Solving Quadratic Programs to High Precision using Scaled Iterative Refinement

TL;DR

This paper introduces a novel iterative refinement algorithm that solves quadratic programs to arbitrary precision, improving solution accuracy beyond traditional floating-point limitations, with proven convergence and practical implementation.

Contribution

It presents a new refinement method for high-precision QP solutions, along with an efficient implementation and reference solutions for benchmarking.

Findings

Proves linear convergence of the refinement algorithm.

Provides publicly available source code implementation.

Offers precise reference solutions for benchmark problems.

Abstract

Quadratic optimization problems (QPs) are ubiquitous, and solution algorithms have matured to a reliable technology. However, the precision of solutions is usually limited due to the underlying floating-point operations. This may cause inconveniences when solutions are used for rigorous reasoning. We contribute on three levels to overcome this issue. First, we present a novel refinement algorithm to solve QPs to arbitrary precision. It iteratively solves refined QPs, assuming a floating-point QP solver oracle. We prove linear convergence of residuals and primal errors. Second, we provide an efficient implementation, based on SoPlex and qpOASES that is publicly available in source code. Third, we give precise reference solutions for the Maros and M\'esz\'aros benchmark library.