condition for to be strongly metrically subregular at .
Corollary 4.9. Let be Banach spaces. Suppose that generalized constraint equation is subsmooth at and that there exists such that Then is strongly metrically subregular at and, more precisely, for any there exists such that
Proof. From Theorem 4.3, we need only show that for some . Since the assumption that is subsmooth at , by Proposition 3.3, for any , there exists such that
Take an arbitrary . By (4.47), there exist , such that . Let . Applying (4.49) with in place of , it follows from this and (4.50) that we have Then, And so, This shows that . The proof is completed.
Corollary 4.10. Suppose that generalized constraint equation is subsmooth at . Then the following statements are equivalent:(i) is strongly metrically subregular at ;(ii)there exists such that ;(iii);(iv);(v) is strongly metrically subregular at for over .
Proof. First, by Corollaries 4.8 and 4.9, it is clear that (i)(ii). Noting that , (ii)(v) is immediate from (i)(ii).
It is clear that (ii)(iii). Noting that and are cones, hence, This shows that (ii)(iv).
It remains to show that (iv)(ii). Suppose that (iv) holds, by the Alaogu theorem, for each , the set is weakly star-closed, it follows from the well-known Baire category theorem and (iv) that Hence, (ii) holds. The proof is completed.
This paper was supported by the National Natural Science Foundations, China, (Grants nos. 11061039, 11061038, and 11261067), and IRTSTYN.
D. Klatte and B. Kummer, Nonsmooth Equations in Optimization, vol. 60, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2002.
B. S. Mordukhovich, Variational Analysis and Generalized Differentiation. I, II, Springer, Berlin, Germany, 2006.
F. H. Clarke, R. J. Stern, and P. R. Wolenski, “Proximal smoothness and the lower- property,” Journal of Convex Analysis, vol. 2, no. 1-2, pp. 117–144, 1995.View at: Google Scholar
F. H. Clarke, Optimization and Nonsmooth Analysis, John Wiley & Sons, New York, NY, USA, 1983.
R. R. Phelps, Convex Functions, Monotone Operators and Differentiability, vol. 1364 of Lecture Notes in Mathematics, Springer, New York, NY, USA, 1989.View at: Publisher Site
R. T. Rockafellar and R. J.-B. Wets, Variational Analysis, vol. 317, Springer, Berlin, Germany, 1998.View at: Publisher Site