Sufficient optimality conditions in bilevel programming

TL;DR

This paper develops first- and second-order sufficient optimality conditions for smooth optimistic bilevel programming problems, using reformulations and directional derivatives to relate conditions to initial problem data.

Contribution

It introduces novel first- and second-order optimality conditions for bilevel problems, leveraging reformulations and second-order derivatives to connect conditions with original data.

Findings

First-order conditions via tangent cone estimates

Second-order conditions using second-order directional derivatives

Applicable to problems with linear or strongly stable lower levels

Abstract

This paper is concerned with the derivation of first- and second-order sufficient optimality conditions for optimistic bilevel optimization problems involving smooth functions. First-order sufficient optimality conditions are obtained by estimating the tangent cone to the feasible set of the bilevel program in terms of initial problem data. This is done by exploiting several different reformulations of the hierarchical model as a single-level problem. To obtain second-order sufficient optimality conditions, we exploit the so-called value function reformulation of the bilevel optimization problem, which is then tackled with the aid of second-order directional derivatives. The resulting conditions can be stated in terms of initial problem data in several interesting situations comprising the settings where the lower level is linear or possesses strongly stable solutions.