On Nonsmooth Semi-Infinite Minimax Programming Problem with -Invexity
We are interested in a nonsmooth minimax programming Problem (SIP). Firstly, we establish the necessary optimality conditions theorems for Problem (SIP) when using the well-known Caratheodory's theorem. Under the Lipschitz -invexity assumptions, we derive the sufficiency of the necessary optimality conditions for the same problem. We also formulate dual and establish weak, strong, and strict converse duality theorems for Problem (SIP) and its dual. These results extend several known results to a wider class of problems.
Convexity plays a central role in many aspects of mathematical programming including analysis of stability, sufficient optimality conditions, and duality. Based on convexity assumptions, nonlinear programming problems can be solved efficiently. There have been many attempts to weaken the convexity assumptions in order to treat many practical problems. Therefore, many concepts of generalized convex functions have been introduced and applied to mathematical programming problems in the literature . One of these concepts, invexity, was introduced by Hanson in . Hanson has shown that invexity has a common property in mathematical programming with convexity that Karush-Kuhn-Tucker conditions are sufficient for global optimality of nonlinear programming under the invexity assumptions. Ben-Israel and Mond  introduced the concept of preinvex functions which is a special case of invexity. Many other concepts of generalized convexity such as -invexity , -convexity , -convexity , -convexity , and -invexity  have also been introduced. With these definitions of generalized invexity on the hand, several authors have been interested recently in the optimality conditions and duality results for different classes of minimax programming problems; see [9–12] for details.
Recently, Antczak and Stasiak  generalized the definition of -invexity notion introduced by Caristi et al. and M. V. Ştefănescu and A. Ştefănescu [14, 15] for differentiable optimization problems to the case of mathematical programming problems with locally Lipschitz functions. They proved sufficient optimality conditions and duality results for nondifferentiable optimization problems involving locally Lipschitz -invex functions. Antczak  also considered a class of nonsmooth minimax programming problems in which functions involved are locally Lipschitz -invex. We point out that this locally Lipschitz -invexity includes the -convexity as a special case and Yuan et al.  defined firstly the -convexity with a convex functional .
Due to a growing number of theoretical and practical applications, semi-infinite programming has recently become one of the most substantial research areas in applied mathematics and operations research. For more details on semi-infinite programming we refer to the survey papers [17–19] and for clear understanding of different aspects of semi-infinite programming we refer to . M. V. Ştefănescu and A. Ştefănescu  considered differentiable Problem with new -invexity. However, the results of this kind of programming can not be used to deal with the concrete nonsmooth semi-infinite minimax programming problem as presenting in Example 9 in Section 3 since the objective function is nondifferentiable at . Therefore, we are interested in dealing with nonsmooth Problem with locally Lipschitz -invexity proposed in , in this paper.
The rest of the paper is organized as follows. In Section 2, we present concepts regarding Lipschtiz -invexity. In Section 3, we present not only necessary but also sufficient optimality conditions for nonsmooth Problem . When the necessary optimality conditions and the -invexity concept are utilized, dual Problem is formulated for the primal and duality results between them are presented in Section 4. Section 5 is our conclusions.
2. Notations and Preliminaries
In this section, we provide some definitions and results that we shall use in the sequel. Let be a subset of and denote , , , and .
Definition 1. A real-valued function is said to be locally Lipschitz on if, for any , there exist a neighborhood of and a positive constant such that
Definition 2 (see ). Let and . If exists, then is said to be the Clarke derivative of at in the direction . If this limit superior exists for all , then is called Clarke differentiable at . The set is called the Clarke subgradient of at .
Note that if a function is locally Lipschitz, then its Clarke subgradient must exist.
The definition of the locally Lipschitz -invexity was introduced by Antczak and Stasiak ; see also the following Definition 3. This generalized invexity was introduced as a generalization of differentiable -invexity notion defined by Caristi et al. and M. V. Ştefănescu and A. Ştefănescu in [14, 17]. The main tool used in the definition of the locally Lipschitz -invexity notion is the above Clarke generalized subgradient (see Definition 2).
Definition 3. Let be a real-valued Lipschitz function on . For fixed , let be convex with respect to the third argument on such that for every and any . If there exists a real-valued function such that holds for all (), then is said to be (strictly) locally Lipschitz -invex at on or shortly (strictly) -invex at on . If is (strictly) locally Lipschitz -invex at any of , then is (strictly) locally Lipschitz -invex on .
Remark 4. In order to define an analogous class of (strictly) locally Lipschitz -incave functions, the direction of the inequality in the definition of these functions should be changed to the opposite one.
In this paper, we deal with the nonsmooth semi-infinite minimax programming Problem with the locally Lipschitz -invexity proposed by Antczak and Stasiak . Here, Problem is where and are compact subsets of some Hausdorff topological spaces, , . Let be the set of feasible solutions of Problem ; in other words, . For convenience, let us define the following sets for every :
If , then represents the index set of the active restrictions at . Note that when is not empty.
Consider the nonlinear programming problem where . A particular case of Problem is the minimax problem in which the functions , are given by respectively. Let . Consider the following unconstrained optimization problem : where . Then, the relationship between Problems and is given in the following lemma.
Condition 1. We assume that (a) the sets and are compact; (b) the function is upper semicontinuous in , and the function is upper semicontinuous in ; (c) the function is locally Lipschitz in and uniformly for in , and the function is locally Lipschitz in and uniformly for in ; (d) the function is regular in ; that is, , where the symbol denotes the derivative with respect to ; also the function is regular in ; (e) the set-valued map is upper semicontinuous in , and the set-valued map is upper semicontinuous in .
Condition 2. For any finite subset , for some , the equality with , , , implies that
Clarke [22, Theorem 2.1] has shown that, under the assumptions (a)–(e) of Condition 1, the maximum function defined by (6) is locally Lipschitz; exists and is given by the formula where denotes the inner product of vectors and . Moreover, the in (6) can be replaced by and the subgradient is given by
Similarly, the maximum function defined by (7) is locally Lipschitz; and are given by respectively.
3. Optimality Conditions
In this section, we establish not only the necessary optimality conditions theorems but also the sufficient optimality conditions theorems for Problem with the functions involved being locally Lipschitz with respect to the variable .
Theorem 6 (necessary optimality conditions). Let be an optimal solution of . One also assume that Condition 1 holds. Then there exist nonnegative integers and with , vectors (), (), and scalars (), () such that
Here, one allow the case, where if , then the set is empty; similarly, if , then the set is empty.
Proof. Let be a local minimizer for Problem . This means that is a local minimizer for Problem , where and are given by (6) and (7), respectively. Therefore, is a local minimizer for Problem .
By Condition 1 and [22, Theorem 2.1], is locally Lipschitzian and regular at , so the function has the same properties. Then, using [21, Propositions and ], we obtain here the equality is used in the fourth equality. Hence, by Caratheodory's theorem, there exist the nonnegative integers and and the scalars () and () such that for some and . Note that, for each , means that there exists such that , and denote this by . Similarly, there exists such that for each . Now the desired inequalities (12) and (13) can be deduced from the above discussion.
Theorem 7 (necessary optimality conditions). Let be an optimal solution of . One also assume that Conditions 1 and 2 hold. Then there exist the nonnegative integers and with , the vectors (), (), and the scalars (), () satisfying (12) and
Proof. By Theorem 6, we need to prove , on the contrary, that is, , then one obtains from (12) and (13) that
respectively. By (19), there exist for satisfying
Now one obtains from the assumptions of Condition 2 that for ; this contradicts to (20), and we obtain the desired results.
Next, we derive a sufficient optimality conditions theorem for Problem under the assumption of -invexity as defined in Definition 3.
Proof. Suppose, contrary to the result, that is not an optimal solution for Problem . Hence, there exists such that
Now, we can write the following statement:
By the generalized invexity assumptions of and , we have
Employing (26) to (25), we have
By (18) and the convexity of , we deduce that
This, together with (40) and the assumption for any , follows that
This is a contradiction to condition (12).
Example 9. Let and . Define
Then, is -invex at for each , is -invex at for each , and
Consider . Since , then we can assume . Therefore, where . Now, from Theorem 8, we can say that is an optimal solution to .
Making use of the optimality conditions of the preceding section, we present dual Problem to the primal one and establish weak, strong, and strict converse duality theorems. For convenience, we use the following notations:
Note that if is empty for some , then define .
Proof. Suppose to the contrary that . Therefore, we obtain
Thus, we obtain
We obtain that
Similar to the proof of Theorem 8, by (46) and the generalized invexity assumptions of and , we have
This follows that
Thus, we have a contradiction to (35). So .
Theorem 11 (strong duality). Let Problem satisfy Conditions 1 and 2; let be an optimal solution of Problem . If the hypothesis of Theorem 10 holds for all -feasible points , then there exists , such that is a -optimal solution, and the two problems and have the same optimal values.
Proof. By Theorem 7, there exists (), satisfying the requirements specified in the theorem, such that is a -feasible solution; then the optimality of this feasible solution for follows from Theorem 10.
Theorem 12 (strict converse duality). Let and be optimal solutions of and , respectively. Suppose that are strictly -invex at for each and are -invex at for each . If then ; that is, is a -optimal solution, and
Proof. Suppose to the contrary that . By the generalized invexity assumptions of and , we have
Therefore, we obtain from (51) and the convexity of that holds for all and . This, together with (35), (49), and follows that while
From the above inequality, we can conclude that there exists , such that or
It follows that
On the other hand, we know from Theorem 10 that
This contradicts to (59).
In this paper, we have discussed a nonsmooth semi-infinite minimax programming Problem . We have extended the necessary optimality conditions for Problem considered in  to the nonsmooth case; we have also extended the sufficient optimality conditions and dual results of Problem addressed by M. V. Ştefănescu and A. Ştefănescu in  to the nonsmooth case under the Lipschitz -invexity assumptions as defined in . More exactly, we have established the necessary optimality conditions theorems for the Problem when using Caratheodory's theorem. Under the Lipschitz -invexity assumptions as defined in , we have derived the sufficiency of the necessary optimality conditions for Problem . In the end, we have constructed a dual model and derived duality results between Problems and . These results extend several known results to a wider class of problems.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
This research is supported by the Natural Science Foundation of Guangdong Province (Grant no. S2013010013101) and the Foundation of Hanshan Normal University (Grant nos. QD20131101 and LQ200905).
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