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

Liang Wu, Wei Xiao, Richard Braatz

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 exact (rather than an upper bound), simple to calculate, and 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.