Sufficiency and Duality in Minimax Fractional Programming with Generalized (φ, Ρ)-invexity

Amongst various important applications, one important application of nonlinear programming is to maximize or minimize the ratio of two functions, commonly called fractional programming. The characteristics of fractional programming problems have been investigated widely [1, 6, 10] and [13]. In noneconomic situations, fractional programming problems arisen in information theory, stochastic programming, numerical analysis, approximation theory, cluster analysis, graph theory, multifacility location theory, decomposition of large-scale mathematical programming problems, goal programming and among others. Recently, some biologists have been studying fractional programming problems to improve the accuracy of melting temperature estimations (cf. Leber et al. [15]). The necessary and sufficient conditions for generalized minimax programming were first developed by Schmitendorf [17]. Later, several authors considered these optimality and duality theorems for minimax fractional programming problems, one can consult [2, 11, 16] and [20]. Antczak [4] proved optimality conditions for a class of generalized fractional minimax programming problems involving B-(p, r)-invexity functions and established duality theorems for various duality models. Later on, Ahmad et al. [3] discussed sufficient optimality conditions and duality theorems for a nondifferentiable minimax fractional programming problem with B-(p, r)invexity.