Abstract

A new class of second order (K, F) pseudoconvex function is introduced with example. A pair of Wolfe type second order nondifferentiable symmetric dual programs over arbitrary cones with square root term is formulated. The duality results are established under second order (K, F) pseudoconvexity assumption. Also a Wolfe type second order minimax mixed integer programming problem is formulated and the symmetric duality results are established under second order (K, F) pseudoconvexity assumption.

1. Introduction

A mathematical programming with two or more objective functions is called multiobjective programming. Often the several objectives are conflicting in nature. Pareto [1] studied multiobjective problems by reducing them to a single objective one. However, the problems were first explicitly defined and studied by Kuhn and Tucker [2]. They also proposed the definition of proper efficiency which was later modified by Geoffrion [3].

In mathematical programming, a pair of primal and dual programs is called symmetric if the dual of the dual is the primal problem. The duality in linear programming is symmetric. It is not so in nonlinear programming in general. Dorn [4], Dantzig et al. [5], and Mond [6] studied symmetric duality in nonlinear programming assuming the kernel function to be convex in and concave in . Subsequently, Mond and Weir [7] presented a distinct pair of symmetric dual nonlinear programs which admits the relaxation of the convexity/concavity assumption to pseudoconvexity/pseudoconcavity. Mond [6] initiated second order symmetric duality of Wolfe type in nonlinear programming and proved the duality theorems under second order convexity. Mangasarian [8] discussed second order duality in nonlinear programming under inclusion condition. Mond [6, page 93] and Mangasarian [8, page 609] also indicated possible computational advantages of the second order dual over the first order dual. This motivated several authors [3, 6, 9–13] in this field. Yang et al. [13] studied second order multiobjective symmetric dual programs and established the duality relations under F-convexity assumptions. Also Yang et al. [12] formulated a pair of Wolfe type second order nondifferentiable symmetric dual programs containing support function and presented the duality results under F convexity.

Recently, Gulati et al. [14] studied Wolfe and Mond-Weir type second order symmetric duality over arbitrary cones and proved the duality results under generalized convexity assumption. Gulati and Geeta [15] studied Mond-Weir type second order symmetric duality in multiobjective programming over cones and established duality results under pseudoconvexity/K-F convexity assumption. Gulati and Verma [16] formulated a pair of Wolfe type nondifferentiable multiobjective symmetric duality and established the duality results under invexity assumption. Gupta and Kailey [17] formulated a pair of Wolfe type second order nondifferentiable multiobjective symmetric dual programs in which the objective function contains support function and proved the duality results under second order F-convexity assumption. Gupta and Kailey [18] presented second order multiobjective symmetric duality involving cone-convex functions. Saini and Gulati [19] presented a pair of Wolfe type nondifferentiable second order symmetric dual programs over arbitrary cones under second order (K, F)-convexity assumption.

In this paper, motivated by Saini and Gulati [19], a new class of second order (K, F) pseudoconvex/second order (K, F) strongly pseudoconvex function is introduced with example. A pair of Wolfe type second order nondifferentiable symmetric dual programs over arbitrary cone containing square root term is formulated. The duality results are established under second order (K, F) pseudoconvexity assumption.

2. Notation and Preliminaries

The following convention for vectors in will be used:

Definition 1. A set of is called a cone if, for each and , we have . Moreover, if is convex, then it is convex cone.

Definition 2. The positive polar cone of is defined as Let , , and be closed convex cones with nonempty interiors having polars , , and , respectively. Let and be open and . Let .

A general multiobjective nonlinear programming problem can be expressed in the following form.

Primal (P). , is a closed convex cone with nonempty interior in .

Definition 3. A feasible point is weakly efficient solution of (P) if there exist no other such that .

Definition 4. A feasible point is efficient solution of (P) if there exist no other such that .

Definition 5. A function is sublinear in its third argument if, for all ,(1) , for all ,(2) , for  all .

Definition 6. Let be thrice differentiable function. is said to be second order F-pseudoconvex at , if , Now, we are in position to give definition of second order (K, F)-pseudoconvex function and second order strongly (K, F) pseudoconvex function.

Definition 7. The thrice differentiable function is said to be second order (K, F) pseudoconvex at , if ,

Definition 8. The thrice differentiable function is said to be second order strongly (K, F) pseudoconvex at , if ,

Definition 9. is second order (K, F) pseudoconcave, if is second order (K, F) pseudoconvex, and is second order strongly (K, F) pseudoconcave, if is second order strongly (K, F) pseudoconvex function.

Example 10. Let Now at So at So is not second order F convex function.
Now at So is not second order strongly (K, F) pseudoconvex. But So is second order (K, F) pseudoconvex.
Now we can define second order (K, F) pseudoconvexity for a multiobjective function:

Definition 11. A thrice differentiable function is said to be second order (K, F)-pseudoconvex at , for fixed , if there exists sublinear function such that

Definition 12. A thrice differentiable function is said to be second order (K, G)-pseudoconvex at , for fixed , if there exists sublinear function such that

Lemma 13 (generalized Schwartz inequality). Let be a positive semidefinite matrix of order . Then, for all .
The equality holds if for some .

3. Wolfe Type Second Order Multiobjective Nondifferentiable Dual Programs

We consider the following pair of second order Wolfe type nondifferentiable multiobjective programming problems with k-objective.

Primal (SWP). Consider

Dual (SWD). Consider where(1) is thrice differentiable function,(2) and are closed convex cones in and with nonempty interiors, respectively,(3) and are positive polar cones of and , respectively,(4) is a closed convex cone in with and ,(5) ,    are vectors in , and ,    are vectors in ,(6) and    are positive semidefinite matrices of order and , respectively.

Now we establish the following theorem.

Theorem 14 (weak duality theorem). Let be a feasible solution for the primal (WP) and let be a feasible solution for the dual (WD). Suppose there exist sublinear functional and satisfying (1) ,  for  all   ,(2) ,  for  all   .Furthermore assume that, for each , is second order (K, F)-pseudoconvex at for fixed and is second order pseudoconcave at for fixed :

Proof. Since is feasible solution for (WD), from dual constraint (5) we have So Again hypothesis (1) implies . Consider Since is sublinear with respect to third argument, Since , the above inequality can be written as So second order (K, F)-pseudoconvexity of at for fixed implies that This implies that, for , Similarly is feasible solution for (WD), so from primal constraint (1) we have So
Again hypothesis (2) implies . Consider Since is sublinear with respect to third argument, Since , the above inequality can be written as So second order (K, F)-pseudoconcavity of at for fixed implies that This implies that, for , Adding (30) and (37), we get Now from Schwartz inequality (Lemma 13), (17), and (21), we have Also from primal constraint (15), we have For , Similarly from dual constraint (19), we have Using (39), (41), and (42) in (38), we obtain that

Theorem 15 (strong duality). Let be weakly efficient solution of (WP) such that(i) is nonsingular,(ii)the matrix is positive definite,(iii)the set is linearly independent,(iv) . Then there exist such that is feasible for (WD) and two objective values of (WP) and (WD) are equal. Also, if the hypotheses of Theorem 14 are satisfied for all feasible solution of (WP) and (WD), then is an efficient solution of (WD).

Proof. Since is weakly efficient solution of (WD), by the Fritz-John necessary optimality condition on convex cone domain given in Bazaraa and Goode [20], there exist such that the following conditions are satisfied at : Since is nonsingular, (46) implies that We claim that . Indeed if , then (55) implies , which contradicts (54).
Hence Since , using (55) in (45), we get which by hypothesis (ii) and (iv) yields From (55) and (58), we obtain Using (58) and (59) and hypothesis (iii) in (45), we get Again using (58), (59), and (60) in (44), we get Let . Then and so (61) implies Also from (56), (59), and , we obtain Thus, from (52), (62), and (63), we obtain that satisfies the dual constraints (19), (20), (21), and (22).
Thus is feasible for (WD).
Let ; then . From (50) and (59), we get This is a condition of Schwartz inequality: In case , from (51) we get . So (65) implies .
In case , we get . So . Hence . Thus in either case So using (48) and (66), we obtained that the two objective values are equal; that is, Now we claim that is an efficient solution of (WD). If this would not be the case, then there would exist a feasible solution such that This is a contradiction to weak duality Theorem 14.
Hence is efficient solution.

Theorem 16 (converse duality theorem). Let be a weakly efficient solution of (WP) such that (i) is nonsingular,(ii)the matrix is positive definite,(iii)the set is linearly independent,(iv) . Then there exist such that is feasible for (WD) and two objective values of (WP) and (WD) are equal. Also, if the hypotheses of Theorem 14 are satisfied for all feasible solution of (WP) and (WD), then is an efficient solution of (WD).

Proof. The proof follows on lines of Theorem 15.

4. Wolfe Type Minimax Mixed Integer Programming

Let and be two arbitrary sets of integers in and , respectively. Throughout this section, we constrained some of the components of the vector variables and to belong to arbitrary sets of integers and , respectively. Then we write , where and . and are the vectors of the remaining components of and , respectively.

Definition 17. Let be elements of an arbitrary vector space. A vector function will be called additively separable with respect to , if there exist vector function (independent of ) and (independent of ) such that .
We consider the following pair of Wolfe type nondifferentiable minimax mixed integer symmetric primal and dual programs:

Primal (WIP). Consider

Dual (WID). Consider where(1) is thrice differentiable function,(2) and are closed convex cones in and with nonempty interiors, respectively,(3) and are positive polar cones of and , respectively,(4) is a closed convex cone in with and ,(5) are vectors in , and are vectors in ,(6) and are positive semidefinite matrices of order and , respectively.

Theorem 18 (symmetric duality). Let be a weakly efficient solution of (WIP). Also (i) is additively separable with respect to or ; that is, = ,(ii) is thrice differentiable in and ,(iii) is nonsingular,(iv)the vector .Furthermore, for any feasible solution in (WIP) and for any feasible solution in (WID), suppose there exist functional and such that(v) is second order pseudoconvex at with respect to for each and , is second order pseudoconcave at with respect to for each ,(vi) , for all , and , for  all .
Then there exist such that is efficient solution for dual and optimal values (WIP) and (WID) are equal.

Proof. Let where and are a feasible region of primal (WIP) and dual (WID), respectively.
Since is additively separable with respect to or (say with respect to ) from definition, it follows that .
Therefore
So the primal (WIP) can be written as or where (WIP0): Similarly the dual (WID) can be written as where (WID0): For any given and , programs (WIP0) and (WID0) are a pair of Wolfe type second order nondifferentiable multiobjective symmetric dual programs studied in Section 3 and hence in view of hypothesis (ii)–(vi), Theorems 14 and 15 become applicable. Therefore , and we obtain . So the two optimal values are equal and is an efficient solution for the dual.

5. Special Cases

(i) If , , then the problems (SWP) and (SWD) can be reduced to the problem proposed by Gulati et al. [14] as follows.

Primal (WP).

Dual (WD).

(ii) If , , , , and , where , , then the problems (SWP) and (SWD) can be reduced to the problem proposed by Yang et al. [12].

Primal (WP).

Dual (WD).

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.