ε-Properly Efficiency of Multiobjective Semidefinite Programming with Set-Valued Functions

Abstract

In this paper, we study the -properly efficiency of multiobjective semidefinite programming with set-valued functions. Firstly, we obtain the scalarization theorems under the condition of the generalized cone-subconvexlikeness. Then, we establish the alternative theorem which contains matrixes and vectors, the -Lagrange multiplier theorems, and the -proper saddle point theorems of the primal programming under some suitable conditions.

1. Introduction

Vector optimization with set-valued functions has been used widely in many fields, such as economics and engineering. In recent years, set-valued optimization problem has aroused extensive concerns among the researchers [1–6]. Many authors focus their attention on how to get the approximate solutions of optimization problems. Chicco et al. [7] proposed a new type of solution based on the upper comprehensive sets and discussed the existence of optimal points in multicriteria situations. Zhao and Yang [8] obtained some useful properties of -Benson proper efficiency. Gutiérrez et al. [9] presented some properties of strict efficient solutions in vector optimization related to the -efficiency notion. In locally convex Hausdorff topological vector spaces, -strongly efficient solutions of vector optimization with set-valued maps were discussed by Wang [10]. Rong and Wu [11] proposed the concept of -weak efficient solutions and derived scalarization results, saddle point theorems, and Lagrangian multipliers theorems.

Semidefinite programming involves optimization problems with a linear objective function over semidefinite constraints. It shares many interesting properties with linear programming. Semidefinite programming unifies several standard problems (linear and quadratic programming) and finds many applications in engineering [12].

By combining approximate solutions of vector optimization problems with multiobjective semidefinite programming, -properly efficiency of multiobjective semidefinite programming with set-valued functions is discussed in this paper. The rest of the paper is organized as follows: In Section 2, we introduce some notations and definitions used throughout the text. In Section 3, we derive scalarization theorems expressing the necessary and sufficient optimality conditions for -properly efficient solutions of the primal programming. Under the condition of the generalized cone-subconvexlikeness, the alternative theorem which contains matrix and vector and the -Lagrange multiplier theorems are established in Section 4. In Section 5, the -proper saddle point results are presented.

2. Preliminaries

Let and Obviously, is a pointed closed convex cone with nonempty interior.

Let be a nonempty subset of ; the generated cone of is defined as and represent the closure and the interior of , respectively. The positive dual cone of is defined as and the strict positive dual cone of is defined as where . Obviously, we see that . Let be a nonempty subset and let ; we define the following two important sets:

Let be a set of real symmetric matrixes of -order and be a set of real symmetric positive semidefinite matrixes of -order. For matrixes , The dot product of and is defined as , where is the trace of the matrix . For two vectors , their dot product denotes .

It is easy to prove the following results.

Proposition 1. (i) Let , ; then .
(ii) Let ; if for any , then .
We now state the following multiobjective semidefinite programming with set-valued functions to be studied in the present paper: where , , and are set-valued functions with nonempty value, and , . We use to represent the feasible set of and assume that is nonempty. Let

Definition 2. A point is said to be an -properly efficient solution of , if An ordered pair is said to be an -properly efficient pair of , if The following assumptions, (H1) and (H2), for will be imposed in the rest of this paper:(H1)((H1.1)–(H1.3)):(H1.1) is a convex set(H1.2), : (H1.3), : (H2)(Slater constraint qualification): for any and , with , there exists , such that  where

Remark 3. is said to be generalized -subconvexlike on , if assumption (H1.1) is true [1]. is said to be -convexlike on , if assumption (H1.2) is true [2].

3. Scalarization Theorems

The following scalar minimization problem of is discussed in this section.

Definition 4. An ordered pair is said to be an -optimal pair of , if there exist and , such that

Lemma 5 (see [13]). Let be a closed convex pointed cone in and be a closed convex cone in with ; then if and only if .

Theorem 6. Let ; if is an -optimal pair of , then is an -properly efficient pair of .

Proof. Taking arbitrarily , we have that and there exist sequences , , and , such that Therefore, we obtainSince is an -optimal pair of , by Definition 4, we have From and , we obtain Therefore, the right side of (17) is nonnegative; sequentially On the other hand, from , we have also Thus Notice that , and we obtain . Then The proof is complete by Definition 2 and (5).

Theorem 7. If is an -properly efficient pair of , and is generalized -subconvexlike on , then there exists , such that is an -optimal pair of .

Proof. If is an -properly efficient pair of , by Definition 2 and (5), we have Since is generalized -subconvexlike on , from Remark 3, we have that is convex. By Lemma 5, we obtain hence there exists . Since , we obtain By Definition 4, is an -optimal pair of

4. ε-Lagrange Multiplier Theorems

The map is defined as , where , It is easy to prove that , denoted by , is a linear operator from to . A linear operator is said to be nonnegative, denoted by , if . Let denote the set of all linear operators , and let denote the set of all nonnegative linear operators; obviously, .

Remark 8. Let a linear operator , and let a matrix ; by Proposition 1(i), we know that the vector .

The scalar valued Lagrangian function of is defined as where , , , and The vector valued Lagrangian function of is defined as where , , and

Now we consider the following two unconstrained optimization problems:

Lemma 9 (see [1]). Let ; then

Lemma 10. Let ; then if and only if

Proof. Let . If , from Lemma 9, we have It follows that there exist and sequences , such that Thus there exist , , and , such thatIn view of and , there exists a natural number , such that , . It is easy to check that if , then From (26) we have This implies This is a contradiction to the assumption.
Conversely, the sufficient condition can be proved by considering .

Lemma 11 (alternative theorem). Suppose that the set-valued functions , , and satisfy assumptions (H1) and (H2) on ; then exactly one of the following statements is true:(i)There exists , such that , , and .(ii)There exists , , such that

Proof. Obviously, (i) and (ii) cannot hold simultaneously. Otherwise, there exist , , , and , such that By Proposition 1, there exists , , such that and this contradicts (ii). Now we show that if (i) is not true, then (ii) is true. LetIt is easy to show that is a convex set and If (i) is not true, then . Otherwise, assume that ; from (32) there exists , such that By Lemma 10, there exists , such that which is inconsistent with the assumed condition. Hence by the separation theorem of convex sets, there exists , such that Thus for any , , , , , , , and , we haveLetting in (36), we obtain hence Letting in (36), we have also . Now letting and in (36), we have which implies that there exists , such thatAt last we find Contrarily, taking in (39), we obtain and this contradicts (H2). Therefore, (ii) holds.

Theorem 12. Let , ; suppose that , , and satisfy (H1) and (H2) on and is generalized -subconvexlike on . If is an -properly efficient pair of , then there exists , such that

Proof. If is an -properly efficient pair of and is generalized -subconvexlike on , by Theorem 7 and Definition 4, there exists , such that Therefore the following conditions are not true: Since is generalized -subconvexlike on , we can easily prove that is generalized -subconvexlike on Hence , , and satisfy (H1) and (H2) on . From Lemma 11, there exists , , such thatLet ; we have , and (44) can be written as This completes the proof.

Theorem 13. Let and If there exists , such thatthen is an -properly efficient pair of .

Proof. If , there exist and , such that and ; then we have , , andBy (46) and (47), we obtain From and Definition 4, we know that is an -optimal pair of . Therefore, is an -properly efficient pair of by Theorem 6.

Theorem 14. Let , , and , and let , , and satisfy (H1) and (H2) on ; is also generalized -subconvexlike on . If is an -properly efficient pair of , then there exist and , such that is an -properly efficient pair of .

Proof. If is an -properly efficient pair of , by Theorem 12, there exists , such thatTaking satisfying , we can define a map as Then , . Let , . From (49) and , we have Since , , and , . By Definition 4, is a -optimal pair of the scalar problem: From , we obtain that is an -properly efficient pair of by Theorem 6.

Theorem 15. Let and ; if there exist and , such that is an -properly efficient pair of , then is an properly efficient pair of .

Proof. If there exist and , such that is an -properly efficient pair of , by Definition 2 and (5), we haveLet ; we have and . Hence, we obtain thenThis implies By the above expression and (53), we have Therefore, is an -properly efficient pair of by Definition 2 and (5).

5. ε-Proper Saddle Point Theorem

In this section, first we give the concept of -proper saddle point of the set-valued Lagrangian function; then we establish an -proper saddle point theorem.

Definition 16. Let , , and , and an ordered group is said to be an -proper saddle point of Lagrangian function if

Theorem 17. Let , , and If is an -proper saddle point of Lagrangian function , then there exist and , such that the following conditions are satisfied:(i).(ii), .(iii).

Proof. By Definition 16, there exist , , and , such thatFrom (60) and (6), we have That is,Taking in (62), since , , and , we have From the above expression and , we obtainWe assert . Contrarily, by Proposition 1(ii), there exists , such that Letting , we define a map as It is easy to check that , and This contradicts (64), so and . Taking in (64), we obtain . Therefore, condition (iii) holds.
Now, we assert Contrarily, there exists , but . By Proposition 1(ii), there exists , such that Letting , the map is defined as Obviously, by , we obtainOn the other hand, taking in (64), we have and this contradicts (68), so .
Taking in (62), since , , and , we haveFrom , we obtainSuppose that and , such that , letting The map is defined as We obtain This contradicts (71); hence . Therefore, condition (ii) holds. From the fact that and (59), condition (i) holds.

Theorem 18. If an order group is an -proper saddle point of Lagrangian function and , then there exist and , such that is an -properly efficient pair of , where .

Proof. By Theorem 17(i), there exist and , such that ; from (5), we have By Theorem 17(iii), ; hence On the other hand, from the fact that , we obtain . Applying Theorem 15 yields the result.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (11462020) and the Natural Science Foundation of Inner Mongolia Department of Public Education (NJZY032).