An Augmented Lagrangian algorithm for nonlinear semidefinite programming applied to the covering problem

Abstract

In this work, we present an Augmented Lagrangian algorithm for nonlinear semidefinite problems (NLSDPs), which is a natural extension of its consolidated counterpart in nonlinear programming. This method works with two levels of constraints; one that is penalized and other that is kept within the subproblems. This is done to allow exploiting the subproblem structure while solving it. The global convergence theory is based on recent results regarding approximate Karush–Kuhn–Tucker optimality conditions for NLSDPs, which are stronger than the usually employed Fritz John optimality conditions. Additionally, we approach the problem of covering a given object with a fixed number of balls with a minimum radius, where we employ some convex algebraic geometry tools, such as Stengle’s Positivstellensatz and its variations, which allows for a much more general model. Preliminary numerical experiments are presented.