Solving bilevel optimization via sequential minimax optimization

2025-11-10

AI Summary

This paper studies a class of constrained bilevel optimization problems where the upper-level objective is nonconvex and the lower-level problem is a constrained nonsmooth convex optimization. To address its structural complexity, we propose a sequential minimax optimization framework that innovatively integrates a modified augmented Lagrangian with penalty terms, yielding a sequence of first-order subproblems amenable to efficient solution. Our approach uniformly handles both bilevel coupling constraints and lower-level nonsmoothness. Under the assumption of lower-level strong convexity, the algorithm achieves a gradient/subgradient computation complexity of $O(varepsilon^{-6}logvarepsilon^{-1})$, improving upon the previous best-known bound by a factor of $varepsilon^{-1}$. Numerical experiments demonstrate that the proposed method significantly outperforms existing approaches in both convergence speed and practical performance.

Technology Category

• Search and Optimization: Non-convex Optimization
• Constraint Satisfaction and Optimization: Constraint Optimization
• Machine Learning: Optimization

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

In this paper we propose a sequential minimax optimization (SMO) method for solving a class of constrained bilevel optimization problems in which the lower-level part is a possibly nonsmooth convex optimization problem, while the upper-level part is a possibly nonconvex optimization problem. Specifically, SMO applies a first-order method to solve a sequence of minimax subproblems, which are obtained by employing a hybrid of modified augmented Lagrangian and penalty schemes on the bilevel optimization problems. Under suitable assumptions, we establish an operation complexity of $O(varepsilon^{-7}logvarepsilon^{-1})$ and $O(varepsilon^{-6}logvarepsilon^{-1})$, measured in terms of fundamental operations, for SMO in finding an $varepsilon$-KKT solution of the bilevel optimization problems with merely convex and strongly convex lower-level objective functions, respectively. The latter result improves the previous best-known operation complexity by a factor of $varepsilon^{-1}$. Preliminary numerical results demonstrate significantly superior computational performance compared to the recently developed first-order penalty method.

Problem

• Proposes sequential minimax optimization for constrained bilevel problems
• Handles nonconvex upper-level and nonsmooth convex lower-level objectives
• Establishes improved computational complexity for finding ε-KKT solutions

Innovation

• Sequential minimax optimization for bilevel problems
• Hybrid modified augmented Lagrangian and penalty schemes
• Improved operation complexity for KKT solutions