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

## TL;DR

This paper introduces a spectral gradient projection algorithm for solving the positive semi-definite Procrustes problem, demonstrating improved convergence and efficiency through theoretical analysis and numerical experiments.

## Contribution

It proposes a novel non-monotone spectral projected gradient method tailored for the PSDP, combining advanced step size strategies and theoretical guarantees.

## Key findings

- The algorithm converges faster than existing methods.
- Numerical results show improved accuracy and efficiency.
- Comparative analysis confirms the method's effectiveness.

## Abstract

This paper addresses the positive semi-definite procrustes problem (PSDP). The PSDP corresponds to a least squares problem over the set of symmetric and semi-definite 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.