A regularized variance-reduced modified extragradient method for stochastic hierarchical games

TL;DR

This paper introduces a novel variance-reduced extragradient method for solving stochastic hierarchical games, providing convergence guarantees and extending to inexact lower-level solutions, with applications in power markets.

Contribution

It develops a regularized, smoothed variance-reduced extragradient algorithm for stochastic hierarchical games, offering new convergence and complexity results, including inexact lower-level problem scenarios.

Findings

Proves almost-sure convergence of the proposed method.

Provides rate and complexity guarantees for hierarchical equilibria.

Demonstrates applicability to power market models with a virtual power plant example.

Abstract

We consider an N-player hierarchical game in which the i-th player's objective comprises of an expectation-valued term, parametrized by rival decisions, and a hierarchical term. Such a framework allows for capturing a broad range of stochastic hierarchical optimization problems, Stackelberg equilibrium problems, and leader-follower games. We develop an iteratively regularized and smoothed variance-reduced modified extragradient framework for iteratively approaching hierarchical equilibria in a stochastic setting. We equip our analysis with rate statements, complexity guarantees, and almost-sure convergence results. We then extend these statements to settings where the lower-level problem is solved inexactly and provide the corresponding rate and complexity statements. Our model framework encompasses many game theoretic equilibrium problems studied in the context of power markets. We present a realistic application to the virtual power plants, emphasizing the role of hierarchical decision making and regularization.