Optimality Conditions and Duality for DC Programming in Locally Convex Spaces

Abstract

Consider the DC programming problem where and are proper convex functions defined on locally convex Hausdorff topological vector spaces and respectively, and is a linear operator from to . By using the properties of the epigraph of the conjugate functions, the optimality conditions and strong duality of are obtained.

1. Introduction

Let and be real locally convex Hausdorff topological vector spaces, whose respective dual spaces, and are endowed with the weak-topologies and . Let , be proper convex functions, and let be a linear operator such that . We consider the primal DC (difference of convex) programming problem

(1.1)

and its associated dual problem

(1.2)

where and are the Fenchel conjugates of and , respectively, and stands for the adjoint operator, where is the subspace of such that if and only if defined by is continuous on . Note that, in general, is not the whole space because is not necessarily continuous.

Problems of DC programming are highly important from both viewpoints of optimization theory and applications. They have been extensively studied in the literature; see, for example, [1–6] and the references therein. On one hand, such problems being heavily nonconvex can be considered as a special class in nondifferentiable programming (in particular, quasidifferentiable programming [7]) and thus are suitable for applying advanced techniques of variational analysis and generalized differentiation developed, for example, in [7–10]. On the other hand, the special convex structure of both plus function and minus function in the objective of (1.1) offers the possibility to use powerful tools of convex analysis in the study of DC Programming.

DC programming of type (1.1) (when is an identity operator) has been considered in the space in paper [5], where the authors obtained some necessary optimality conditions for local minimizers to (1.1) by using refined techniques and results of convex analysis. In this paper, we extend these results to DC programming in topological vector spaces and also derive some new necessary and/or sufficient conditions for local minimizers to (1.1). Finally, we consider the strong duality of problem (1.1); that is, there is no duality gap between the problem and the dual problem and has at least an optimal solution.

In this paper we study the optimality conditions and the strong duality between and in the most general setting, namely, when and are proper convex functions (not necessarily lower semicontinuous) and is a linear operator (not necessarily continuous). The rest of the paper is organized as follows. In Section 2 we present some basic definitions and preliminary results. The optimality conditions are derived in Section 3, and the strong duality of DC programming is obtained in Section 4.

2. Notations and Preliminary Results

The notation used in the present paper is standard (cf. [11]). In particular, we assume throughout the paper that and are real locally convex Hausdorff topological vector spaces, and let denote the dual space, endowed with the weak-topology By we will denote the value of the functional at , that is, . The zero of each of the involved spaces will be indistinctly represented by

Let be a proper convex function. The effective domain and the epigraph of are the nonempty sets defined by

(2.1)

The conjugate function of is the function defined by

(2.2)

If is lower semicontinuous, then the following equality holds:

(2.3)

Let . For each , the -subdifferential of at is the convex set defined by

(2.4)

When , we put . If in (2.4), the set is the classical subdifferential of convex analysis, that is,

(2.5)

Let , the following inequality holds (cf. [11, Theorem (ii)] ):

(2.6)

Following [12],

(2.7)

The Young equality holds

(2.8)

As a consequence of that,

(2.9)

The following notion of Cartesian product map is used in [13].

Definition 2.1.

Let be nonempty sets and consider maps and . We denote by the map defined by

(2.10)

3. Optimality Conditions

Let denote the identity map on . We consider the image set of through the map , that is,

(3.1)

By [14, Proposition 4.1] and the well-known characterization of optimal solution to DC problem, we obtain the following lemma.

Lemma 3.1.

Let be proper convex fucntions on , and let . Then is a local minimizer of if and only if, for each

(3.2)

Especially, if is a local minimizer of , then

(3.3)

Theorem 3.2.

The following statements are equivalent:

(i)

(ii)For each and each ,

(3.4)

Moreover, is a local optimal solution to problem if and only if for each ,

(3.5)

Proof.

(i)(ii). Suppose that (i) holds. Let , and , then for each ,

(3.6)

Therefore, . Hence, .

Conversely, let . Then . By (i),

(3.7)

Therefore, there exists such that and . Noting that , then

(3.8)

This implies thanks to (2.6). Thus, and . Hence, (3.4) is seen to hold.

(ii)(i). Suppose that (ii) holds. To show (i), it suffices to show that . To do this, let and . By (2.7), there exists such that and From (3.4), there exists such that . Since , it follows from (2.6) that

(3.9)

that is . Hence, and so .

By the well-known characterization of optimal solution to DC problem (see Lemma 3.1), is a local optimal solution to problem if and only if, for each ,

(3.10)

Obviously, holds automatically. The proof is complete.

Let . Define

(3.11)

Theorem 3.3.

The following statements are equivalent:

(i),

(ii)For each and each ,

(3.12)

Moreover, is a local optimal solution to problem if and only if, for each ,

(3.13)

Proof.

(i)(ii). Suppose that (i) holds. Let and . Then one has

(3.14)

Hence, By the given assumption,

(3.15)

Therefore, there exists such that and . Hence, , this means and so . Consequently, . This completes the proof because the converse inclusion holds automatically.

(ii)(i). Suppose that (ii) holds. To show (i), it suffice to show that . To do this, let and . By (2.7), there exists such that and . From (3.12), there exists such that . Since , it follows from (2.6) that

(3.16)

that is . Hence, and so .

Similar to the proof of (3.5), one has that (3.13) holds.

4. Duality in DC Programming

This section is devoted to study the strong duality between the primal problem and its Toland dual, namely, the property that both optimal values coincide and the dual problem has at least an optimal solution.

Given , we consider the DC programming problem given in the form

(4.1)

and the corresponding dual problem

(4.2)

Let denote the optimal values of problems and , respectively, that is

(4.3)

In the special case when , problems and are just the problem and .

Before establishing the relationship between problems and , we give useful formula for computing the values of conjugate functions. The formula is an extension of a well-known result, called Toland duality, for DC problems. In this section, we always assume that and are everywhere subdifferentible.

Proposition 4.1.

Let . Then the conjugate function of is given by

(4.4)

Proof.

By the definition of conjugate function, it follows that

(4.5)

Next, we prove that

(4.6)

Suppose on the contrary that that is, there exists such that

(4.7)

Let and , then

(4.8)

From this, it follows that

(4.9)

which is contradiction to (4.7), and so (4.4) holds.

Following from Proposition 4.1, we obtain the following proposition.

Proposition 4.2.

For each ,

(4.10)

Proof.

Let . Since , it follows from (4.4) that

(4.11)

Remark 4.3.

In the special case when and , formula (4.10) was first given by Pshenichnyi (see [10]) and related results on duality can be found in [15–17].

Proposition 4.4.

For each ,

(i)if is an optimal solution to problem , then is an optimal solution to problem ;

(ii)suppose that and are lower semicontinuous. If is an optimal solution to problem , then is an optimal solution to problem .

Proof.

• (i) Let be an optimal solution to problem and let . Then . It follows from (3.5) that . By the Young equality, we have

  

  (4.12)

Therefore,

(4.13)

By (4.10), is an optimal solution to problem .

• (ii) Let be an optimal solution to problem and . Then and hence thanks to Theorem 3.3. By the Young equality, we have

  

  (4.14)

Since the functions and are lower semicontinuous, it follows from (2.3) that and . Hence, by the above two equalities, one has

(4.15)

By (4.10), is an optimal solution to problem .

Obviously, if is continuous, then and so for each . By Propositions 4.2 and 4.4, we get the following strong duality theorem straightforwardly.

Theorem 4.5.

For each ,

(i)suppose that is continuous. If the problem has an optimal solution, then and has an optimal solution;

(ii)suppose that and are lower semicontinuous. If the problem has an optimal solution, then and has an optimal solution.

Corollary 4.6.

() If the problem has an optimal solution, then and has an optimal solution.

()Suppose that and are lower semicontinuous. If the problem has an optimal solution, then and has an optimal solution.

Remark 4.7.

As in [13], if and has an optimal solution, then we say the converse duality holds between and .

Example 4.8.

Let and let Define by

(4.16)

Then the conjugate functions and are

(4.17)

Obviously, and attained the infimun at , and attained the infimum at . Hence, . It is easy to see that and . Therefore, Proposition 4.4 is seen to hold and Theorem 4.5 is applicable.