Complementarity by Construction: A Lie-Group Approach to Solving Quadratic Programs with Linear Complementarity Constraints

TL;DR

This paper introduces a Lie-group based on-manifold optimization method for solving linear complementarity quadratic programs (LCQPs) in robotics, with an open-source solver called Marble that improves convergence.

Contribution

It leverages the Lie group structure of complementarity constraints to develop a novel on-manifold solver with a well-behaved retraction map, enabling better solutions for LCQPs.

Findings

Marble is competitive on benchmark problems.

Marble successfully solves robotics problems where existing methods fail.

The approach avoids classical issues with complementarity constraints.

Abstract

Many problems in robotics require reasoning over a mix of continuous dynamics and discrete events, such as making and breaking contact in manipulation and locomotion. These problems are locally well modeled by linear complementarity quadratic programs (LCQPs), an extension to QPs that introduce complementarity constraints. While very expressive, LCQPs are non-convex, and few solvers exist for computing good local solutions for use in planning pipelines. In this work, we observe that complementarity constraints form a Lie group under infinitesimal relaxation, and leverage this structure to perform on-manifold optimization. We introduce a retraction map that is numerically well behaved, and use it to parameterize the constraints so that they are satisfied by construction. The resulting solver avoids many of the classical issues with complementarity constraints. We provide an open-source solver, Marble, that is implemented in C++ with Julia and Python bindings. We demonstrate that Marble is competitive on a suite of benchmark problems, and solves a number of robotics problems where existing approaches fail to converge.