Necessary and sufficient optimality conditions are established for a class of nondifferentiable minimax fractional programming problems with square root terms. Subsequently, we apply the optimality conditions to formulate a parametric dual problem and we prove some duality results. | Yugoslav Journal of Operations Research Vol 19 (2009), Number 1, 49-61 DOI: OPTIMALITY AND DUALITY FOR A CLASS OF NONDIFFERENTIABLE MINIMAX FRACTIONAL PROGRAMMING PROBLEMS Antoan BĂTĂTORESCU University of Bucharest, Bucharest batator@ Miruna BELDIMAN Institute of Mathematical Statistics and Applied Mathematics, Romanian Academy, Bucharest Iulian ANTONESCU “Mircea cel Bătrân” Naval Academy, Constanţa, Romania iulanton@ Roxana CIUMARA Academy of Economic Studies, Bucharest marinrox@ Received: December 2007 / Accepted: May 2009 Abstract: Necessary and sufficient optimality conditions are established for a class of nondifferentiable minimax fractional programming problems with square root terms. Subsequently, we apply the optimality conditions to formulate a parametric dual problem and we prove some duality results. Keywords: Fractional programming, generalized invexity, optimality conditions, duality. 1. INTRODUCTION Let us consider the following continuous differentiable mappings: f : R n × R m → R, h : R n × R m → R, g : R n → R p , Ψ : R + → R , 50 A. Batatorescu, M. Beldiman, I. Antonescu, R. Ciumara / On Nondifferentiable Minimax where d Ψ ( x) def = Ψ '( x) > 0 , and g = ( g1 ," , g p ). We denote dx P = { x ∈ R n | g j ( x) ≤ 0, j = 1, 2," , p} () and consider the compact subset Y ⊆ R m . Let Br , r = 1, β , and Dq , q = 1, δ , be n × n positive semi definite matrices such that for each ( x, y ) ∈ P × Y , we have: β f ( x, y ) + ∑ xT Br x ≥ 0, r =1 δ h( x, y ) − ∑ xT Dq x > 0. q =1 In this paper we consider the following non differentiable minimax fractional programming problem: β δ ⎡⎛ ⎞⎤ ⎞ ⎛ inf sup Ψ ⎢⎜ f ( x, y ) + ∑ xT Br x ⎟ ⎜ h( x, y ) − ∑ xT Dq x ⎟ ⎥ . x∈P y∈Y r =1 q =1 ⎢⎣⎝ ⎠ ⎝ ⎠ ⎥⎦ (P) For β = δ = 1, and Ψ ≡ 1, this problem was studied by Lai et al. [3], and further, if B1 = D1 = 0, (P) is a differentiable minimax fractional programming problem which has .
