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Optimality Conditions and Duality for DC Programming in Locally Convex Spaces
Journal of Inequalities and Applications volume 2009, Article number: 258756 (2009)
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 weaktopologies and . Let , be proper convex functions, and let be a linear operator such that . We consider the primal DC (difference of convex) programming problem
and its associated dual problem
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 weaktopology 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
The conjugate function of is the function defined by
If is lower semicontinuous, then the following equality holds:
Let . For each , the subdifferential of at is the convex set defined by
When , we put . If in (2.4), the set is the classical subdifferential of convex analysis, that is,
Let , the following inequality holds (cf. [11, Theorem (ii)] ):
Following [12],
The Young equality holds
As a consequence of that,
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
3. Optimality Conditions
Let denote the identity map on . We consider the image set of through the map , that is,
By [14, Proposition 4.1] and the wellknown 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
Especially, if is a local minimizer of , then
Theorem 3.2.
The following statements are equivalent:
(i)
(ii)For each and each ,
Moreover, is a local optimal solution to problem if and only if for each ,
Proof.
(i)(ii). Suppose that (i) holds. Let , and , then for each ,
Therefore, . Hence, .
Conversely, let . Then . By (i),
Therefore, there exists such that and . Noting that , then
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
that is . Hence, and so .
By the wellknown characterization of optimal solution to DC problem (see Lemma 3.1), is a local optimal solution to problem if and only if, for each ,
Obviously, holds automatically. The proof is complete.
Let . Define
Theorem 3.3.
The following statements are equivalent:
(i),
(ii)For each and each ,
Moreover, is a local optimal solution to problem if and only if, for each ,
Proof.
(i)(ii). Suppose that (i) holds. Let and . Then one has
Hence, By the given assumption,
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
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
and the corresponding dual problem
Let denote the optimal values of problems and , respectively, that is
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 wellknown 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
Proof.
By the definition of conjugate function, it follows that
Next, we prove that
Suppose on the contrary that that is, there exists such that
Let and , then
From this, it follows that
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 ,
Proof.
Let . Since , it follows from (4.4) that
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,
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
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
Then the conjugate functions and are
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.
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The author wish to thank the referees for careful reading of this paper and many valuable comments, which helped to improve the quality of the paper.
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Wang, X. Optimality Conditions and Duality for DC Programming in Locally Convex Spaces. J Inequal Appl 2009, 258756 (2009). https://doi.org/10.1155/2009/258756
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DOI: https://doi.org/10.1155/2009/258756
Keywords
 Optimality Condition
 Local Minimizer
 Related Result
 Dual Problem
 Lower Semicontinuous