A Spectral Gradient Projection Method for the Positive Semi-definite Procrustes Problem

Harry Oviedo

doi:10.15446/recolma.v55n1.99100

Abstract

This paper addresses the positive semi-deffnite procrustes problem (PSDP). The PSDP corresponds to a least squares problem over the set of symmetric and semi-deffnite positive matrices. These kinds of problems appear in many applications such as structure analysis, signal processing, among others. A non-monotone spectral projected gradient algorithm is proposed to obtain a numerical solution for the PSDP. The proposed algorithm employs the Zhang and Hager's non-monotone technique in combination with the Barzilai and Borwein's step size to accelerate convergence. Some theoretical results are presented. Finally, numerical experiments are performed to demonstrate the effectiveness and efficiency of the proposed method, and comparisons are made with other state-of-the-art algorithms.

Highlights

The positive semi-definite Procrustes problem (PSDP) is defined as follows: given two rectangular matrices A, B ∈ Rn×m, we want to find a symmetric and positive semi–definite matrix X∗ ∈ Rn×n that solves the following optimization problem min F (X) s.t
With the purpose of accelerating the convergence of the gradient projection scheme (4), we adopt the non-monotone globalization technique proposed by Zhang and Hager in [18] combined with the Barzilai and Borwein step sizes [3] which usually accelerate the convergence of gradient-based methods
In order to avoid the calculation of such spectral decomposition, in each step, we propose the following idea: first note that if the symmetric part of Yk = Xk − τk∇F (Xk) is positive definite this matrix coincides with its projection over S+(n)

Summary

Introduction

The positive semi-definite Procrustes problem (PSDP) is defined as follows: given two rectangular matrices A, B ∈ Rn×m, we want to find a symmetric and positive semi–definite matrix X∗ ∈ Rn×n that solves the following optimization problem min F (X) s.t. In [8], a method called “AN-FGM” is proposed which is a semi-analytic approach that reduces the problem (1) to the case when A is diagonal and uses the FGM to address a more easy problem, this proposal looks quite efficient to deal with problems where A is ill-conditioned Another alternative to compute a numerical solution of problem (1), has been studied in [1], where the authors propose an algorithm called “Parallel tangents” that is based on the gradient projection method that incorporates an over-relaxation step. One drawback of this parallel tangents method is that it does not guarantee optimal convergence.

Notations and Important Tools

A feasible Update Scheme

Numerical Experiments

Method

Conclusions