Decision Rule Approaches for Pessimistic Bilevel Linear Programs under Moment Ambiguity with Facility Location Applications

TL;DR

This paper develops novel distributionally robust optimization methods for pessimistic bilevel linear programs with moment ambiguity, providing computational approaches and demonstrating their effectiveness on facility location problems.

Contribution

It introduces new reformulations and algorithms for solving pessimistic bilevel programs under moment ambiguity, including SDP and copositive programming approaches.

Findings

Effective upper bounds via SDP and copositive reformulations.

Exact solutions for discrete ambiguity sets.

Enhanced computational efficiency with cutting-plane algorithms.

Abstract

We study a pessimistic stochastic bilevel program in the context of sequential two-player games, where the leader makes a binary here-and-now decision, and the follower responds a continuous wait-and-see decision after observing the leader's action and revelation of uncertainty. Only the information of the mean, covariance, and support is known. We formulate the problem as a distributionally robust (DR) two-stage problem. The pessimistic DR bilevel program is shown to be equivalent to a generic two-stage distributionally robust stochastic (nonlinear) program with both a random objective and random constraints under proper conditions of ambiguity sets. Under continuous distributions, using linear decision rule approaches, we construct upper bounds on the pessimistic DR bilevel program based on (1) 0-1 semidefinite programming (SDP) approximation and (2) an exact 0-1 copositive programming reformulations. When the ambiguity set is restricted to discrete distributions, an exact 0-1 SDP reformulation is developed, and explicit construction of the worst-case distribution is derived. To further improve the computation of the proposed 0-1 SDPs, a cutting-plane framework is developed. Moreover, based on a mixed-integer linear programming approximation, another cutting-plane algorithm is proposed. Extensive numerical studies are conducted to demonstrate the effectiveness of the proposed approaches on a facility location problem.