Reformulations of Quadratic Programs for Lipschitz Continuity

TL;DR

This paper introduces a method to reformulate quadratic programs as second-order cone problems to ensure Lipschitz continuity, independence from constraint structure, and enable closed-form solutions for faster computation.

Contribution

The authors propose a novel reformulation of quadratic programs into SOCPs that guarantees Lipschitz continuity and admits a closed-form solution, regardless of constraint structure.

Findings

Reformulation ensures Lipschitz continuity of the solution.

The approach is independent of constraint linear independence.

Closed-form solutions enable faster computation.

Abstract

Optimization-based controllers often lack regularity guarantees, such as Lipschitz continuity, when multiple constraints are present. When used to control a dynamical system, these conditions are essential to ensure the existence and uniqueness of the system's trajectory. Here we propose a general method to convert a Quadratic Program (QP) into a Second-Order Cone Problem (SOCP), which is shown to be Lipschitz continuous. Key features of our approach are that (i) the regularity of the resulting formulation does not depend on the structural properties of the constraints, such as the linear independence of their gradients; and (ii) it admits a closed-form solution, which is not available for general QPs with multiple constraints, enabling faster computation. We support our method with rigorous analysis and examples.