This repository contains the implementation and examples from our paper "Reformulations of Quadratic Programs for Lipschitz Continuity." In the paper, we present a systematic method to guarantee Lipschitz continuity in optimization-based controllers by reformulating Quadratic Programs (QPs) as Second-Order Cone Programs (SOCPs).

🔍 Motivation

Optimization-based controllers, such as those derived from constrained QPs, are powerful tools for controlling dynamical systems. However, when multiple constraints are involved, the resulting control law $\pi_{\rm qp}$ may lack regularity properties like Lipschitz continuity. This lack of continuity can result in:

• Discontinuous control inputs,
• chattering behavior in closed-loop systems,
• Non-unique or ill-posed system trajectories.

Previous work typically verifies Lipschitz continuity of QP solutions by analyzing constraint qualifications. However, this analysis is often nontrivial and difficult to generalize. Moreover, little attention has been given to handling cases where the required constant qualifications fail. To address this, we propose a novel SOCP reformulation that guarantees Lipschitz continuity of the control law, regardless of the structure of the constraint matrices.

🧩 Key Features

✅ Guaranteed Lipschitz continuity of the optimizer $\pi_{\rm socp}$

✅ Closed-form solution, unlike generic multi-constraint QPs

✅ Faster computation, enabling real-time deployment

✅ Independent of constraint qualifications (e.g., no need for Linear Independence Constraint Qualifications (LICQs))

📁 In This Repo

• The main script is main.ipynb, which includes two working examples.
• Additional notebooks example_1.nb and example_2.nb provide more detailed analysis and visualizations of the two examples in the main script.

📈 How to Use

To get started, open the main script main.ipynb. This notebook demonstrates how to use the LipschitzControllers module to convert general QPs into our Lipschitz-continuous SOCP formulation. Two example problems are included to illustrate the performance and structure of the controller.

Examples

We provide two examples whose original QPs admit non-Lipschitz continuous minimizers (as shown in the animations below) and show that our reformulations do admit Lipschitz continuous minimizers. For more detailed explanations about these examples, please refer to our paper.

Example 1

Example 2