PLMEs and Disjunctive Decompositions for Bilevel Optimization

Abstract: This paper studies bilevel polynomial optimization in which lower level constraining functions depend linearly on lower level variables. We show that such a bilevel program can be reformulated as a disjunctive program using partial Lagrange multiplier expressions (PLMEs). An advantage of this approach is that branch problems of the disjunctive program are easier to solve. In particular, since the PLME can be easily obtained, these branch problems can be efficiently solved by polynomial optimization techniques. Solving each branch problem either returns infeasibility or gives a candidate local or global optimizer for the original bilevel optimization. We give necessary and sufficient conditions for these candidates to be global optimizers, and sufficient conditions for the local optimality. Numerical experiments are also presented to show the efficiency of the method.