Duality for nondifferentiable multiobjective higher-order symmetric programs over cones involving generalized ( F , α , ρ , d ) -convexity

Gupta et al. Journal of Inequalities and Applications 2012, 2012:298 http://www.journalofinequalitiesandapplications.com/content/2012/1/298 RESEARCH Open Access Duality for nondifferentiable multiobjective higher-order symmetric programs over cones involving generalized (F, α, ρ, d)-convexity SK Gupta1* , N Kailey2 and Sumit Kumar3 * Correspondence: 1 Department of Mathematics, Indian Institute of Technology, Patna, 800 013, India Full list of author information is available at the end of the article Abstract In this paper, a pair of Wolfe type higher-order symmetric nondifferentiable multiobjective programs over arbitrary cones is formulated and appropriate duality relations are then established under higher-order-K-(F, α , ρ , d)-convexity assumptions. A numerical example which is higher-order K-(F, α , ρ , d)-convex but not higher-order K-F-convex has also been illustrated. Special cases are also discussed to show that this paper extends some of the known works that have appeared in the literature. MSC: 90C29; 90C30; 49N15 Keywords: higher-order symmetric duality; support function; multiobjective programming; cones; efficient solutions 1 Introduction Mangasarian [] introduced the concept of second- and higher-order duality for nonlinear problems. He has also indicated that the study is significant due to the computational advantage over the first-order duality as it provides tighter bounds for the value of the objective function when approximations are used. Motivated by the concept in [], several researchers [–] have worked in this field. Multiobjective optimization has a large number of applications. As an example, it is generally used in goal programming, risk programming etc. Optimality conditions for multiobjective programming problems can be found in Miettinen [] and Pardalos et al. []. Recently, Chinchuluun and Pardalos [] discussed recent developments in multiobjective optimization. These include optimality conditions, applications, global optimization techniques, the new concept of epsilon pareto optimal solutions and heuristics. Chen [] considered a pair of symmetric higher-order Mond-Weir type nondifferentiable multiobjective programming problems and established usual duality results under higher-order F-convexity assumptions. Gulati and Gupta [] proved duality theorems for a pair of Wolfe type higher-order nondifferentiable symmetric dual programs. Ahmad et al. [] formulated a general Mond-Weir type higher-order dual for a nondifferentiable multiobjective programming problem and established usual duality results. Gulati and Geeta [] pointed out certain omissions in some papers on symmetric duality in multiobjective programming and discussed their corrective measures. Later on, Ahmad and Husain [] and Gulati et al. [] formulated second-order multiobjective symmetric dual programs with cone constraints and established duality results under second© 2012 Gupta et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Gupta et al. Journal of Inequalities and Applications 2012, 2012:298 http://www.journalofinequalitiesandapplications.com/content/2012/1/298 Page 2 of 16 order invexity assumptions. An omission in the strong duality theorem in Yang et al. [] has been rectified in Gupta and Kailey []. The concept of (F, ρ)-convexity was introduced by Preda [] as an extension of Fconvexity [] and ρ-convexity []. Yang et al. [] formulated several second-order duals for scalar programming problem and proved duality results involving generalized Fconvex functions. Zhang and Mond [] extended the class of (F, ρ)-convex functions to second-order and obtained duality relations for multiobjective dual problems. Motivated by various concepts of generalized convexity, Liang et al. [] introduced a unified formulation of generalized convexity, called (F, α, ρ, d)-convexity and obtained corresponding optimality conditions and duality relations for a single objective fractional problem. This was later on extended to multiobjective fractional programming problem in Liang et al. []. Inspired by the concept given in [, , ], Ahmad and Husain [] introduced second-order (F, α, ρ, d)-convex functions and proved duality relations for Mond-Weir type second-order multiobjective problems. In the recent work of Ahmad and Husain [], an attempt is made to remove certain omissions and inconsistencies in the work of Mishra and Lai []. Agarwal et al. [] achieved duality results for a pair of Mond-Weir type multiobjective higher-order symmetric dual programs over arbitrary cones under higher-order K F-convexity assumptions. Recently, Agarwal et al. [] have filled some gap in the work of Chen [] and proved a strong duality theorem for Mond-Weir type multiobjective higherorder nondifferentiable symmetric dual programs. In this paper, we formulate a pair of symmetric higher-order Wolfe type nondifferentiable multiobjective programs over arbitrary cones and prove weak, strong and converse duality theorems under higher-order-K -(F, α, ρ, d)-convexity assumptions. We also give a nontrivial example of a function lying in the class of higher-order K -(F, α, ρ, d)-convex but not in the class of higher-order K -F-convex. Our study extends some of the known results that appeared in [, , , ]. 2 Notations and preliminaries Consider the following multiobjective programming problem: K-minimize subject to φ(x)   x ∈ X  = x ∈ S : –g(x) ∈ C , (P) where S ⊆ Rn is open, φ : S → Rk , g : S → Rm , K is a closed convex pointed cone in Rk with int K = φ and C is a closed convex cone in Rm with nonempty interior. Definition  [] The positive polar cone C * of C is defined as   C * = z : ξ T z  , for all ξ ∈ C . Definition  [] A point x̄ ∈ X  is a weak efficient solution of (P) if there exists no other x ∈ X  such that φ(x̄) – φ(x) ∈ int K. Gupta et al. Journal of Inequalities and Applications 2012, 2012:298 http://www.journalofinequalitiesandapplications.com/content/2012/1/298 Definition  [] A point x̄ ∈ X  is an efficient solution of (P) if there exists no other x ∈ X  such that φ(x̄) – φ(x) ∈ K\{}. Definition  [, , ] Let D be a compact convex set in Rn . The support function of D is defined by   S(x | D) = max xT y : y ∈ D . A support function, being convex and everywhere finite, has a subdifferential, that is, there exists z ∈ Rn such that S(y | D)  S(x | D) + zT (y – x) for all y ∈ D. The subdifferential of S(x | D) is given by   ∂S(x | D) = z ∈ D : zT x = S(x | D) . For any set S ⊂ Rn , the normal cone to S at a point x ∈ S is defined by   NS (x) = y ∈ Rn : yT (z – x)   for all z ∈ S . It can be easily seen that for a compact convex set D, y is in ND (x) if and only if S(y | D) = xT y, or equivale (...truncated)