On sequential optimality conditions associated with the safeguarded augmented Lagrangian algorithmic scheme

Author(s)

Renan William Prado

• Document type: Doctoral Thesis
• Press: Campinas, SP.
• Institution: Universidade Estadual de Campinas (UNICAMP). Instituto de Matemática, Estatística e Computação Científica
• Defense date: 2023-03-27

Examining board members

Sandra Augusta Santos; Maria Laura Schuverdt; Leonardo Delarmelina Secchin; Gabriel Haeser; Roberto Andreani

• Advisor: Sandra Augusta Santos; Lucas Eduardo Azevedo Simões

Abstract

The safeguarded augmented Lagrangian method (SALM) is widely used to solve general nonlinear programming problems. Due to its ability to calculate approximate Karush-Kuhn-Tucker (KKT) points, the method is known to converge to KKT points under weak constraint qualifications. Such a success is explained by the sequential optimality conditions. They are practical conditions that, when ensured by the sequence generated by the algorithm, provide a convergence analysis closer to the real behavior of the method, accurately reflecting the capabilities of the algorithm. Thus, a way to ensure new properties for a method is to perform its detailed global convergence analysis through sequential optimality conditions. With that in mind, and aiming to improve the global convergence theory of the SALM, it is interesting to clarify how the optimality conditions that the method already guarantees are related to each other, as well as to ensure which conditions generated by other algorithms are also guaranteed by the SALM, indicating that the method, possibly, guarantees better candidates for minimizers than the other algorithms addressed in the analysis. Aiming at such questions, this work had two goals: to show that two new sequential optimality conditions are associated with the SALM, one previously established for the non-smooth context, called ?-Approximate Smooth Optimality Condition (?-ASOC), and another derived for the quadratic penalty method, called Strong AKKT (SAKKT); the second goal was to clarify the relationship between the newly devised conditions and the main conditions already studied in the literature. To ensure that the second goal was met, it was shown that the complementary approximate KKT (CAKKT) and the strong approximate gradient projection (SAGP) conditions are guaranteed by the ?-ASOC, that the positive approximate KKT (PAKKT) conditions are not related to the ?-ASOC and that, under mild assumptions, SAGP is equivalent to the CAKKT conditions. Improving previous results, this work shows that a mild modification of the SAKKT conditions, still guaranteed by the SALM, is stronger than other known sequential optimality conditions, namely, CAKKT, PAKKT and the ?-ASOC. In short, the SALM global convergence results were clarified and a more complete relationship between the sequential optimality conditions guaranteed by the method was given (AU)

• FAPESP's process: 19/18859-4 - A continuous optimization method with stopping criterion based on a new sequential optimality condition
• Grantee: Renan Willian Prado
• Support Opportunities: Scholarships in Brazil - Doctorate (Direct)