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Revista Colombiana de Matemáticas
Print version ISSN 0034-7426
Rev.colomb.mat. vol.55 no.1 Bogotá Jan./June 2021 Epub Nov 17, 2021
https://doi.org/10.15446/recolma.v55n1.99100
ORIGINAL ARTICLES
A Spectral Gradient Projection Method for the Positive Semi-definite Procrustes Problem
Un Método de Gradiente Proyectado Espectral para el Problema de Mínimos Cuadrados Matricial Semi-definido Positivo
1 Fundação Getulio Vargas - Escola de Matemática Aplicada, FGV\EMAp, Brazil, Rio de Janeiro, Brasil
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 eficiency of the proposed method, and comparisons are made with other state-of-the-art algorithms.
Keywords: Non-monotone algorithm; Constrained optimization; Symmetric positive semi-definite constraints; Least-Square problems
En este artículo abordamos el problema de mínimos cuadrados lineales sobre el conjunto de matrices simétricas y definidas positivas (PSDP). Esta clase de problemas surge en un gran número de aplicaciones tales como análisis de estructuras, procesamiento de señales, análisis de componentes principales, entre otras. Para resolver este tipo de problemas, proponemos un método de gradiente proyectado espectral no-monótono. El algoritmo propuesto usa la técnica de globalización no-monótona de Zhang y Hager, en combinación con los tamaños de paso de Barzilai y Borwein para acelerar la convergencia del método. Además, presentamos y comentamos algunos resultados teóricos concernientes al algoritmo desarrollado. Finalmente, llevamos a cabo varios experimentos numéricos con el fin de demostrar la efectividad y la eficiencia del nuevo enfoque, y realizamos comparaciones con algunos métodos existentes en la literatura.
Palabras clave: Algoritmo no-monótono; optimización con restricciones; restricciones simétricas y semi definidas positivas; problema de mínimos cuadrados
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Received: June 12, 2020; Accepted: March 06, 2021
This is an open-access article distributed under the terms of the Creative Commons Attribution License
