Derivative-free optimization methods for finite minimax problems

Derivative-free optimization focuses on designing methods to solve optimization problems without the analytical knowledge of the function. In this paper we consider the problem of designing derivative-free methods for finite minimax problems: minx maxi=1,2,...N{fi(x)}. In order to solve the problem efficiently, we seek to exploit the smooth substructure within the problem. Using ideas developed in [BLO02, BLO05], we create the idea of a robust simplex gradient descent direction, and use it to accelerate convergence. Convergence is proven by showing that the resulting algorithm fits into the Directional Direct-Search framework. Numerical tests demonstrate the algorithm’s effectiveness on finite minimax problems.