EIQP: Execution-time-certified and Infeasibility-detecting QP Solver

TL;DR

This paper introduces EIQP, a novel infeasible interior-point method for convex quadratic programming that certifies solution feasibility or detects infeasibility within a guaranteed time, suitable for real-time control applications.

Contribution

It proposes a new infeasible IPM with exact, data-independent iteration complexity for real-time convex QP solving, including infeasibility detection, with accessible implementation.

Findings

Achieves $O(rac{ ext{log}(rac{n+1}{ ext{epsilon}})}{- ext{log}(1-rac{0.414213}{ ext{sqrt}(n+1)})})$ iteration complexity

Provides a simple, line-search-free implementation with MATLAB, Julia, and Python interfaces

Demonstrates effectiveness in real-time control scenarios

Abstract

Solving real-time quadratic programming (QP) is a ubiquitous task in control engineering, such as in model predictive control and control barrier function-based QP. In such real-time scenarios, certifying that the employed QP algorithm can either return a solution within a predefined level of optimality or detect QP infeasibility before the predefined sampling time is a pressing requirement. This article considers convex QP (including linear programming) and adopts its homogeneous formulation to achieve infeasibility detection. Exploiting this homogeneous formulation, this article proposes a novel infeasible interior-point method (IPM) algorithm with the best theoretical $O(\sqrt{n})$ iteration complexity that feasible IPM algorithms enjoy. The iteration complexity is proved to be \textit{exact} (rather than an upper bound), \textit{simple to calculate}, and \textit{data independent}, with the value $\left\lceil\frac{\log(\frac{n+1}{\epsilon})}{-\log(1-\frac{0.414213}{\sqrt{n+1}})}\right\rceil$ (where $n$ and $\epsilon$ denote the number of constraints and the predefined optimality level, respectively), making it appealing to certify the execution time of online time-varying convex QPs. The proposed algorithm is simple to implement without requiring a line search procedure (uses the full Newton step), and its C-code implementation (offering MATLAB, Julia, and Python interfaces) and numerical examples are publicly available at https://github.com/liangwu2019/EIQP.