Implementation of augmented Lagrangian methods with first-order information

• Grant number: 24/22384-0
• Support Opportunities: Scholarships in Brazil - Post-Doctoral
• Start date: April 01, 2025
• End date: March 31, 2027
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Applied Mathematics
• Principal Investigator: Ernesto Julián Goldberg Birgin
• Grantee: Diaulas Murize Santana Vieira Marcondes
• Host Institution: Instituto de Matemática e Estatística (IME). Universidade de São Paulo (USP). São Paulo , SP, Brazil
• Associated research grant: 23/08706-1 - Numerical optimization, AP.TEM

Abstract

In recent years there has been increasing interest in very large optimization problems, such as applications related to the Internet, machine learning, telecommunications and the financial market. In many problems we don't have access to the Hessian matrix and, in large problems, the use of Hessians requires substantial computational effort. To get around this problem, we need to use first-order methods, which only use evaluations of the objective function, the constraints and their gradients. In this project, we will develop augmented Lagrangian methods for nonlinear optimization that use only first-order information, both in solving the subproblems and for possible accelerations. In the intended approach, the subproblems are minimization problems in boxes and the acceleration is related to solving a non-linear system. The methods developed for these tasks have value in their own right, since minimization problems in boxes and non-linear systems also have multiple practical applications. In this project we intend to produce efficient implementations of the proposed methods, as well as asymptotic convergence and complexity theory. We are particularly interested in the worst-case complexity theory for a box minimization method based on the active constraints technique and using a truncated Newton-type method approximating the Hessian-vector product inside the faces. We are also interested in the efficient resolution of a KKT system using only first-order information. (AU)