Solutions of Two-stage Stochastic Minimax Problems

TL;DR

This paper studies a new class of two-stage stochastic minimax problems with nonconvex-concave first-stage and strongly convex-concave second-stage objectives, establishing theoretical properties and proposing an effective solution algorithm.

Contribution

It introduces a novel class of stochastic minimax problems, analyzes their properties, and develops an inexact parallel proximal gradient algorithm with convergence guarantees.

Findings

The second-stage minimax value function has specific properties.

The SAA approach converges as sample size increases.

The proposed algorithm is effective in numerical experiments.

Abstract

This paper introduces a class of two-stage stochastic minimax problems where the first-stage objective function is nonconvex-concave while the second-stage objective function is strongly convex-concave. We establish properties of the second-stage minimax value function and solution functions, and characterize the existence and relationships among saddle points, minimax points, and KKT points. We apply the sample average approximation (SAA) to the class of two-stage stochastic minimax problems and prove the convergence of the KKT points as the sample size tends to infinity. An inexact parallel proximal gradient descent ascent algorithm is proposed to solve this class of problems with the SAA. Numerical experiments demonstrate the effectiveness of the proposed algorithm and validate the convergence properties of the SAA approach.