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

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.