Abstract
In this paper, we derive necessary and sufficient optimality conditions for a general minimax programming problem involving some classes of generalized convexities with the tool-right upper-Dini-derivative. Moreover, using the concept of optimality conditions, Mond-Weir type duality theory has been developed for such a minimax programming problem.
MSC:26A51, 49J35, 90C32.
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1 Introduction
The minimax approach to optimization theory certainly is not new. It takes its origins in von Neumann’s game theory. The broad spectrum of existing results and applications of minimax theory in the field of optimization is captured in the book edited by Du et al. [1]. Starting with work of Schmittendorf [2], minimax programming problems have been studied by several authors; for example, see [3–9] and the references cited therein.
Convexity is a sufficient but not necessary condition for many important results of mathematical programming, since there are diverse extensions of the notion of convexity bearing the same properties. Moreover, it is well known that a function is convex iff its restriction to each line segment in its domain is convex. This property inspired Ortega and Rheinboldt [10] to introduce an important generalization of convex functions by replacing a line segment joining two points by a continuous arc and called them arcwise connected functions defined on arcwise connected sets.
Following the idea of arcwise convexity, Avriel and Zang [11] introduced Q-connected (QCN) functions and P-connected (PCN) functions and also they have discussed necessary and sufficient local-global minimum properties of these functions. Some elementary properties of these functions in terms of their directional derivatives have been studied by Bhatia and Mehra [12]. Bhatia and Mehra [12] also established optimality conditions for scalar-valued nonlinear programming problems involving these functions.
To relax the definition of arcwise convexity in terms of directional derivative recently Yuan and Liu [13] introduced the concept of -right upper-Dini-derivative locally arcwise connected with respect to the arc H and established optimality and duality results for a nonlinear multiobjective programming problem. In this paper, we use generalized convex functions, in terms of the right upper-Dini-derivative to derive necessary and sufficient optimality conditions for a general minimax programming problem and duality results for its Mond-Weir type dual model.
This paper is structured as follows: Some preliminary concepts and properties regarding generalized convex functions are given in Section 2. In Section 3, we establish necessary and sufficient optimality conditions for a general minimax programming problem involving generalized convex functions. In Section 4, we establish appropriate duality theorems for a Mond-Weir type dual problem. Finally, in Section 5 we summarize our main results and also point out some further research opportunities.
2 Preliminaries
Let denote the n-dimensional Euclidean space, its nonnegative orthant and . For a nonempty set Q in a topological vector pace E, denote the closure of Q and
denotes the dual cone of Q, where is the dual space of E.
For some nonempty subset Y, let denote the product space in a product topology. Then the topological dual space of is the generalized finite sequence space consisting of all the functions with finite support [14]. The set denote the convex cone of all nonnegative functions on Y. Then the topological dual of is given by
Now, we recall some well-known results and concepts which will be used in the sequel.
Definition 2.1 [15]
A set is said to be an arcwise connected set if, for every , , there exists a continuous vector-valued function , called an arc, such that
Definition 2.2 [13]
Let φ be a real-valued function defined on an arcwise connected set . Let , and be the arc connecting and in X. The right upper-Dini-derivative of φ with respect to at is defined as follows:
Using this upper-Dini-derivative concept, Yuan and Liu [13] introduced a class of functions, which called -right upper-Dini-derivative function. For convenience, we use the following notations.
Definition 2.3 [13]
A set is said to be locally arcwise connected at if for any and there exist a positive number , with , and a continuous arc such that for any .
The set X is locally arcwise connected on X if X is locally arcwise connected at any .
Definition 2.4 [13]
Let be a locally arcwise connected set and be a real function defined on X. The function φ is said to be -right upper-Dini-derivative locally arcwise connected with respect to H at , if there exist real functions , such that
If φ is -right upper-Dini-derivative locally arcwise connected with respect to H at for any , then φ is called -right upper-Dini-derivative locally arcwise connected with respect to H on X.
Remark 2.1 It revealed by an example given in [13] that there exists a function, which is -right upper-Dini-derivative locally arcwise connected but neither d-ρ--invex [16] nor d-invex [17] nor directional differentially B-arcwise connected [15].
Now we define the notions of ρ-generalized-pseudo-right upper-Dini-derivative locally arcwise connected, strictly ρ-generalized-pseudo-right upper-Dini-derivative locally arcwise connected and ρ-generalized-quasi-right upper-Dini-derivative locally arcwise connected functions.
Definition 2.5 The function is said to be ρ-generalized-pseudo-right upper-Dini-derivative locally arcwise connected (with respect to H) at , if there exists a real function such that
equivalently
The function is said to be ρ-generalized-pseudo-right upper-Dini-derivative locally arcwise connected (with respect to H) on X if it is ρ-generalized-pseudo-right upper-Dini-derivative locally arcwise connected (with respect to H) at any .
The following example shows that there exists a function which is ρ-generalized-pseudo-right upper-Dini-derivative locally arcwise connected but not -right upper-Dini-derivative locally arcwise connected with respect to the arc H.
Example 2.1 Let and the function be defined by
For any, , defining the arc by
Note that, by the definition of right upper-Dini-derivative by (1), for we have
Let be defined by
Now, for , it follows that
This means that φ is ρ-generalized-pseudo-right upper-Dini-derivative locally arcwise connected (with respect to H) at . But φ is not a -right upper-Dini-derivative locally arcwise connected with respect to same arc H and ρ at because for and , we can see that
Definition 2.6 The function is said to be strictly ρ-generalized-pseudo-right upper-Dini-derivative locally arcwise connected (with respect to H) at , if there exists a real function such that
equivalently
The function is said to be strictly ρ-generalized-pseudo-right upper-Dini-derivative locally arcwise connected (with respect to H) on X if it is strictly ρ-generalized-pseudo-right upper-Dini-derivative locally arcwise connected (with respect to H) at any .
Definition 2.7 The function is said to be ρ-generalized-quasi-right upper-Dini-derivative locally arcwise connected with respect to H at , if there exists a real function such that
equivalently
The function is said to be ρ-generalized-quasi-right upper-Dini-derivative locally arcwise connected (with respect to H) on X if it is ρ-generalized-quasi-right upper-Dini-derivative locally arcwise connected (with respect to H) at any .
The next example shows that there exists a function which is ρ-generalized-quasi-right upper-Dini-derivative locally arcwise connected but neither -right upper-Dini-derivative locally arcwise connected nor ρ-generalized-pseudo-right upper-Dini-derivative locally arcwise connected with respect to the arc H.
Example 2.2 Let and the function be defined by
For any, , defining the arc by
Clearly, for we have
Let be defined by
Now, we can easily verify that φ is ρ-generalized-quasi-right upper-Dini-derivative locally arcwise connected (with respect to H) at . However, for , and , we can deduce that
and
Hence, φ is neither -right upper-Dini-derivative locally arcwise connected nor ρ-generalized-pseudo-right upper-Dini-derivative locally arcwise connected with respect to same arc H and ρ at .
Definition 2.8 [13]
A function is called preinvex (with respect to ) on X if there exists a vector-valued function η such that
holds for all and any .
Definition 2.9 [13]
A function is said to be convexlike if for any and , there is such that
Remark 2.2 The convex and the preinvex functions are convexlike functions.
In the next section we will use the following version of Theorem 2.3 from [9].
Lemma 2.1 Let and , where X and Y are arbitrary nonempty sets. Let the pair be convexlike on X. Assume that for some neighborhood U of 0 in and a constant , the set is a nonempty closed subset of , where
Then exactly one of the following systems is solvable:
-
(I)
, , ,
-
(II)
∃ an integer , scalars , , and vectors , , such that and .
3 Optimality conditions
Consider the following general minimax programming problem:
where , , X is an open arcwise connected subset of , Y is a compact subset of and is continuous on Y for every . denote the set of feasible solutions of (P).
For , we define
In view of the continuity of on Y and compactness of Y, it is clear that is nonempty compact subset of Y, . Throughout this paper we assume that the right upper-Dini-derivatives of the functions , , with respect to an arc at exist , , , and is continuous on Y, , . Also assume that , is continuous on X.
The following lemma can be proved without difficulty on the same lines as in Lemma 3.1 (Mehra and Bhatia [9]).
Lemma 3.1 Let be an optimal solution of (P). Then the system
has no solution .
We now prove the following theorem by using Lemmas 2.1 and 3.1, which gives the necessary optimality conditions for an optimal solution of problem (P).
Theorem 3.1 (Necessary optimality conditions)
Let be an optimal solution of (P). Further, let , , be convexlike functions of x on X and let there exist a neighborhood U of 0 in and a constant such that is a nonempty closed set, where
Then there exist an integer , scalars , , , , and vectors , , such that
Proof If is an optimal solution of (P) then, by Lemma 3.1, the system (2) has no solution . But the assumption of Lemma 2.1 also holds and since the system (2) has no solution , we obtain the result that there exist an integer , scalars , , , , and vectors , , such that
and
If we put , for , by (3) and (4) we obtain the required result. □
Now, we prove the following sufficient optimality conditions for the considered minimax problem (P) under generalized convexity with upper-Dini-derivative concept.
Theorem 3.2 (Sufficient optimality conditions)
Let and there exist an integer , scalars , , , , , and vectors , , such that ,
Also, assume that
-
(i)
for , is -right upper-Dini-derivative locally arcwise connected (with respect to H) at ,
-
(ii)
for , is -right upper-Dini-derivative locally arcwise connected (with respect to H) at ,
-
(iii)
.
Then is an optimal solution of (P).
Proof Suppose to the contrary that is not an optimal solution of (P). Then there exists an such that
Further, as , we have
Also, , , we have
Thus, from the above three inequalities, we get
Using , and , we obtain
For , , , we have , which in view of (6) implies that
Now, by (7) and (8) we obtain
On the other hand, from the assumptions that , and , are and -right upper-Dini-derivative locally arcwise connected (with respect to H) at , we have
From (10) and (11) together with , , and , , we get
By (5) and using , , it follows that
which is a contradiction to (9). Hence is an optimum solution for (P) and the theorem is proved. □
Theorem 3.3 (Sufficient optimality conditions)
Let and there exist an integer , scalars , , , , and vectors , , such that the conditions (5) and (6) of Theorem 3.2 hold. Also, assume that
-
(i)
is -generalized-pseudo-right upper-Dini-derivative locally arcwise connected (with respect to H) at ,
-
(ii)
is -generalized-quasi-right upper-Dini-derivative locally arcwise connected (with respect to H) at ,
-
(iii)
.
Then is an optimal solution of (P).
Proof Suppose to the contrary that is not an optimal solution of (P) and following the proof of Theorem 3.2, we have
which by -generalized-pseudo-right upper-Dini-derivative locally arcwise connected (with respect to H) of at , we have
For , , , we have , which in view of (6) implies that
which by -generalized-quasi-right upper-Dini-derivative locally arcwise connected (with respect to H) of at , we have
By (12) and (13), we get
where the last inequality is according to . Therefore,
which is a contradiction to (5). Hence is an optimum solution for (P) and the theorem is proved. □
Theorem 3.4 (Sufficient optimality conditions)
Let and there exist an integer , scalars , , , , , and vectors , , such that the conditions (5) and (6) of Theorem 3.2 hold. Also, assume that
-
(i)
is strictly -generalized-pseudo-right upper-Dini-derivative locally arcwise connected (with respect to H) at ,
-
(ii)
is -generalized-quasi-right upper-Dini-derivative locally arcwise connected (with respect to H) at ,
-
(iii)
.
Then is an optimal solution of (P).
Proof The proof follows along similar lines as the proof of Theorem 3.3 and hence is omitted. □
Theorem 3.5 (Sufficient optimality conditions)
Let and there exist an integer , scalars , , , , , and vectors , , such that the conditions (5) and (6) of Theorem 3.2 hold. Also, assume that
-
(i)
is -generalized-quasi-right upper-Dini-derivative locally arcwise connected with respect to H at ,
-
(ii)
for , , is -generalized-quasi-right upper-Dini-derivative locally arcwise connected and is strictly -generalized-pseudo-right upper-Dini-derivative locally arcwise connected with respect to H at , with ,
-
(iii)
.
Then is an optimal solution of (P).
Proof Suppose to the contrary that is not an optimal solution of (P) and following the proof of Theorem 3.2, we have
which by -generalized-quasi-right upper-Dini-derivative locally arcwise connected of with respect to H at , we have
Since for and for , we have
which by -generalized-quasi-right upper-Dini-derivative locally arcwise connected of , , and strictly -generalized-pseudo-right upper-Dini-derivative locally arcwise connected of with respect to H at , we have
Since , , , and for , from (15) and (16), we get
By (14) and (17), we get
where the last inequality is according to . Therefore,
which is a contradiction to (5). Hence is an optimum solution for (P) and the theorem is proved. □
4 Duality
This section deals with the duality theorems for the following Mond-Weir type dual (D) of minimax problem (P):
where = { is an integer, , , , for some , }, and denotes the set of all satisfying
If for a triplet in the set is empty then we define the supremum over it to be −∞.
Theorem 4.1 (Weak duality)
Let x and be feasible solutions of (P) and (D), respectively. Assume that
-
(i)
is -generalized-pseudo-right upper-Dini-derivative locally arcwise connected (with respect to H) at z,
-
(ii)
is -generalized-quasi-right upper-Dini-derivative locally arcwise connected (with respect to H) at z,
-
(iii)
.
Then
Proof Suppose to the contrary that
Thus, we have
It follows from , , and , that
which by -generalized-pseudo-right upper-Dini-derivative locally arcwise connected (with respect to H) of at z, we have
For , , , we have , which in view of (19) implies that
which by -generalized-quasi-right upper-Dini-derivative locally arcwise connected (with respect to H) of at z, we have
By (20) and (21), we get
where the last inequality is according to . Therefore,
which is a contradiction to (18). Hence the theorem is proved. □
Theorem 4.2 (Strong duality)
Let be an optimal solution of (P). Assume that the conditions of Theorem 3.1 are satisfied. Then there exist , and such that is a feasible solution of (D) and the two objectives have same values. If, in addition, the assumption of weak duality Theorem 4.1 hold for all feasible solutions of (D), then is an optimal solution of (D).
Proof Since is an optimal solution for (P) and all the conditions of Theorem 3.1 are satisfied, there exist , and such that is a feasible solution of (D) and the two objective values are equal. The optimality of for (D) thus follows from Theorem 4.1. □
Theorem 4.3 (Strict converse duality)
Let and be optimal solutions of (P) and (D), respectively. Assume that the hypothesis of Theorem 4.2 is fulfilled. Also, assume that
-
(i)
is strictly -generalized-pseudo-right upper-Dini-derivative locally arcwise connected (with respect to H) at ,
-
(ii)
is -generalized-quasi-right upper-Dini-derivative locally arcwise connected (with respect to H) at ,
-
(iii)
.
Then .
Proof Suppose to the contrary that . According to Theorem 4.2, we know that there exist , and such that is a feasible solution of (D) and
Thus, we have
It follows from , , and , that
Now proceeding on the same lines as in Theorem 4.1, we get
which is a contradiction to (18). Hence the theorem is proved. □
5 Conclusion
In this study we have established necessary and sufficient optimality conditions under generalized convexity using the tool-right upper-Dini-derivative for a general minimax programming problem. Mond-Weir type duality theory is also obtained. These results can be extended to the following semiinfinite minimax programming problem (SIP) with the tool-right upper-Dini-derivative:
where is a nonempty open arcwise connected set, Y is a compact metrizable topological space, is a real-valued function defined on X. and are compact subsets of complete metric spaces, for each , is a real-valued function defined on X for all , for each , is a real-valued function defined on X for all , for each and , , and are continuous real-valued functions defined, respectively, on and for all . We shall investigate this semiinfinite programming problem in subsequent papers.
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Acknowledgements
The research of the second and fourth author is financially supported by King Fahd University of Petroleum and Minerals, Saudi Arabia under the Internal Research Project No. IN131026.
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All authors carried out the proof. All authors conceived of the study and participated in its design and coordination. All authors read and approved the final manuscript.
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Jayswal, A., Ahmad, I., Kummari, K. et al. On minimax programming problems involving right upper-Dini-derivative functions. J Inequal Appl 2014, 326 (2014). https://doi.org/10.1186/1029-242X-2014-326
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DOI: https://doi.org/10.1186/1029-242X-2014-326