Continuous Optimization Second-order multiobjective symmetric duality with cone constraints T.R. Gulati a, * , Himani Saini a , S.K. Gupta b a Department of Mathematics, Indian Institute of Technology, Roorkee 247 667, India b Department of Mathematics, Indian Institute of Technology, Patna 800 013, India article info Article history: Received 18 June 2008 Accepted 26 December 2009 Available online 13 January 2010 Keywords: Multiobjective symmetric duality g-bonvexity/g-pseudobonvexity Cones Efficient solutions Properly efficient solutions abstract In this paper, we formulate Wolfe and Mond–Weir type second-order multiobjective symmetric dual problems over arbitrary cones. Weak, strong and converse duality theorems are established under g-bon- vexity/g-pseudobonvexity assumptions. This work also removes several omissions in definitions, models and proofs for Wolfe type problems studied in Mishra [9]. Moreover, self-duality theorems for these pairs are obtained assuming the function involved to be skew symmetric. Ó 2010 Elsevier B.V. All rights reserved. 1. Introduction Mangasarian [8] introduced the concept of second-order duality for nonlinear problems. Its study is significant due to computa- tional advantage over first-order duality as it provides tighter bounds for the value of the objective function when approxima- tions are used [7,8,11]. Bector and Chandra [2] introduced the con- cept of bonvex functions. Mond [11] established second-order duality for nonlinear programs under second-order convexity assumptions. Later on, Bector and Chandra [1] formulated sec- ond-order symmetric dual programs in the spirit of Mond and Weir [12] and established appropriate duality results involving pseudo- bonvex functions. Recently, Yang et al. [16,18] studied second-or- der symmetric dual programs and established duality relations under F-convexity assumptions. Devi [4] formulated a pair of second-order symmetric dual non- linear programming problems over arbitrary cones under g- pseudobonvexity assumptions. Mishra [9] formulated a similar second-order model for multiobjective problems, proved a weak duality theorem and stated that the proofs of strong and converse duality theorems follow on the lines of corresponding theorems in Devi [4]. Since the models and proofs in [4] contain several errors (see [5]), the same have been carried over in Mishra [9], who even did not observe that the proofs of strong and converse duality the- orems in [4] are different, while in symmetric duality the state- ment and the proof of the converse duality theorem go exactly as for the strong duality theorem. This paper is organized as follows. In the next section we pres- ent some relevant preliminaries. In Section 3, we formulate a pair of Wolfe type second-order multiobjective symmetric dual prob- lems with cone constraints and establish weak and strong duality theorems under g-bonvexity assumptions. These two sections also serve to remove several omissions in definitions, models and proofs in [9]. Section 4 contains duality relations for Mond-Weir type symmetric dual models under g-pseudobonvexity assump- tions. Self-duality results for these pairs have been stated in Sec- tion 5. The last section contains an appendix. 2. Preliminaries Let C 1 and C 2 be closed convex cones with nonempty interiors in R n and R m , respectively. For i ¼ 1; 2; C  i , called the polar cone of C i , is defined as follows : C  i ¼fz : x T z5 0 for all x 2 C i g: Suppose that S 1 # R n and S 2 # R m are open sets such that C 1  C 2  S 1  S 2 . Definition 2.1. [14]. A twice differentiable function f : S 1  S 2 ! R is said to be g 1 -bonvex in the first variable at u 2 S 1 , if there exists a function g 1 : S 1  S 1 ! R n such that for x 2 S 1 ; v 2 S 2 ; r 2 R n , f ðx; v Þ f ðu; v Þ=g T 1 ðx; uÞ½r x f ðu; v Þþ r xx f ðu; v Þr 1 2 r T r xx f ðu; v Þr and f ðx; yÞ is said to be g 2 -bonvex in the second variable at v 2 S 2 , if there exists a function g 2 : S 2  S 2 ! R m such that for u 2 S 1 ; y 2 S 2 ; p 2 R m , 0377-2217/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2009.12.024 * Corresponding author. Tel.: +91 9837106279. E-mail address: trgmaiitr@rediffmail.com (T.R. Gulati). European Journal of Operational Research 205 (2010) 247–252 Contents lists available at ScienceDirect European Journal of Operational Research journal homepage: www.elsevier.com/locate/ejor