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. Keywords Augmented Lagrangian • Nonlinear semidefinite programming • Covering problem • Convex algebraic geometry Communicated by Natasa Krejic.

A Shape-Newton approach to the problem of covering with identical balls

“…In this section, we prove that the second-order derivatives of G, as defined in (2), are given by (5,7,8,9,10). In [6] we have built appropriate bi-Lipschitz mappings T t in order to use integration by substitution for the differentiation of G(x + tδx, r) and G(x, r + tδr).…”

“…From the practical point of view, underlying partitions that lead to exact calculations might be implemented using power diagrams [4,21]. We observe that formulae (5,7,8,9,10) are valid for general sets A satisfying Assumptions 1 and 2, but the exact numerical computation of G, ∇G and ∇ 2 G requires A to be a union of non-overlapping convex polygons. The exact calculation of ∇G and ∇ 2 G can actually be performed for any set A such that the intersections of ∂A with circles can be computed analytically.…”

“…The Minkowski island fractal shown in Figure 9, with Vol(A) = 16, is composed by p = 16 unit squares. Each square can be represented by its bottom-left corners V = {(3, 0), (1, 1), (3, 1), (4, 1), (0, 2), (1, 2), (2, 2), (3, 2), (2, 3), (3,3), (4, 3), (5,3), (1,4), (2,4), (4,4), (2, 5)}.…”

“…None of the mentioned works addresses the problem in a unified way as a continuous optimization problem, nor do they present first-or second-order derivatives of the functions that define the problem. In [5], the problem of covering an arbitrary region is modeled as a nonlinear semidefinite programming problem using convex algebraic geometry tools. The introduced model describes the covering problem without resorting to discretizations, but it depends on some polynomials of unknown degrees whose coefficients are difficult to compute, limiting the applicability of the method.…”

An Augmented Lagrangian Method for Cardinality-Constrained Optimization Problems

“…For NLPs it has been shown that this variant possesses strong convergence properties even under very mild assumptions [4,6]. It has since been applied to solve various other problems, including MPCC [7,26], quasi-variational inequalities [27], generalized Nash equilibrium problems [16,29], and semidefinite programming [8,14].…”