Robust Global Solutions of Bilevel Polynomial Optimization Problems with Uncertain Linear Constraints

TL;DR

This paper develops a novel approach to find robust global solutions for bilevel polynomial optimization problems with uncertain linear constraints, using sum of squares and semidefinite programming techniques.

Contribution

It introduces a new polynomial characterization and reformulation method for robust bilevel problems with uncertainty, enabling global solution computation.

Findings

Characterizes robust solutions via sum of squares polynomials.

Reformulates the problem as a single non-convex polynomial optimization.

Demonstrates solution via semidefinite programming with numerical examples.

Abstract

This paper studies, for the first time, a bilevel polynomial program whose constraints involve uncertain linear constraints and another uncertain linear optimization problem. In the case of box data uncertainty, we present a sum of squares polynomial characterization of a global solution of its robust counterpart where the constraints are enforced for all realizations of the uncertainties within the prescribed uncertainty sets. By characterizing a solution of the robust counterpart of the lower-level uncertain linear program under spectrahedral uncertainty using a new generalization of Farkas' lemma, we reformulate the robust bilevel program as a single level non-convex polynomial optimization problem. We then characterize a global solution of the single level polynomial program by employing Putinar's Positivstellensatz of algebraic geometry under coercivity of the polynomial objective function. Consequently, we show that the robust global optimal value of the bilevel program is the limit of a sequence of values of Lasserre-type hierarchy of semidefinite linear programming relaxations. Numerical examples are given to show how the robust optimal value of the bilevel program can be calculated by solving semidefinite programming problems using the Matlab toolbox YALMIP.