Second-order duality for a nondifferentiable minimax fractional programming under generalized α-univexity

Gupta Sk, Dangar D, Kumar, Kumar Sumit

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In this paper, we concentrate our study to derive appropriate duality theorems for two types of second-order dual models of a nondifferentiable minimax fractional programming problem involving second-order α -univex functions. Examples to show the existence of α -univex functions have also been illustrated. Several known results including many recent works are obtained as special cases. MSC: 49J35, 90C32, 49N15.

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• Publié par: biomed
• Publié le: 01 janvier 2012
• Nombre de lectures: 17

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Gupta et al.Journal of Inequalities and Applications2012,2012:187 http://www.journalofinequalitiesandapplications.com/content/2012/1/187

R E S E A R C HOpen Access Second-order duality for a nondifferentiable minimax fractional programming under generalizedα-univexity

1* 12 SK Gupta, D Dangarand Sumit Kumar

* Correspondence: [e-mail] 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, we concentrate our study to derive appropriate duality theorems for two types of second-order dual models of a nondifferentiable minimax fractional programming problem involving second-order α -univex functions. Examples to show the existence of α -univex functions have also been illustrated. Several known results including many recent works are obtained as special cases. MSC:49J35; 90C32; 49N15 Keywords:minimax programming; fractional programming; nondifferentiable programming; second-order duality; α -univexity

1 Introduction After Schmitendorf [], who derived necessary and sufficient optimality conditions for static minimax problems, much attention has been paid to optimality conditions and du- ality theorems for minimax fractional programming problems [–]. For the theory, al- gorithms, and applications of some minimax problems, the reader is referred to []. In this paper, we consider the following nondifferentiable minimax fractional program- ming problem:

T/ f(x,y) + (x Bx) Minimizeψ(x) =sup T/ y∈Y h(x,y) – (x Dx) subject tog(x)≤,

(P)

l nl nl whereYis a compact subset ofR,f(∙,∙) :R×R→R,h(∙,∙) :R×R→Rare twice n ln m continuously differentiable onR×Randg(∙) :R→Ris twice continuously differen- n T/ tiable onR,B, andDare an×npositive semidefinite matrix,f(x,y) + (x Bx)≥, and T/n h(x,y) – (x Dx for each () >x,y)∈J×Y, whereJ={x∈R:g(x)≤}. Motivated by [, , ], Yang and Hou [] formulated a dual model for fractional mini- max programming problem and proved duality theorems under generalized convex func- tions. Ahmad and Husain [] extended this model to nondifferentiable and obtained du- ality relations involving (F,α,ρ,d)-pseudoconvex functions. Jayswal [] studied duality theorems for another two duals of (P) underα-univex functions. Recently, Ahmadet al. [] derived the sufficient optimality condition for (P) and established duality relations for

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