Scholarship 22/15197-3 - Otimização - BV FAPESP
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Computational optimization methods for problems with simple convex constraints

Grant number: 22/15197-3
Support Opportunities:Scholarships in Brazil - Master
Start date until: March 01, 2023
End date until: February 28, 2025
Field of knowledge:Physical Sciences and Mathematics - Mathematics - Applied Mathematics
Principal Investigator:Roberto Andreani
Grantee:Elivandro Oliveira Grippa
Host Institution: Instituto de Matemática, Estatística e Computação Científica (IMECC). Universidade Estadual de Campinas (UNICAMP). Campinas , SP, Brazil
Associated research grant:18/24293-0 - Computational methods in optimization, AP.TEM

Abstract

The spectral gradient method (or BBR -- Barzilai-Borwein-Raydan) developed by Raydan, and based on Barzilai and Borwein's seminal proposal for quadratic minimization, proved to be an effective tool for solving large-scale unconstrained problems. Since then, several similar methods have appeared in the literature, a topic that remains very active among researchers. Other important methods for unconstrained optimization are those that conjugate directions (also called conjugate gradients -- CG). Since the classic method of Hestenes and Stiefel for quadratics, extensions to general functions have been proposed in the literature. The main one, considered the most effective, is the Hager and Zhang method known as CG_descent. In its current version, it implements quasi-Newton steps to correct the loss of orthogonality during the minimization process. Recently, another such method, called CG_OPT 2.0, was proposed by Liu, Liu and Dai, for which the authors report favorable numerical tests against CG_descent. In this project, we consider the main BBR-type methods and the two GC-type methods mentioned. Firstly, we intend to review the BBR methods, studying in detail the convergence theory and numerically comparing them on CUTEst problems. In a second moment, we intend to study the conjugate gradient methods for non-quadratic functions CG_descent and CG_OPT 2.0. We also consider the recent Delayed Weighted Gradient (DWGM) method for quadratics, proposed by Oviedo, which has been attracting the attention of this supervisor candidate. In fact, in 2022 an extension for general strongly convex functions was published with contributors, and a possible hybridization using DWGM is under discussion. This is a topic that seems fruitful and long-lived. Therefore, the idea is that the student reaches the end of his master's degree with sufficient knowledge and experience to contribute, in a possible doctorate, with such research. Finally, it is worth mentioning that the candidate has prerequisites in non-linear optimization, has experience with numerical practice and read some of the references of this project during his undergraduate course.

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