A Regularized Variance-Reduced Modified Extragradient Method for Stochastic Hierarchical Games

Shisheng Cui (), Uday V. Shanbhag () and Mathias Staudigl ()
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Shisheng Cui: Beijing Institute of Technology
Uday V. Shanbhag: University of Michigan
Mathias Staudigl: University of Mannheim

Journal of Optimization Theory and Applications, 2025, vol. 206, issue 1, No 11, 53 pages

Abstract: Abstract We consider an N-player hierarchical game in which the ith 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 study of virtual power plants, emphasizing the role of hierarchical decision making and regularization. Preliminary numerics suggest that empirical behavior compares well with theoretical guarantees.

Keywords: Hierarchical games; Variational inequality problems; Tikhonov regularization; Iterative smoothing; Variance-reduction; Virtual power plants (VPPs); 65K15; 90C15; 90C30; 90C33 (search for similar items in EconPapers)
Date: 2025
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DOI: 10.1007/s10957-025-02683-8

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