Similarity Relations Based Numerical Algorithm for Solving Maximin Problems

In this paper, we address the maximin optimization problem and introduce an algorithm to solve it. The core objective is to maximize a given function expressed as a minimum of the values of finite linear functions. This paper introduces an algorithm for optimizing such functions, taking into account that they are subject to gradient discontinuity. To achieve this, we propose an optimization technique that combines the effectiveness of the steepest descent method with a tailored strategy to handle gradient disruptions. Our approach involves constructing a search direction by forming a linear combination of gradients from neighboring functions. The key innovation lies in the assignment of weights to these gradients based on a defined similarity relation. This allows the algorithm to adaptively weigh the contributions of different gradients, addressing the challenges posed by gradient discontinuity.