Finding Robust Global Optimal Values of Bilevel Polynomial Programs with Uncertain Linear Constraints

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.

The robust bilevel continuous knapsack problem with uncertain coefficients in the follower’s objective

“…Due to the hardness of deterministic bilevel optimization, it is not surprising that relatively few articles dealing with bilevel optimization problems under uncertainty, in particular using the robust optimization approach, have been published so far. In [8], the authors consider bilevel problems with linear constraints and a linear follower's objective, while the leader's objective is a polynomial. The robust counterpart of the problem, with interval uncertainty in the leader's and the follower's constraints, is solved via a sequence of semidefinite programming relaxations.…”