Optimality conditions and DC-Dinkelbach-type algorithm for generalized fractional programs with ratios of difference of convex functions - Ghazi, Abdelouafi; Roubi, Ahmed (Optimization letters, 2021)
In this paper, we develop optimality conditions and propose an algorithm for generalized fractional programming problems whose objective function is the maximum of finite ratios of difference of convex (dc) functions, with dc constraints, that we will call later, DC-GFP. Such problems are generally nonsmooth and nonconvex. We first give in this work, optimality conditions for such problems, by means of convex analysis tools. For solving DC-GFP, the use of Dinkelbach-type algorithms conducts to nonconvex subproblems, which causes the failure of the latter since it requires finding a global minimum for these subprograms. To overcome this difficulty, we propose a DC-Dinkelbach-type algorithm in which 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 show that every cluster point of the sequence of optimal solutions of these subproblems satisfies necessary optimality conditions of KKT type. Finally we end with some numerical tests to illustrate the behavior of our algorithm.