A Lagrange Multiplier Expression Method for Bilevel Polynomial Optimization

TL;DR

This paper introduces a novel method for solving bilevel polynomial optimization problems using polynomial optimization relaxations based on KKT conditions and Moment-SOS techniques, with proven convergence and demonstrated efficiency.

Contribution

The paper develops a new approach combining polynomial relaxations, KKT conditions, and Moment-SOS methods for bilevel polynomial optimization, with convergence analysis and numerical validation.

Findings

Method effectively solves bilevel polynomial problems.

Convergence of the algorithm is theoretically established.

Numerical experiments confirm the method's efficiency.

Abstract

This paper studies bilevel polynomial optimization problems. To solve them, we give a method based on polynomial optimization relaxations. Each relaxation is obtained from the Kurash-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-SOS relaxations are used to solve the polynomial optimizattion 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.