APP-Hom Method for Box Constrained Quadratic Programming

TL;DR

This paper introduces the APP-Hom method, combining an accelerated proximal point algorithm with a homotopy approach, to efficiently solve box constrained quadratic programming problems, including non-convex cases, with significant step reduction and computational efficiency.

Contribution

The paper develops a novel APP-Hom algorithm that integrates an accelerated proximal point method with a homotopy technique, enhancing efficiency for solving BQP problems.

Findings

APP algorithm requires fewer steps than PP algorithm.

Homotopy method is highly efficient for strictly-convex BQP.

APP-Hom performs well on diverse BQP applications.

Abstract

In this paper, based on a $Q$-linear convergence analysis and an estimate of the linear convergence factor of the proximal point (PP) algorithm for solving box constrained quadratic programming (BQP) problems, an accelerated proximal point (APP) algorithm for solving BQP problems is presented. To solve the strictly convex BQP problems in each step of the APP algorithm, an efficient homotopy method, which tracks the solution path of a parametric quadratic program, is given. The algorithm with APP algorithm as outer iteration and the homotopy method as inner iteration is named by APP-Hom. The inner homotopy method is efficient by implementing, a warm-start technique based on the accelerated proximal gradient (APG) method, an $\varepsilon$-relaxation technique for checking prime and dual feasibility and determining/correcting the active set. Numerical tests for randomly generated dense and sparse BQPs, BQPs arising from image deblurring, BQPs in SVM, as well as discretized obstacle problem, elastic-plastic torsion problem, and the journal bearing problem show that the APP algorithm takes much less steps than the PP algorithm, the homotopy method is very efficient for strictly-convex BQP, and in consequence, that the APP-Hom is very efficient for non-convex BQP.