Outer Convexification of Constraints

Abstract

In this chapter, we take a closer look at combinatorial issues raised by applying outer convexification to constraints that directly depend on a discrete control. We describe several undesirable phenomena that arise when treating such problems with a Sequential Quadratic Programming (SQP) method, and give explanations based on the violation of Linear Independence Constraint Qualification (LICQ). We show that the Nonlinear Programs (NLPs) can instead be treated favorably as Mathematical Programs with Vanishing Constraints (MPVCs), a class of challenging problems with complementary constraints and nonconvex feasible set. We give a summary of a Lagrangian framework for MPVCs due to [3] and others, which allows for the design of derivative based descent methods for the efficient solution of MPVCs in the next chapter.