Error Bound for Conic Inequality in Hilbert Spaces

Abstract

We consider error bound issue for conic inequalities in Hilbert spaces. In terms of proximal subdifferentials of vector-valued functions, we provide sufficient conditions for the existence of a local error bound for a conic inequality. In the Hilbert space case, our result improves and extends some existing results on local error bounds.

1. Introduction

Let be a Banach space and let be a proper lower semicontinuous function. Consider the following inequality: Let and . Recall that inequality has a local error bound at if there exist such that where and denotes the open ball centered at of radius . In the variational analysis literature, sensitivity analysis of mathematical programming and convergence analysis of some algorithms of optimization problems are deeply tied to the notion of error bound. Since Hoffman's pioneering work [1], the study of error bounds has received extensive attention in the mathematical programming literature (for details, see [2–9] and references therein). For us, the work of Ioffe in this area will be particularly important. The author in his seminal work [10] first characterized error bound (under a different name) in terms of the subdifferentials and gave the following interesting result: if is locally Lipschitz at and there exist such that then has a local error bound at , where denotes the Clarke-Rockafellar subdifferential of at (for its definition, see Section 2). Results of the same flavour have sprung up. In 2010, in considering a general closed multifunction in place of and with the subdifferential replaced by the coderivative of , Zheng and Ng [11] extended the above-mentioned result to a generalized equation defined by a closed multifunction. Recently, Zheng and Ng [12] once again extended Ioffe's classic result to the conic inequality case in Asplund spaces in terms of the conic subdifferential defined by Fréchet normal cone. In this paper, we will extend Ioffe's result to the conic inequality case in the Hilbert space setting.

Let be Banach spaces with ordered by a closed convex cone . Let and be a function, where is the union of   with an abstract infinity . In this paper, we consider the following conic inequality: Indeed, many constraint systems in optimization can be viewed as special cases of . For example, when , , and , reduces to where each is a real-valued function on . Another important case is the following: when and , reduces to constraint systems of semi-infinite optimization problems; of course, one can also view the constraint systems in conic optimization problems as examples of . Hence conic inequality systems exist quite extensively. For such a kind of systems, we consider their error bounds. Let denote the solution set of ; that is, We say that has a local error bound at if there exist such that It is clear that (5) reduces to (1) when and . The main aim of this paper is to extend the above-mentioned Ioffe’s classic result on error bounds to conic inequality with being any pair of Hilbert spaces in terms of proximal subdifferentials of vector-valued functions defined by proximal normal cone with a kind of variational behavior of “order two.” In particular, we have improved Zheng and Ng's result in the Hilbert space setting.

2. Preliminaries

In this section, we summarize some fundamental tools of variational analysis and nonsmooth optimization using basically standard terminology and notation; see, for example, [13–15], for more details.

Let be a Banach space with topological dual and let . For a proper lower semicontinuous function , we denote by and the domain and the epigraph of , respectively; that is, Throughout the paper, the symbol always denotes the convergence relative to the distance induced by the norm while the arrow signifies the weak* convergence in the dual space .

Recall that the Fréchet subdifferential of at is defined as (see [15, 16]) Let denote the proximal subdifferential of at ; that is, For a closed subset of and , define, respectively, the Fréchet normal cone and the proximal normal cone and of to as where denotes the indicator function of (i.e., if and otherwise). Clearly, if and only if each of the following two conditions holds.(1)There exist such that (2)There exists such that Let denote the Mordukhovich (limiting) normal cone of to ; that is, if and only if there exist sequences and such that

It is well known that Let be a Banach space and let be a closed convex cone, which defines a preorder in as follows: . Let usually is called the dual cone of . Let denote an abstract infinity and . For a vector-valued function , the epigraph of with respect to the ordering cone is defined by In the vector-valued setting, in view of (13), we will use the following kinds of subdifferentials of : In the special case when and , we have , The following lemma will be useful in our analysis later.

Lemma 1. Let be spaces with being a closed convex cone and suppose that is a function such that is closed. Then, for any and ,

Proof. Let . Then it follows from (11) that there exists such that Noting that for any and that one has This means that , which implies in turn that and thus completes the proof of the lemma.

In establishing our main results, it will be crucial to present the following lemmas (cf. Chapter 1, Proposition 2.11 and Theorem 6.1 in [14] and Chapter 8, Theorem in [16]).

Lemma 2. Let be a Hilbert space and let be proper lower semicontinuous and suppose that is twice continuously differentiable at . Then where denotes the derivative of at .

Lemma 3. Let be a nonempty closed subset of a Hilbert space and let be such that . Then there exists satisfying the following properties.(i)The set of closest points in to is the singleton .(ii) is Fréchet differentiable at and (iii).

Lemma 4. Let be a Hilbert space and let be a proper lower semicontinuous function. Let and be such that . Then, for any , there exist and in with having the following property:

3. Main Results

As an extension of Ioffe's classic results on error bounds to conic inequality in Asplund spaces, Zheng and Ng [12] proved Theorem I.

Theorem I. Let be Asplund spaces with being a closed convex cone. Let be a function such that its -epigraph is closed. Let and be such that Then conic inequality (CIE) has a local error bound at .

It is well known that for any . So condition (28) will be much easier to be satisfied if is replaced by . Taking this fact into account, Theorem 6 provides a sharper result with replaced by and gives a relationship between the modulus of error bound and corresponding radius which were not mentioned in Theorem I. Its proof, which is slightly different from the one of Theorem I in [12], is based on a smooth variational principle rather than the Ekeland variational principle. Since the proof of Theorem 6 proceeds by contradiction, we give the following proposition to describe quantitative properties for a point that violates (5).

Proposition 5. Let be Hilbert spaces with being a closed convex cone. Let be a function such that its -epigraph is closed. Let , and be such that Then there exist satisfying the following properties:

Proof. It follows from (29) that there exists such that Define the function by for all . Then is lower semicontinuous (due to the closedness of ) and This and Lemma 4 imply that there exist such that From (36), it is easy to verify that (30) holds. Applying (35) and leads us to the following: which also implies that and thus verifies (31), while (37) together with gives us and so . It follows from this and (39) that Furthermore, (37) implies that where It follows from (40) and Lemma 3 that is twice continuously differentiable at and So applying Lemma 2 to (41) leads us to the inclusions which justifies (32) and thus completes the proof of the proposition.

With the preparation that we have done, the proof of our main result is now straightforward.

Theorem 6. Let be Hilbert spaces with being a closed convex cone. Let be a function such that its -epigraph is closed. Let and be such that Then, one has

Proof. The proof proceeds by contradiction. Namely, suppose to the contrary that (46) is not true; then, there exists such that and consequently it follows from Proposition 5 that there exist such that (30), (31), and (32) hold. Then, one has which implies Since , we can take such that . Then, by Lemma 1, one has It follows from (32) and (50) that Noting that , this implies that . Indeed, due to the definition of proximal normal cone, there exists such that for all .
Namely, for all .
Setting in the above inequality, one has Noting that and is a cone, one has which means that Let Then and so . Combining this with (30) implies that which contradicts (45) and (49). The proof is complete.

Letting in Theorem 6, we have the following global error bound result.

Corollary 7. Let , be spaces with being a closed convex cone. Let be a function such that its -epigraph is closed. Let and , be such that Then, one has

Taking Theorems I and 6 and Corollary 7 into consideration, we arrive at the following remarks.

Remark 8. Since the class of Asplund space is more extensive than that of Hilbert space from the framework of spaces, Theorem I has a wider range of applications than Theorem 6.
The subdifferential in Theorem 6 expresses a kind of variational behavior of “order two,” while in Theorem I is of “order one.” In general, is smaller than , and, furthermore, may have measure zero even for a smooth function. In [17], Clarke et al. have constructed a function on whose proximal subgradient is nonempty on a set that is small in the sense of measure and category. For the sake of convenience, we present this interesting result to show this kind of somewhat surprising phenomenon as follows.
Let be the (continuous) function of period 4 which satisfies for , for , and define Clearly is continuous, and thus defined by is on . Following the excellent proof of Clarke, we arrive at the conclusion that both and have measure zero. For more details, we refer the reader to Theorem 6.1 in [17].
To our knowledge, we do not know whether the assumption of Hilbert spaces in Theorem 6 and Corollary 7 can be extended to more general Banach spaces or not, for example, whether these results can be extended to () spaces.

Under the assumption somewhat stronger than that for Theorem 6, we have the following stability version regarding local error bounds for .

Theorem 9. Let be Hilbert spaces and suppose that is continuous and that is a closed convex cone such that , where denotes the interior. Let be such that . Then, there exists such that, for any , the corresponding conic inequality (CIE) has a local error bound at each .

Proof. By Theorem 6, it suffices to show that there exist , such that for all . If it is not the case, we can find a sequence in converging to and satisfying By the definition of , this means that there exist and such that Without loss of generality, we assume that converges to with the weak* topology (taking a subsequence if necessary). Taking the continuity assumption of into consideration, it follows from (63) that We claim that . Granting this, one has , which contradicts the assumption that . It remains to show that . Since , there exist and such that . Hence, and so . Since . This shows that , which completes the proof.

It is important to note that plays an important role in the validity of Theorem 9, as the following example shows. Hence the assumption of in Theorem 9 is not superfluous.

Example 10 (failure of having a local error bound). Let , , , and for all . Then is continuous, and .

It is well known and easy to check that . Take a sequence in , where Then converges to and and that it follows that Hence the conic inequality does not have a local error bound at . This situation occurs, of course, because is empty.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This research was supported by the National Natural Science Foundation of China (Grant nos. 11371312 and 11261067) and IRTSTYN.