This paper deals with minimax fractional programs whose objective functions are the maximum of finite ratios of convex functions, with arbitrary convex constraints set. For such 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. We give necessary optimality conditions for such problems, by means of convex analysis tools. We then propose a method, based on solving approximately a sequence of parametric convex problems, which acts as dc (difference of convex functions) algorithm, if the parameter is positive and as Dinkelbach algorithm if not. We show that every cluster point of the sequence of optimal solutions of these subproblems satisfies necessary optimality conditions of KKT criticality type, that are also of Clarke stationarity type. Finally we end with some numerical tests to illustrate the behaviour of the algorithm.

本文处理极小极小分数规划，其目标函数是凸函数的有限比的最大值，具有任意凸约束集。对于此类问题，Dinkelbach 型算法无法工作，因为参数子问题可能是非凸的，而后者需要这些子问题的全局最优解。我们通过凸分析工具为此类问题提供必要的最优性条件。然后，我们提出了一种方法，该方法基于近似求解一系列参数凸问题，如果参数为正，则作为 dc（凸函数差）算法，如果不是，则作为 Dinkelbach 算法。我们证明了这些子问题的最优解序列的每个聚类点都满足 KKT 临界类型的必要最优性条件，也属于克拉克平稳类型。最后，我们以一些数值测试结束，以说明算法的行为。