A study on sequential optimality conditions for nonlinear conic programming with applications in the safeguarded augmented Lagrangian method

Abstract

An important class of constrained optimization problems that has recently received attention is the class of nonlinear conic programming problems. This class of problems is difficult to solve. For this reason, it can only be obtained solution candidates in practice. Despite this, it has been observed that when optimization problems are treated naturally, optimization algorithms have a better practical performance, finding better candidate solutions. Simultaneously, nonlinear conic programming is a natural way to formulate several optimization problems. Thus, providing an efficient algorithm to directly solve such an important class of problems also provides better solution candidates.Aiming to provide an efficient algorithm to solve nonlinear conic programming problems as naturally as possible, this project proposes modifying the safeguarded augmented Lagrangian method in order to reduce its runtime. Inspired by the Inexact Penalty Decomposition Method, our proposal incorporates sequential optimality conditions to improve the existing method, maintaining its global convergence theory. We plan to perform numerical experiments to validate these improvements. (AU)