CCOpt: an Open-Source Solver for Large-Scale Mathematical Programs with Complementarity Constraints

TL;DR

CCOpt is an open-source Julia package that offers advanced algorithms for solving large-scale Mathematical Programs with Complementarity Constraints, improving speed and robustness over existing methods.

Contribution

The paper introduces CCOpt, a Julia-based solver with novel interior-point algorithms, regularization techniques, and interfaces, tailored for challenging MPCC problems.

Findings

Substantial speedups over standard implementations and commercial solvers.

Effective handling of large-scale MPCCs in power systems and control applications.

Algorithmic improvements often yield an order of magnitude performance gains.

Abstract

This paper presents the Julia package CCOpt, built on top of the interior-point solver MadNLP. CCOpt implements a suite of algorithms for Mathematical Programs with Complementarity Constraints (MPCCs). The solver additionally comes with interfaces for use in Matlab, Python, and C++. MPCCs have recently gained renewed attention in engineering optimization, as complementarity provides a powerful modeling tool for nonsmooth functions and logical conditions. These problems are inherently challenging since their nonlinear programming reformulations violate classical regularity conditions at all feasible points, complicating both theoretical analysis and numerical treatment. Consequently, specialized algorithms are required to handle this degeneracy, and several approaches have been proposed. We implement a toolbox of methods, including relaxation and penalty approaches, as well as a crossover to recently proposed active-set methods. Our solver is based on nonlinear interior-point algorithms that couple the relaxation or penalty parameter with the barrier parameter, yielding substantial speedups compared to standard implementations. Both monotone and nonmonotone strategies for updating this joint parameter update are proposed and investigated. In addition, we propose regularization techniques that improve the conditioning of the KKT system for small relaxation parameters, enhancing robustness and computational efficiency. The implementation is validated on the classical MacMPEC benchmark, large-scale problems in security-constrained optimal power flow, optimal control of nonsmooth systems, as well as on quadratic programs with complementarity constraints arising in model predictive control. This benchmarking reveals an algorithmically driven improvement of often an entire order of magnitude over other methods, including commercial solvers.