A lagrange multiplier expression method for bilevel polynomial optimization

This paper studies bilevel polynomial optimization. We propose a method to solve it globally by using polynomial optimization relaxations. Each relaxation is obtained from the Karush–Kuhn–Tucker (KKT) conditions for the lower level optimization and the exchange technique for semi-infinite programming. For KKT conditions, Lagrange multipliers are represented as polynomial or rational functions. The Moment–sum-of-squares relaxations are used to solve the polynomial optimization relaxations. Under some general assumptions, we prove the convergence of the algorithm for solving bilevel polynomial optimization problems. Numerical experiments are presented to show the efficiency of the method.