Black-Box Followers, White-Box Leaders: Partial Zeroth-Order Methods for MPECs

2026-05-15

AI Summary

This work addresses equilibrium-constrained optimization problems where the leader knows its own objective function but can only observe the follower’s response through point queries. To tackle this setting, the authors propose the Partial Zeroth-Order Optimization Scheme (PZOS), which combines exact gradient information of the leader with zeroth-order estimates of the follower’s Jacobian matrix, grounded in a novel partial Goldstein subdifferential theory. By leveraging known structural information while applying zeroth-order approximation solely to the unknown follower response, PZOS substantially reduces gradient estimation variance and ensures convergence. Empirical results demonstrate that, in routing toll design and security attack–defense games, PZOS converges faster and yields higher-quality solutions than fully black-box methods, exhibiting robust performance even under low query budgets.

Abstract

We study mathematical programs with equilibrium constraints, in which a leader knows their own cost function, but lacks a model of the followers' response. Instead, the leader can only query this response at specific points. While this setting precludes the use of gradient-based methods, existing zeroth-order approaches treat the composed objective entirely as a black box, deploying zeroth-order tools across both the leader and follower. Such approaches are inefficient, as they discard information the leader already possesses about their own cost function. In this work we instead propose to deploy zeroth-order tools only where they are truly needed: to handle the unknown, non-smooth followers' response. Specifically, we first propose PZOS, an algorithm that combines exact partial gradients of the leader's cost with zeroth-order Jacobian estimates of the followers' response in a chain-rule-inspired manner, and establish that it achieves a strictly lower variance bound than the black-box baseline. Second, we introduce the partial Goldstein subdifferential, a stationarity notion tailored to this composite structure, and prove convergence of our algorithm to both standard and partial Goldstein stationary points. Finally, we validate our method on two application domains -- toll optimization in routing games and defense-attack investment in security games -- demonstrating consistent improvements over black-box baselines in convergence speed, objective value, and estimator variance, with robust performance even under few queries per iteration.

Problem

• Mathematical Programs with Equilibrium Constraints
• Zeroth-Order Optimization
• Black-Box Optimization
• Leader-Follower Games
• Partial Gradient Information

Innovation

• partial zeroth-order optimization
• mathematical programs with equilibrium constraints
• Goldstein subdifferential
• gradient estimation
• bilevel optimization