Partial Lagrange Multiplier Expressions and Disjunctive Decompositions for Bilevel Optimization

This paper studies bilevel polynomial optimization in which lower-level constraint functions depend linearly on lower-level variables. We show that such bilevel program can be reformulated as a disjunctive program by using Karush-Kuhn-Tucker (KKT) conditions with a sparse type of Lagrange multipliers. This kind of Lagrange multipliers can be conveniently represented by polynomials, for which we call partial Lagrange multiplier expressions (PLMEs). By doing this, each branch problem of the disjunctive program can be solved efficiently 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.