A Sequential Convex Programming Approach to Solving Quadratic Programs and Optimal Control Problems with Linear Complementarity Constraints

TL;DR

This paper introduces a sequential convex programming method with a homotopy approach for efficiently solving quadratic programs with linear complementarity constraints, addressing their nonconvexity and computational challenges.

Contribution

It presents a novel penalty reformulation and a convex subproblem approach that simplifies solving complex quadratic programs with complementarity constraints.

Findings

Convex subproblems have a constant Hessian, enabling efficient solutions.

The method guarantees descent at each iteration.

Numerical experiments demonstrate potential computational speedups.

Abstract

Mathematical programs with complementarity constraints are notoriously difficult to solve due to their nonconvexity and lack of constraint qualifications in every feasible point. This work focuses on the subclass of quadratic programs with linear complementarity constraints. A novel approach to solving a penalty reformulation using sequential convex programming and a homotopy on the penalty parameter is introduced. Linearizing the necessarily nonconvex penalty function yields convex quadratic subproblems, which have a constant Hessian matrix throughout all iterates. This allows solution computation with a single KKT matrix factorization. Furthermore, a globalization scheme is introduced in which the underlying merit function is minimized analytically, and guarantee of descent is provided at each iterate. The algorithmic features and possible computational speedups are illustrated in a numerical experiment.