Implicit Primal-Dual Interior-Point Methods for Quadratic Programming

TL;DR

This paper proposes a novel primal-dual interior-point method for quadratic programming that implicitly satisfies complementarity, improves numerical stability, and enables efficient high-precision solutions for large-scale problems.

Contribution

It introduces an implicit complementarity approach using a retraction map and softplus function, enhancing spectral boundedness and numerical stability of interior-point methods.

Findings

Spectrally bounded KKT system improves numerical stability.

Eliminates need for factorization at every iteration.

Enables high-precision solutions with low-precision arithmetic.

Abstract

This paper introduces a new method for solving quadratic programs using primal-dual interior-point methods. Instead of handling complementarity as an explicit equation in the Karush-Kuhn-Tucker (KKT) conditions, we ensure that complementarity is implicitly satisfied by construction. This is achieved by introducing an auxiliary variable and relating it to the duals and slacks via a retraction map. Specifically, we prove that the softplus function has favorable numerical properties compared to the commonly used exponential map. The resulting KKT system is guaranteed to be spectrally bounded, thereby eliminating the most pressing limitation of primal-dual methods: ill-conditioning near the solution. These attributes facilitate the solution of the underlying linear system, either by removing the need to compute factorizations at every iteration, enabling factorization-free approaches like indirect solvers, or allowing the solver to achieve high accuracy in low-precision arithmetic. Consequently, this novel perspective opens new opportunities for interior-point methods, especially for solving large-scale problems to high precision.