Solving bilevel problems with products of upper- and lower-level variables

TL;DR

This paper introduces an iterative algorithm to solve a specific class of bilevel programming problems with linear upper levels and bilinear lower levels, addressing their NP-hardness through duality gap penalization and linearization.

Contribution

The paper presents a novel iterative method tailored for bilevel problems with bilinear lower-level terms, combining duality gap penalization and linearization techniques.

Findings

Algorithm effectively solves the targeted bilevel problems.

Numerical example demonstrates practical applicability.

Addresses NP-hardness in a specific bilevel problem class.

Abstract

Bilevel programming problems frequently arise in real-world applications across various fields, including transportation, economics, energy markets and healthcare. These problems have been proven to be NP-hard even in the simplest form with linear upper and lower-level problems. This paper addresses a specific type of bilevel programming problem where the upper-level is linear, and the lower level includes bilinear terms involving product of variables from both levels. We propose a new iterative algorithm that addresses this specific class of bilevel problems by penalizing the duality gap and linearizing the bilinear terms. The effectiveness of the algorithm is argued and demonstrated through a numerical example.