Duality theory for optimistic bilevel optimization

TL;DR

This paper develops duality theory for optimistic bilevel optimization using value function reformulation, establishing weak and strong duality results under various conditions without requiring convexity.

Contribution

It introduces duality results for bilevel optimization via Fenchel-Lagrange methods, including conditions for strong duality without convexity or Slater assumptions.

Findings

Weak and strong duality results established

Duality concepts extended to non-convex bilevel problems

Strong duality achieved under generalized Slater conditions

Abstract

In this paper, we exploit the so-called value function reformulation of the bilevel optimization problem to develop duality results for the problem. Our approach builds on Fenchel-Lagrange-type duality to establish suitable results for the bilevel optimization problem. First, we overview some standard duality results to show that they are not applicable to our problem. Secondly, via the concept of partial calmness, we establish weak and strong duality results. In particular, Lagrange, Fenchel-Lagrange, and Toland-Fenchel- Lagrange duality concepts are investigated for this type of problems under some suitable conditions. Thirdly, based on the use of some regularization of our bilevel program, we establish sufficient conditions ensuring strong duality results under a generalized Slater-type condition without convexity assumptions and without the partial calmness condition. Finally, without the Slater condition, a strong duality result is constructed for the bilevel optimization problem with geometric constraint.