Incremental Methods with Acceleration and Exact Smoothing of Convex Optimization Problems: A Unified Approach Through First-Order Inexact Methods

• Grant number: 23/15603-4
• Support Opportunities: Scholarships in Brazil - Doctorate
• Start date: March 01, 2024
• End date: February 29, 2028
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Applied Mathematics
• Principal Investigator: Ernesto Julián Goldberg Birgin
• Grantee: Gabriel Rodrigues Silva Grillo
• Host Institution: Instituto de Matemática e Estatística (IME). Universidade de São Paulo (USP). São Paulo , SP, Brazil
• Associated research grant: 18/24293-0 - Computational methods in optimization, AP.TEM
• Associated scholarship(s): 25/06013-4 - Efficient Iterative Reconstruction Algorithms for Magnetic Resonance Imaging, BE.EP.DR

Abstract

The present project addresses two important questions currently open in the literature regarding large-scale convex optimization using first-order moment-accelerated algorithms. The first objective is to obtain a (perhaps exact) way of smoothing completely non-smooth composite problems that has a theory compatible with practical results, as currently in imaging applications experience shows that existing theoretical accuracy limits are clearly very pessimistic. Furthermore, in the partially smooth composite case, we wish to address the question of how to achieve a moment-accelerated prox-incremental method that has a satisfactory convergence theory. This type of technique is important in the case where the gradient of the smooth term of the objective function has a sum structure of a large number of terms and is of high computational cost. Both objectives are useful in applications to variational regularization of inverse problems in images.