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.

在本文中，我们开发了最优性条件，并提出了一种针对广义分数规划问题的算法，该算法的目标函数是具有dc约束的凸（dc）函数之差的有限比率的最大值，此后我们将其称为DC-GFP。这样的问题通常是不光滑和不凸的。我们首先通过凸分析工具给出此类问题的最优条件。为了解决DC-GFP，使用Dinkelbach型算法处理非凸子问题，这会导致后者失败，因为它需要为这些子程序找到全局最小值。为了克服这个困难，我们提出了一种DC-Dinkelbach型算法，其中我们通过凸函数高估了这些子问题中的目标函数，并通过后者的内部凸子集高估了约束，导致凸子问题。我们表明，这些子问题的最优解序列的每个聚类点都满足KKT类型的必要最优性条件。最后，我们通过一些数值测试来说明我们算法的行为。