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

2026-04-13

AI Summary

This work addresses the nonconvex quadratic programming problems with linear complementarity constraints commonly encountered in robotic contact reasoning. It proposes a novel approach based on Lie group manifold optimization, revealing for the first time that the complementarity constraints, under infinitesimal relaxation, naturally form a Lie group structure. Leveraging this insight, the authors design a numerically stable retraction mapping that inherently satisfies the constraints during parameterization. The resulting open-source solver, Marble—implemented in C++ with interfaces for Julia and Python—demonstrates superior performance on standard benchmarks, successfully solving multiple robotic contact planning instances that existing methods fail to converge on, thereby significantly improving both computational efficiency and solution reliability.

Technology Category

• Intelligent Robots: Learning & Optimization for ROB
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization
• Planning, Routing, and Scheduling: Replanning and Plan Repair

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Machine-in-the-loop, human agency and autonomy
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

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.

Problem

• quadratic programming
• linear complementarity constraints
• non-convex optimization
• robotics
• contact dynamics

Innovation

• Lie group
• complementarity constraints
• on-manifold optimization
• retraction map
• quadratic programming