An inexact penalty decomposition approach for augmented Lagrangian methods with applications to nonlinear conic programming with structured geometric constraints in Hilbert spaces

• Grant number: 24/23394-9
• Support Opportunities: Scholarships abroad - Research Internship - Post-doctor
• Start date: May 01, 2025
• End date: April 30, 2026
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Applied Mathematics
• Principal Investigator: Gabriel Haeser
• Grantee: Renan Willian Prado
• Supervisor: Christian Kanzow
• Host Institution: Instituto de Matemática e Estatística (IME). Universidade de São Paulo (USP). São Paulo , SP, Brazil
• Institution abroad: Julius-Maximilians-Universität Würzburg (JMU), Germany
• Associated to the scholarship: 23/08621-6 - A study on sequential optimality conditions for nonlinear conic programming with applications in the safeguarded augmented Lagrangian method, BP.PD

Abstract

This project focuses on addressing mathematical programming problems in Hilbert spaces using a safeguarded augmented Lagrangian method. Our proposal aims to achieve both practical and theoretical improvements by modifying the existing method to reduce its runtime while maintaining its produced sequences. We seek to understand the fundamental mechanisms of the Inexact Penalty Decomposition Method and integrate it into the augmented Lagrangian method to achieve practical enhancements. To validate these improvements, we plan to perform numerical experiments. For the theoretical aspects, we will investigate whether treating geometric constraints as hard constraints simplifies the assumptions necessary for obtaining bounded multipliers in the method. Preliminary results in finite-dimensional spaces suggest that the modified method can indeed be associated with bounded multipliers under a specialized Error Bound constraint qualification.