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

TL;DR

This paper introduces a novel partial zeroth-order method for mathematical programs with equilibrium constraints, leveraging known leader information while efficiently handling unknown followers' responses.

Contribution

The authors propose PZOS, a new algorithm that combines exact partial gradients with zeroth-order Jacobian estimates, improving efficiency over black-box approaches.

Findings

PZOS achieves lower variance bounds than black-box baselines.

The method converges to standard and partial Goldstein stationary points.

Empirical validation shows faster convergence and better objective values.

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.