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Nonsmooth Multiobjective Fractional Programming with Local Lipschitz Exponential -Invexity
We study nonsmooth multiobjective fractional programming problem containing local Lipschitz exponential -invex functions with respect to and . We introduce a new concept of nonconvex functions, called exponential -invex functions. Base on the generalized invex functions, we establish sufficient optimality conditions for a feasible point to be an efficient solution. Furthermore, employing optimality conditions to perform Mond-Weir type duality model and prove the duality theorems including weak duality, strong duality, and strict converse duality theorem under exponential -invexity assumptions. Consequently, the optimal values of the primal problem and the Mond-Weir type duality problem have no duality gap under the framework of exponential -invexity.
Convexity plays an important role in mathematical programming problems, some of which are sufficient optimality conditions or duality theorems. The sufficient optimality conditions and duality theorems are being studied by extending the concept of convexity. One of the most generalizations of convexity of differentiable function in optimality theory was introduced by Hanson . Then the characteristics of invexity—an invariant convexity—were applied in mathematical programming (cf. [1–7]). Besides, the concept of invexity of differentiable functions has been extended to the case of nonsmooth functions (cf. [8–17]). After Clarke  defined generalized derivative and subdifferential on local Lipschitz functions, many practical problems are described under nonsmooth functions. For example, Reiland  used the generalized gradient of Clarke  to define nondifferentiable invexity for Lipschitz real valued functions. Later on, with generalized invex Lipschitz functions, optimality conditions and duality theorems were established in nonsmooth mathematical programming problems (cf. [8–17]). Indeed, problems of multiobjective factional programming have various types of optimization problems, for example, financial and economic problems, game theory, and all optimal decision problems. In multiobjective programming problems, when the necessary optimality conditions are established, the conditions for searching an optimal solution will be employed. That is, extra reasonable assumptions for the necessary optimality conditions are needed in order to prove the sufficient optimality conditions. Moreover, these reasonable assumptions are various (e.g., generalized convexity, generalized invexity, set-value functions, and complex functions). When the existence of optimality solution is approved in the sufficient optimality theorems, the optimality conditions to investigate the duality models could be employed. Then the duality theorems could be proved. The better condition is that there is no duality gap between primal problems and duality problems.
In this paper, we focus a system of nondifferentiable multiobjective nonlinear fractional programming problem as the following form: subject to with where is a separable reflexive Banach space in the Euclidean -space , , , , and are locally Lipschitz functions on . Without loss of generality, we may assume that , for all , .
In this paper, we introduce a new class of Lipschitz functions, namely, exponential --invex Lipschitz functions which are motivated from the results of Antczak , Clarke , and Reiland . We employ this exponential --invexity and necessary optimality conditions to establish the sufficient optimality conditions on a nondifferentiable multiobjective fractional programming problem . Using optimality conditions, we construct Mond-Weir duality model for the primal problem and prove that the duality theorems have the same optimal value as the primal problem involving --invexity.
2. Definitions and Preliminaries
Let denote Euclidean space, and let denote the order cone. For cone partial order, if , in , we define:(1) if and only if for all ;(2) if and only if for all ;(3) if and only if for all ;(4) if and only if and for some .
Definition 1. Let be an open subset of . The function is said to be locally Lipschitz at if there exists a positive real constant and a neighborhood of such that where is an arbitrary norm in .
For any vector in , the generalized directional derivative of at in the direction in Clarke’s sense  is defined by
The generalized subdifferential of at is defined by the set where is the dual space of and stands for the dual pair of and .
Evidently, for any and in . If is a convex function, then is coincid with usual subdifferential .
Definition 2 (see ). is said to be regular at if for any , the one-side directional derivative exists and .
Lemma 3 (see ). Let and be Lipschitz near , and suppose . Then is Lipschitz near and one has If and are regular at , then equality holds to the above , that is, the subdifferential is singleton and is regular at .
Let be a local Lipschitz function. For , we define If , we say that the problem has constraint qualification at (cf. ).
On the basis of the definition for invex functions of Lipschitz functions in Reiland , we modified Antczak’s generalized --invex with respect to and for differentiable to nondifferentiable case for a class of locally Lipschitz exponential --invex functions as follows.
Definition 4. Let , be arbitrary real numbers. A locally Lipschitz function is said to be exponential --invex (strictly) at with respect to w.r.t. (for brevity) if there exists a function with property only if in and a function such that for each , the following inequality holds for : If or is zero, then (8) can give some modification by using the limit of or .(i)If , in (8), then we deduce that (ii)If , then (8) becomes (iii)If , , then (10) holds.
Remark 5. All theorems in our work will be described only in the case of and . We omit the proof of other cases like in (i), (ii), and (iii).
A feasible solution to is said to be an efficient solution to if there is no such that .
3. Optimality Conditions
In this section, we establish some sufficient optimality conditions. The necessary optimality conditions to the primal problem given by  and the subproblems of , for , given by  are used in our theorem.
Lemma 6 (see ). is an optimal solution to problem if and only if solves , where is as the following problem:
Theorem 7 (see , necessary optimality conditions). If is an optimal solution of and has a constraint qualification, for , , then, there exist and such that where
For convenience, let where .
Now, we give a useful lemma whose simple proof is omitted in this paper.
Lemma 8. If , where is a real function, then .
The sufficient optimality conditions can be deduced from the converse of necessary optimality conditions with extra assumptions. Since the sufficient optimality theorem is various depending on extra assumptions, the duality model is also various. We establish the sufficient optimality conditions and duality theorems involving the exponential --invexity.
Theorem 9. Let be a feasible solution of such that there exist , satisfying the conditions (13)~(16) at . Furthermore, suppose that any one of the conditions and hold: (a) is an exponential --invex function at in w.r.t. and , (b) is an exponential --invex function at in w.r.t. and , and is an exponential --invex function at in w.r.t. the same function and but not necessarily, equal to .
Then, is an efficient solution to problem .
Proof. Suppose that is -feasible. By expression (13), there exist , , and , such that
and that is a zero vector of .
From the above expression, the dual pair of If is not an efficient solution to problem , then there exists -feasible such that that is, Thus, we have From relations , (14), and (16), we obtain where .
If hypothesis (a) holds, is an exponential --invexity w.r.t. and at for all . Then by Definition 4, we have that the following inequality holds. Because of equality (20) and inequality (25), we obtain According to Lemma 8 and , we have
Equation (23) along with (24) yields which contradicts inequality (27).
If hypothesis (b) holds, is an exponential --invex function w.r.t. and at for all , that is, -feasible. Then by Definition 4, we have the following inequality:
From inequalities (24) and (29), we have By inequality (30) and multiplying (20) by , it yields that
Since is an exponential --invex function w.r.t. and at for all , that is, -feasible then by Definition 4, we have From inequalities (31) and (32), we obtain By Lemma 8 and , we get
If is not an efficient solution to problem , then we reduce inequality (23) in the same way. But inequality (34) contradicts inequality (23). Hence, the proof is complete.
4. Mond-Weir Type Duality Model
In order to propose Mond-Weir type duality model, it is convenient to restate the necessary conditions in Theorem 7 as the following form. Mainly, we use the expressions (13) and (15) to get Then putting in the above expression, we obtain Consequently, from inequality (14), it yields that where . For simplicity, we write still by . Then the result of Theorem 7 can be restated as the following theorem.
Theorem 10 (necessary optimality conditions). If is an efficient solution to and satisfyies constraint qualification in , , then, there exist , such that
For any , if we use instead of satisfying the necessary conditions (38)~(40) as the constraints of a new dual problem, namely, Mond-Weir type dual , then it constitutes by a maximization programming problem with the same objective function as the problem , and we use the necessary optimality conditions of as the constraint of the new problem . Precisely, we can state this dual problem as the maximization problem as the following form: subject to the resultant of necessary condition in Theorem 10:
Then we can derive the following weak duality theorem between and .
Theorem 11 (weak duality). Let and be -feasible and -feasible, respectively. Denote a function by
with . Suppose that is an exponential --invex function at w.r.t. and .
Proof. Let and be and -feasible, respectively. From expression (38), there exist , , and , to satisfy
Then, the dual pair of yields Since is an exponential --invex function w.r.t. and at , we have the following inequality: By the above inequality and equality (48), we obtain According to Lemma 8 and , we have
We want to prove that .
Suppose on the contrary that . Then and there is some index such that Then by , we have Since , it follows from (43), (44), and (54) that This implies that which contradicts inequality (51), and the proof of theorem is complete.
Theorem 12 (strong duality). Let be the efficient solution of problem satisfying the constraint qualification at in , . Then there exist and such that -feasible. If the hypotheses of Theorem 11 are fulfilled, then is an efficient solution to problem . Furthermore, the efficient values of and are equal.
Proof. Let be an efficient solution to problem . Then there exist , such that satisfies (42)~(44) that is, is a feasible solution for the problem . Actually, is also an efficient solution of .
Suppose on the contrary that if were not an efficient solution to , then there exists a feasible solution of such that and there is a , It follows that which contradicts the weak duality Theorem 11. Hence, is an efficient solution of and the efficient values of and are clearly equal.
Theorem 13 (strict converse duality). Let and be the efficient solutions of and , respectively. Denote a function by with . If is a strictly exponential --invex function at w.r.t. and for all optimal vectors in and in , respectively, then and the efficient values of and are equal.
Proof. Suppose that . From expression (42), there exist , , and , such that
It follows that the dual pair in becomes From Theorem 12, we see that there exist and such that is the efficient solution of and By inequality (43) and equality (62), it becomes Eliminating the dominators in (63), we get or According to the above equality and by the property (44), reduces to From relations , (44), (66), and , we obtain Hence, we reduce Since is a strictly exponential --invex function w.r.t. and at , we have From (68) and (69), we obtain This contradicts equality (61). Hence, the proof of theorem is complete.
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