Reformulations of Quadratic Programs for Lipschitz Continuity

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 cf{QP} into a cf{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 under some assumptions, which is not available for general cp{QP} with multiple constraints, enabling faster computation. We support our method with rigorous analysis and examples.