Bilevel optimization with a multi-objective lower-level problem: Risk-neutral and risk-averse formulations

TL;DR

This paper introduces new formulations and algorithms for bilevel optimization problems with conflicting objectives at the lower level, addressing both risk-neutral and risk-averse cases in deterministic and stochastic settings.

Contribution

It develops novel risk-aware formulations and gradient algorithms for stochastic bilevel problems with multi-objective lower levels, a topic previously underexplored.

Findings

Proposed risk-neutral and risk-averse formulations for bilevel problems.

Developed gradient-based algorithms for deterministic and stochastic cases.

Addressed computational challenges in multi-objective lower-level optimization.

Abstract

In this work, we propose different formulations and gradient-based algorithms for deterministic and stochastic bilevel problems with conflicting objectives in the lower level. Such problems have received little attention in the deterministic case and have never been studied from a stochastic approximation viewpoint despite the recent advances in stochastic methods for single-level, bilevel, and multi-objective optimization. To solve bilevel problems with a multi-objective lower level, different approaches can be considered depending on the interpretation of the lower-level optimality. An optimistic formulation that was previously introduced for the deterministic case consists of minimizing the upper-level function over all non-dominated lower-level solutions. In this paper, we develop new risk-neutral and risk-averse formulations, address their main computational challenges, and develop the corresponding deterministic and stochastic gradient-based algorithms.