Solving mathematical programs with complementarity constraints arising in nonsmooth optimal control

TL;DR

This paper reviews solution methods for MPCCs from nonsmooth optimal control problems, introduces a new benchmark set called nosbench, and evaluates relaxation-based algorithms, highlighting their current limitations and areas for improvement.

Contribution

It introduces the nosbench benchmark collection of 603 MPCCs from nonsmooth OCPs and evaluates various relaxation-based solution strategies on this dataset.

Findings

Scholtes' relaxation with IPOPT solves 73.8% of problems

Current methods do not always generalize well to nonsmooth MPCCs

Highlights the need for improved algorithms and software for MPCCs

Abstract

This paper examines solution methods for mathematical programs with complementarity constraints (MPCC) obtained from the time-discretization of optimal control problems (OCPs) subject to nonsmooth dynamical systems. The MPCC theory and stationarity concepts are reviewed and summarized. The focus is on relaxation-based methods for MPCCs, which solve a (finite) sequence of more regular nonlinear programs (NLP), where a regularization/homotopy parameter is driven to zero. Such methods perform reasonably well on currently available benchmarks. However, these results do not always generalize to MPCCs obtained from nonsmooth OCPs. To provide a more complete picture, this paper introduces a novel benchmark collection of such problems, which we call nosbench. The problem set includes 603 different MPCCs and we split it into a few representative subsets to accelerate the testing. We compare different relaxation-based methods, NLP solvers, homotopy parameter update and relaxation parameter steering strategies. Moreover, we check whether the obtained stationary points allow first-order descent directions, which may be the case for some of the weaker MPCC stationarity concepts. In the best case, the Scholtes' relaxation [Scholtes, 2002] with IPOPT [W\"achter and Biegler, 2006] as NLP solver manages to solve 73.8 % of the problems. This highlights the need for further improvements in algorithms and software for MPCCs.