Extension of the value function reformulation to multiobjective bilevel optimization

TL;DR

This paper extends the value function reformulation to multiobjective bilevel optimization, introducing a frontier map approach to derive necessary optimality conditions for complex vector-valued problems.

Contribution

It introduces a frontier map concept for multiobjective lower-level problems and develops a new constraint qualification for deriving optimality conditions.

Findings

The frontier map generalizes the value function for multiobjective problems.

A tractable constraint qualification is established.

Necessary optimality conditions are derived as a natural extension of scalar bilevel problems.

Abstract

We consider a multiobjective bilevel optimization problem with vector-valued upper- and lower-level objective functions. Such problems have attracted a lot of interest in recent years. However, so far, scalarization has appeared to be the main approach used to deal with the lower-level problem. Here, we utilize the concept of frontier map that extends the notion of optimal value function to our parametric multiobjective lower-level problem. Based on this, we build a tractable constraint qualification that we use to derive necessary optimality conditions for the problem. Subsequently, we show that our resulting necessary optimality conditions represent a natural extension from standard optimistic bilevel programs with scalar objective functions.