Augmented Lagrangian methods for constrained optimization using differentiable exact penalty

• Grant number: 22/06745-7
• Support Opportunities: Scholarships in Brazil - Doctorate
• Start date: September 01, 2022
• Status: Discontinued
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Applied Mathematics
• Principal Investigator: Roberto Andreani
• Grantee: Mariana da Rosa
• 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
• Associated scholarship(s): 24/15980-5 - Integration of Continuous and Discrete Optimization Methods in the Distance Geometry Problem with Data Uncertainties, BE.EP.DR

Abstract

Augmented Lagrangian methods are effective tools for solving general constrained optimization problems. One of its drawbacks, however, is the excessive increase in the penalty parameter, which can generate numerical instabilities, especially in non-convex problems. One of the strategies that aim to reduce this problem employs exact penalty functions. However, the usual exact penalty functions are non-differentiable, which in practice makes it difficult to use efficient algorithms for smooth optimization. This project aims to circumvent this problem by using differentiable exact penalty functions, which were initially proposed by Di Pillo and Grippo. First, we intend to extend the theory of the exact penalty method described in Andreani et al (2012) assuming only the constant rank of the Jacobian matrix of the constraints. Such an extension represents an important improvement in the current theory, conceived with the hypothesis of linear independence because it allows the implementation of the method to solve degenerate problems. Second, a hybridization of this extension with the Powell-Hestenes-Rockafellar augmented Lagrangian method, in this case, Algencan, will be proposed. The aim is to obtain a functional implementation that increases the effectiveness of Algencan. Thirdly, the theoretical conditions for the boundedness of the Algencan penalty parameter will be studied. In addition to extending the literature results, this last topic has the potential to produce a more efficient implementation of the hybrid method. (AU)