Total Lagrange duality for DC infinite optimization problems
© Fang and Chen; licensee Springer. 2013
Received: 8 August 2013
Accepted: 30 August 2013
Published: 8 November 2013
We present some total Lagrange duality results for inequality systems involving infinitely many DC functions. By using properties of the subdifferentials of involved functions, we introduce some new notions of constraint qualifications. Under the new constraint qualifications, we provide necessary and/or sufficient conditions for the stable total Lagrange duality to hold.
The optimal values of problems and are denoted by and , respectively.
Usually, there is a so-called duality gap between the optimal values of primal problem and its Lagrange dual problem . A challenge in convex analysis is to give sufficient conditions which guarantee the strong Lagrange duality, that is, and the dual problem has an optimal solution. Several sufficient and/or necessary conditions were given in the past in order to eliminate the above-mentioned duality gap, see, for example, [1–3, 5] and the references therein. In particular, the authors in  established a complete characterization for the strong Lagrange duality under assumption that f and are not necessarily convex, and in , the authors considered the optimization problem , but with and , being DC (difference of two convex functions) functions, and they obtained some complete characterizations for the weak and strong Lagrange dualities. As pointed in , problems of DC programming are highly important from both viewpoints of optimization theory and applications, and they have been extensively studied in the literature (cf. [15–24] and the references therein).
However, without assuming the lower semicontinuity of g and , problems (1.8) and are in general not equivalent.
Clearly, the strong Lagrange duality ensures the total Lagrange duality, but the converse does not necessarily hold in general. To our knowledge, not many results are known to provide complete characterizations for the total Lagrangian duality for the DC optimization problem (1.3). Except the works in paper  by Fang et al., where, assuming in addition that , , a complete characterization was established for the stable total Lagrangian duality for problem (1.3), that is, the characterization for (1.9) to hold for in place of f with any . However, the approaches in  do not work for the DC optimization problem (1.3).
In this paper, we do not impose any topological assumption on C or on f, g, and , that is, C is not necessarily closed, and f, g, , are not necessarily lsc, and , are necessarily differentiable. One of our main aims in the present paper is to use these constraint qualifications (or their variations) involving subdifferentials, which have been studied and extensively used, see, for example, [2, 3, 6, 12, 26], to provide characterizations for the total Lagrangian duality. Most of results obtained in the present paper seem new and are proper extensions of the results in  in the special case when , . In particular, both our dual problem and the regularity conditions introduced here are defined in terms of subdifferential of the convex functions f, g, and rather than those of the DC functions and , which are different from the consideration in .
The paper is organized as follows. The next section contains the necessary notations and preliminary results. In Section 3, we provide some characterizations for the weak Lagrange dualities and the total Lagrangian dualities to hold.
2 Notations and preliminaries
The notations used in this paper are standard (cf. ). In particular, we assume throughout the whole paper that X is a real locally convex space, and let denote the dual space of X. For and , we write for the value of at x, that is, . Let Z be a set in X. The closure of Z is denoted by clZ. If , then clW denotes the weak∗ closure of W. For the whole paper, we endow with the product topology of and the usual Euclidean topology.
3 The total Lagrange dualities
To make the dual problem considered here well defined, we further assume that clg and , are proper. Then . For the whole paper, any elements and are understood as and , respectively.
for any with and . In particular, in the case when , problem , as well as its dual problem , are reduced to problem , and its dual problem as defined in (1.3) and (1.5), respectively.
Let and denote the optimal values of and , respectively. For each , we use to denote the optimal solution set of . In particular, we write for . Obviously, for each , . This section is devoted to the study of characterizing the total Lagrange dualities. Unlike the convexity case, the cases for DC optimization problems are more complicated. We begin with the following definition, where the notations of the weak Lagrange duality and the stable weak Lagrange duality were introduced in .
the weak Lagrange duality holds if ;
the stable weak Lagrange duality holds if for each ;
the stable -total Lagrange duality holds if, for each , and problem has an optimal solution provided that . In particular, in the case when , the stable -total duality is called the stable total duality.
Thus, if g and , , are lsc, then the weak Lagrange duality holds. The following proposition provides a weaker condition for the weak Lagrange duality to hold.
Proposition 3.1 Let . Suppose that g and each are lsc at . Then the weak Lagrange duality holds.
the last inequality holds because and . This implies that . Hence, by (3.11), one gets and the proof is complete. □
Form (2.9), if , then g and each are lsc at . Hence, the following corollary follows from Proposition 3.1 directly.
Corollary 3.1 Let . If , then .
Motivating by , we introduce the following condition (LSC) to characterize the relationships between and and the weak Lagrange duality.
- (a)Since and , it follows that . Hence, by (2.5), the family satisfies the (LSC) if and only if(3.15)
Obviously, if g and , are lsc, then the (LSC) holds. But the converse is not true, in general, as to be shown by Example 3.1 below.
This implies that the (LSC) holds. However, the function g is not lsc at .
in terms of the (LSC). For this purpose, we first give the following lemma by the definition of conjugate functions. The proof is standard (cf. [, Lemma 4.1]), and so we omit it.
Proposition 3.2 The family satisfies the (LSC) if and only if (3.16) holds. Consequently, if the (LSC) holds, then the weak Lagrange duality holds.
Proof Suppose that the (LSC) holds. Then (3.14) holds. Let . To show that , it suffices to show that by (3.10). To do this, suppose, on the contrary, that . Then there exists such that . Thus, by (3.18), , and so by (3.14). It follows from (3.17) that . This contradicts and completes the proof of the inequality .
Conversely, suppose that (3.16) holds. By Remark 3.1(b), it suffices to show that (3.15) holds. To do this, let . Then, by (3.18), , and so , thanks to (3.16). Hence, by (3.17), . Therefore, (3.15) is proved. The proof is complete. □
The following proposition provides an estimate for the subdifferential of the DC function in terms of the subdifferentials of the convex functions involved.
Hence, , and inclusion (3.21) holds. □
Considering the possible inclusions among , and , we introduce the following definition.
- (a)the quasi weakly basic constraint qualification (the quasi (WBCQ)) at if(3.24)
- (b)the weakly basic constraint qualification (the (WBCQ)) at if(3.25)
We say that the family satisfies the quasi (WBCQ) (resp. the (WBCQ)) if it satisfies the quasi (WBCQ) (resp. the (WBCQ)) at each point .
- (a)The following implication holds:
In the special case, when , , the quasi (WBCQ) and (WBCQ) are reduced to the (WBCQ) f for the family introduced in .
For our main theorems in this section, the following lemma is helpful.
where the last equality holds because of (3.27) and . Since , it follows that (3.26) holds. The proof is complete. □
The following theorem provides a sufficient condition and a necessary condition for the stable -total Lagrange duality.
The family satisfies the (WBCQ).
The stable -total Lagrange duality holds between and .
The family satisfies the quasi (WBCQ).
Then (i) ⇒ (ii) ⇒ (iii).
thanks to the assumed (WBCQ). Thus, by Lemma 3.2, we get that there exists such that (3.26) holds for each . Moreover, we have that by Corollary 3.1. Thus, and is an optimal solution of . This implies that the stable -total Lagrange duality holds.
as is arbitrary. Hence, . Therefore, (3.24) holds, and the proof is complete. □
Theorem 3.2 below provides sufficient conditions ensuring the stable total Lagrange duality.
Theorem 3.2 Suppose that the family satisfies the (WBCQ), and that the stable weak Lagrange duality holds between and . Then the stable total Lagrange duality holds.
Proof Let . Suppose that . Let . Then and hence by the assumed (WBCQ). Thus, Lemma 3.2 is applied to get that there exists such that (3.26) holds for each . This together with the stable weak Lagrange duality implies that , and is an optimal solution of . Thus, the stable total Lagrange duality holds, and the proof is complete. □
In the case when , , by Theorem 3.1, we have the following corollary, which was given in [, Theorem 5.2].
The first author was supported in part by the National Natural Science Foundation of China (grant 11101186) and supported in part by the Scientific Research Fund of Hunan Provincial Education Department (grant 13B095). The second author was supported in part by the National Natural Science Foundation of China (11001289) and the Key Project of Chinese Ministry of Education (211151).
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