HIGHER-ORDER SYMMETRIC DUALITY FOR MULTIOBJECTIVE PROGRAMMING WITH CONE CONSTRAINTS

. In this work, a pair of higher-order symmetric dual multiobjective optimization problems is formulated. Weak, strong and converse duality theorems are established under suitable assumptions. Some examples are also given to illustrate our main results. Furthermore, certain deﬁciencies in the formulations and the proof of the work of Kassem [Applied Mathematics and Computation, 209 (2009), 405–409] are pointed out.


1.
Introduction. Symmetric duality for nonlinear optimization problems was first introduced by Dorn [8] in the sense that the dual of the dual is recast in the form of the primal. Subsequently, first-order symmetric duality for differentiable optimization problems has been studied by many researchers like Dantzig et al. [6], Bazaraa and Goode [4], Chandra and Kumar [5] and Mond and Weir [15].
Motivated by the concept of second-and higher-order duality in nonlinear programming problems introduced by Mangasarian [13], several researchers have worked extensively in this field, due to the computational advantage that it provides better tighter bounds for the primal problem than the first-order duality. For example, Yang et al. [20] discussed higher-order symmetric duality of multiobjective optimization problems under invexity conditions. Padhan and Nahak [16] established higher-order duality results for a pair of Wolfe type and Mond-Weir type higher-order multiobjective symmetric dual problems under higher-order invexity and higher-order pseudoinvexity assumptions. Agarwal et al. [1] presented strong duality theorem for a pair of Mond-Weir type nondifferentiable multiobjective higher-order symmetric dual problems over arbitrary cones under higher-order K-F -convexity assumptions. Ahmad [2] formulated a unified higher-order symmetric dual for a nondifferentiable multiobjective programming problem and presented duality results under suitable assumptions. Suneja and Louhan [18] established weak, strong and converse duality results for a pair of Wolfe type higher-order symmetric dual problems over cones under the assumption of higher-order coneinvexity. They also obtained weak, strong and converse duality results for the pair of Mond-Weir type higher-order symmetric dual problems. Kassem [12] studied a pair of higher-order multiobjective symmetric dual problems under generalized cone-pseudoinvexity. However, (i) In [12], the objective functions of higher-order symmetric dual models are scalars. But, in the weak, strong and converse duality theorems, all functions are treated as vectors. If so, it makes the formulas (1) "−[∇ y f (x, y) + ∇ p h(x, y, p)] ∈ C + 2 " and (4) "∇ u f (u, v)+∇ q g(u, v, r) ∈ C + 1 " in [12] unreasonable. Because the left terms are matrices. This leads to incorrect formulations of the primal and the dual problems.
(ii) From an algebraic point of view, the statement in [12] that For the sake of improving these results of Kassem [12], in this paper we present a modified pair of higher-order symmetric dual models, and establish various types of duality results under appropriate assumptions. This fills in some gaps in the work of Kassem [12].
The rest of this paper is organized as follows. In Section 2, some preliminaries are provided. New concepts of higher-order strong K−pseudoinvexity and higher-order strict K−pseudoinvexity are presented. Moreover, an example is constructed to show the relationship between higher-order strong K−pseudoinvexity and higherorder strict K−pseudoinvexity. In Section 3, we introduce a new pair of higher-order symmetric dual models. Under the assumption of higher-order generalized coneinvexity and other suitable conditions, weak, strong and converse duality theorems of higher-order symmetric dual problems are also derived. Furthermore, an example is provided to illustrate the results of strong duality theorem in this section. Finally, we draw some conclusions in Section 4.

2.
Preliminaries. Let R n denote n−dimensional Euclidean space with the nonnegative orthant R n + . Let K be a closed convex cone in R k . The positive dual cone A general multiobjective optimization problem can be expressed in the following form: where f : R n → R k , g : R n → R m and each component is differentable; ∅ = C ⊆ R n is convex; K and Q are closed convex cones in R k and R m , respectively. The solution discussed in this paper is defined in the sense of efficiency as given below: The following Fritz John type necessary optimality conditions for efficient solutions from [17] will be needed.
Ifx is an efficient solution of (MOP), then there existλ ∈ K + and µ ∈ Q + with (λ,μ) = 0 such that In the rest of this paper, we assume that The following definitions are based on Definition 2.4 of Padhan and Nahak [16].
Definition 2.4. Let η : R m × R m → R m be a vector valued function. The vector valued function f (x, ·) is said to be higher-order strictly K−pseudoinvex at y ∈ R m with respect to h and η, if for all v ∈ R m and p ∈ R m , Remark 1. (i) It is easy to see that when K = R + and h(x, y, p) ≡ 0, both higherorder strong K−pseudoinvexity and higher-order strict K−pseudoinvexity reduce to pseudoinvexity as considered in Chandra and Kumar [5].
(ii) If K is a pointed convex cone, then higher-order strong K−pseudoinvexity of f (x, ·) at y ∈ R m implies higher-order strict K−pseudoinvexity of f (x, ·) at y ∈ R m with respect to the same functions h and η, but the converse fails. This can be seen via the following example.
First, we present weak duality results, which state that the objective value of any feasible solution of (M P ) is not less than those of (M D ) under some appropriate conditions, such as higher-order strong cone-pseudoinvexity and higher-order strict cone-pseudoinvexity. (i) f (·, v) is higher-order strongly K−pseudoinnvex at u with respect to g and η 1 ; (ii) −f (x, ·) is higher-order strictly K−pseudoinnvex at y with respect to −h and η 2 ; Proof. By contradiction, suppose that

LIPING TANG, XINMIN YANG AND YING GAO
By hypothesis (iv), we have as (9) holds. Combining (10) and (18), we get which along with λ ∈ intK + yields that It follows from the higher-order strict K−pseudoinvexity of −f (x, ·) at y with respect to −h and η 2 that According to (17) and (19), we obtain Then, by the higher-order strong K−pseudoinvexity of f (·, v) at u with respect to g and η 1 , we derive which together with λ ∈ intK + implies On the other hand, by (13) and hypothesis (iii), we get which along with (14) yields that This contradicts (20), and hence we conclude that Remark 3. (i) If conditions (i) and (ii) of Theorem 3.1 are replaced by (i ) f (·, v) is higher-order strictly K−pseudoinnvex at u with respect to g and η 1 , (ii ) −f (x, ·) is higher-order strongly K−pseudoinnvex at y with respect to −h and η 2 , respectively, the conclusion of Theorem 3.1 still holds. The proof is similar to those of Theorem 3.1.
In what follows, we move our attention to strong duality between the pair (M P ) and (M D ). This is the issue that how to get efficient solutions of higher-order symmetric dual problem (M D ) from those of primal problem (M P ).  Proof. Since (x,ȳ,λ,p) is an efficient solution to (M P ), it follows from Lemma 2.2 that there exist α ∈ K + , β ∈ C 2 , µ ∈ R + and δ ∈ R + such that (α, β, µ, δ) = 0.