Second-order methods for discontinuous composite problems

• Grant number: 24/20168-8
• Support Opportunities: Scholarships in Brazil - Doctorate
• Start date: March 01, 2025
• End date: February 28, 2029
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Applied Mathematics
• Principal Investigator: Paulo José da Silva e Silva
• Grantee: Gabriel Belém Barbosa
• Host Institution: Instituto de Matemática, Estatística e Computação Científica (IMECC). Universidade Estadual de Campinas (UNICAMP). Campinas , SP, Brazil
• Associated research grant: 23/08706-1 - Numerical optimization, AP.TEM
• Associated scholarship(s): 26/11005-3 - Scalable Second-Order and Hybrid Methods for Optimization with Discontinuous Regularizations, BE.EP.DR

Abstract

This project aims to investigate and advance the field of nonconvex composite optimization, focusing on the recent incorporation of second-order information into iterative models. Two recent sources stand out in this area: Forward-backward envelope for the sum of two nonconvex functions: further properties and nonmonotone line-search algorithms [Themeles 2018] and The Indefinite Proximal Gradient Method [Leconte 2024]. Several real-world problems, including sparse signal recovery, image processing, and portfolio optimization, can be modeled in the form of the problem of interest. This work will also involve extensive numerical experimentation on several types of problems to evaluate the performance of the proposed algorithms. The experiments will focus on comparing the performance of different line search strategies with or without relaxed assumptions, especially nonmonotone variants, the impact of incorporating quasi-Newton information into the subproblem and the effectiveness of each update. New hybrid strategies also possibly will be analysed. The project aims contribute to advance the understanding of methods such as ZeroFPR [Themeles 2018] and indefinite proximal gradient [Leconte 2024] in the treatment of non-convex composite optimization problems, to provide practical guidance in the selection of appropriate line search strategies and quasi-Newton updates for different problem structures, and to develop efficient and robust algorithms to solve a broader class of non-convex composite optimization problems, potentially leading to improved solutions in important application areas. (AU)