Directional first order approach for a class of bilevel programs

TL;DR

This paper introduces a directional first order method for a class of bilevel programs that relaxes the convexity requirement of the lower level, providing new optimality conditions and handling nonconvex cases.

Contribution

It proposes a novel directional first order approach for bilevel programs without convexity assumptions, characterizing the lower level via directional conditions.

Findings

The approach characterizes the lower level by its first order condition in a directional neighborhood.

Established directional necessary optimality conditions under common constraint qualifications.

Demonstrated the method on a nonconvex bilevel problem where classical methods fail.

Abstract

In this paper, we study a class of bilevel optimization program (BP), where the feasible set of the lower level program is independent of the upper level variable. For bilevel programs it is known that the first order approach requires the convexity of the lower level program while reformulations involving the value function results in difficult optimization problems. In this paper we propose a directional first order approach which does not require the convexity of the lower level program. First, under some reasonable assumptions, we show that the lower level program can be equivalently characterized by its first order condition over a directional neighborhood. Next, for the resulting single level optimization problem, under common constraint qualifications, we establish directional necessary optimality conditions. Finally, an example of BP with nonconvex lower level program is given, where we demonstrate the failure of the classical first order approach and derive necessary optimality conditions using its directional counterpart.