Solving Quadratic Programs with Slack Variables via ADMM without Increasing the Problem Size

TL;DR

This paper introduces a modified ADMM approach for quadratic programs with slack variables that avoids increasing problem size, leading to faster numerical solutions.

Contribution

A novel ADMM scheme that handles slack variables without enlarging the problem, maintaining standard structure and improving computational efficiency.

Findings

The method is mathematically equivalent to ADMM with slack variables.

Numerical experiments demonstrate significant speedups.

The approach preserves the original problem structure.

Abstract

Proximal methods such as the Alternating Direction Method of Multipliers (ADMM) are effective at solving constrained quadratic programs (QPs). To tackle infeasible QPs, slack variables are often introduced to ensure feasibility, which changes the structure of the problem, increases its size, and slows down numerical resolution. In this letter, we propose a simple ADMM scheme to tackle QPs with slack variables without increasing the size of the original problem. The only modification is a slightly different projection in the z-update, while the rest of the algorithm remains standard. We prove that the method is equivalent to applying ADMM to the QP with additional slack variables, even though slack variables are not added. Numerical experiments show speedups of the approach.