Difference of Convex Functions Programming Approaches for Minimax Fractional Optimization Problems: Optimality Conditions and Resolution Algorithms

Abstract

This work deals with scalar and vector minimax fractional programs whose objective functions are the maximum of the quotients of difference of convex (DC) functions. These problems are generally nonsmooth and nonconvex. We give optimality conditions and develop algorithms to find a solution to such problems. We begin our study by the particular generalized fractional programming problems with ratios of convex functions, and convex constraints. We then consider the more general case of minimax fractional programs with ratios of DC functions, and DC constraints. Optimality conditions and algorithms are also developed for vector fractional programs with ratios of DC functions, and DC constraints. For such scalar and vector problems, Dinkelbach-type algorithms fail to work since the parametric subproblems may be nonconvex, whereas the latter need a global optimal solution of these subproblems. To overcome this difficulty, we overestimate the objective function in these subproblems by a convex function, and the constraints set by an inner convex subset of the latter, which leads to convex subproblems. We establish optimality conditions of Karush-Kuhn-Tucker type for these various problems, and show that our algorithms can find points that satisfy these necessary optimality conditions. Finally, we give some numerical tests on various problems to evaluate the efficiency of the proposed algorithms.