- Research Article
- Open Access

# Optimality Conditions and Duality for DC Programming in Locally Convex Spaces

- Xianyun Wang
^{1}Email author

**2009**:258756

https://doi.org/10.1155/2009/258756

© XianyunWang. 2009

**Received:**10 February 2009**Accepted:**25 September 2009**Published:**11 October 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.

## Keywords

- Optimality Condition
- Local Minimizer
- Related Result
- Dual Problem
- Lower Semicontinuous

## 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

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 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

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.

## 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 well-known characterization of optimal solution to DC problem, we obtain the following lemma.

Lemma 3.1.

Theorem 3.2.

The following statements are equivalent:

Proof.

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

Obviously, holds automatically. The proof is complete.

Theorem 3.3.

The following statements are equivalent:

Proof.

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

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 well-known result, called Toland duality, for DC problems. In this section, we always assume that and are everywhere subdifferentible.

Proposition 4.1.

Proof.

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

Following from Proposition 4.1, we obtain the following proposition.

Proposition 4.2.

Proof.

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.

(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 .

- (i)

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.

(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.

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.

## Declarations

### Acknowledgment

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.

## Authors’ Affiliations

## References

- An LTH, Tao PD:
**The DC (difference of convex functions) programming and DCA revisited with DC models of real world nonconvex optimization problems.***Annals of Operations Research*2005,**133:**23–46. 10.1007/s10479-004-5022-1MathSciNetView ArticleMATHGoogle Scholar - Bot RI, Wanka G:
**Duality for multiobjective optimization problems with convex objective functions and D.C. constraints.***Journal of Mathematical Analysis and Applications*2006,**315**(2):526–543. 10.1016/j.jmaa.2005.06.067MathSciNetView ArticleMATHGoogle Scholar - Dinh N, Nghia TTA, Vallet G:
**A closedness condition and its applications to DC programs with convex constraints.***Optimization*2008,**1:**235–262.MathSciNetMATHGoogle Scholar - Dinh N, Vallet G, Nghia TTA:
**Farkas-type results and duality for DC programs with convex constraints.***Journal of Convex Analysis*2008,**15**(2):235–262.MathSciNetMATHGoogle Scholar - Horst R, Thoai NV:
**DC programming: overview.***Journal of Optimization Theory and Applications*1999,**103**(1):1–43. 10.1023/A:1021765131316MathSciNetView ArticleMATHGoogle Scholar - Martínez-Legaz J-E, Volle M:
**Duality in D.C. programming: the case of several D.C. constraints.***Journal of Mathematical Analysis and Applications*1999,**237**(2):657–671. 10.1006/jmaa.1999.6496MathSciNetView ArticleMATHGoogle Scholar - Demyanov VF, Rubinov AM:
*Constructive Nonsmooth Analysis, Approximation & Optimization*.*Volume 7*. Peter Lang, Frankfurt, Germany; 1995:iv+416.MATHGoogle Scholar - Mordukhovich BS:
*Variational Analysis and Generalized Differentiation. I: Basic Theory, Grundlehren der Mathematischen Wissenschaften*.*Volume 330*. Springer, Berlin, Germany; 2006:xxii+579.Google Scholar - Mordukhovich BS:
*Variational Analysis and Generalized Differentiation. II: Application, Grundlehren der Mathematischen Wissenschaften*.*Volume 331*. Springer, Berlin, Germany; 2006:i–xxii and 1–610.Google Scholar - Rockafellar RT, Wets RJ-B:
*Variational Analysis, Grundlehren der Mathematischen Wissenschaften*.*Volume 317*. Springer, Berlin, Germany; 1998:xiv+733.Google Scholar - Zalinescu C:
*Convex Analysis in General Vector Spaces*. World Scientific, River Edge, NJ, USA; 2002:xx+367.View ArticleMATHGoogle Scholar - Jeyakumar V, Lee GM, Dinh N:
**New sequential Lagrange multiplier conditions characterizing optimality without constraint qualification for convex programs.***SIAM Journal on Optimization*2003,**14**(2):534–547. 10.1137/S1052623402417699MathSciNetView ArticleMATHGoogle Scholar - Bot RI, Wanka G:
**A weaker regularity condition for subdifferential calculus and Fenchel duality in infinite dimensional spaces.***Nonlinear Analysis: Theory, Methods & Applications*2006,**64**(12):2787–2804. 10.1016/j.na.2005.09.017MathSciNetView ArticleMATHGoogle Scholar - Mordukhovich BS, Nam NM, Yen ND:
**Fréchet subdifferential calculus and optimality conditions in nondifferentiable programming.***Optimization*2006,**55**(5–6):685–708. 10.1080/02331930600816395MathSciNetView ArticleMATHGoogle Scholar - Singer I:
**A general theory of dual optimization problems.***Journal of Mathematical Analysis and Applications*1986,**116**(1):77–130. 10.1016/0022-247X(86)90046-6MathSciNetView ArticleMATHGoogle Scholar - Singer I:
**Some further duality theorems for optimization problems with reverse convex constraint sets.***Journal of Mathematical Analysis and Applications*1992,**171**(1):205–219. 10.1016/0022-247X(92)90385-QMathSciNetView ArticleMATHGoogle Scholar - Volle M:
**Concave duality: application to problems dealing with difference of functions.***Mathematical Programming*1988,**41**(2):261–278.MathSciNetView ArticleMATHGoogle Scholar

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