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

Arun L Bishop, Micah Reich, Zac Manchester

Keywords: robotics, optimization, complementarity, Lie groups

TL;DR: A Lie group approach to solving quadratic programs with linear complementarity constraints for robotics optimization problems with continuous and discrete elements.

Abstract: Many problems in robotics require reasoning over a mix of continuous dynamics and discrete events, such as making and breaking contact in manipulation. These problems are locally well modeled by linear complementarity quadratic programs (LCQPs), which extend QPs to complementarity constraints. While expressive, LCQPs are non-convex, and few solvers exist for computing good local solutions. In this work, we observe that complementarity constraints form a Lie group under relaxation which can be used to perform on-manifold optimization. We introduce a retraction map that is numerically well behaved, and use it to parameterize the constraints such 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++, is competitive on a suite of benchmark problems, and solves robotics problems where existing approaches fail.

Email Sharing: We authorize the sharing of all author emails with Program Chairs.

Data Release: We authorize the release of our submission and author names to the public in the event of acceptance.

Submission Number: 26