Ademir A. Ribeiro

A bundle-filter method for nonsmooth convex constrained optimization

For solving nonsmooth convex constrained optimization problems, we propose an algorithm which combines the ideas of the proximal bundle methods with the filter strategy for evaluating candidate points. The resulting algorithm inherits some attractive features from both approaches. On the one hand, it allows effective control of the size of quadratic programming subproblems via the compression and aggregation techniques of proximal bundle methods. On the other hand, the filter criterion for accepting a candidate point as the new iterate is sometimes easier to satisfy than the usual descent condition in bundle methods. Some encouraging preliminary computational results are also reported.

Global Convergence of Filter Methods for Nonlinear Programming

We present a general filter algorithm that allows a great deal of freedom in the step computation. Each iteration of the algorithm consists basically in computing a point which is not forbidden by the filter, from the current point. We prove its global convergence, assuming that the step must be efficient, in the sense that, near a feasible nonstationary point, the reduction of the objective function is “large.” We show that this condition is reasonable, by presenting two classical ways of performing the step which satisfy it. In the first one, the step is obtained by the inexact restoration method of Martinez and Pilotta. In the second, the step is computed by sequential quadratic programming.

Global convergence of slanting filter methods for nonlinear programming

In this paper, we present a general algorithm for nonlinear programming which uses a slanting filter criterion for accepting the new iterates. Independently of how these iterates are computed, we prove that all accumulation points of the sequence generated by the algorithm are feasible. Computing the new iterates by the inexact restoration method, we prove stationarity of all accumulation points of the sequence.

Numerical comparison of merit function with filter criterion in inexact restoration algorithms using hard-spheres problems

Abstract Inexact Restoration methods have been introduced for solving nonlinear programming problems. Each iteration is composed of two phases. The first one reduces a measure of infeasibility, while in the second one the objective function value is reduced in a tangential approximation of the feasible set. The point obtained from the second phase is compared with the current point either by means of a merit function or by using a filter criterion. A comparative numerical study about these criteria by using a family of Hard-Spheres Problems is presented.

Local convergence of filter methods for equality constrained non-linear programming

In Gonzaga et al. [A globally convergent filter method for nonlinear programming, SIAM J. Optimiz. 14 (2003), pp. 646–669] we discuss general conditions to ensure global convergence of inexact restoration filter algorithms for non-linear programming. In this article we show how to avoid the Maratos effect by means of a second-order correction. The algorithms are based on feasibility and optimality phases, which can be either independent or not. The optimality phase differs from the original one only when a full Newton step for the tangential minimization of the Lagrangian is efficient but not acceptable by the filter method. In this case a second-order corrector step tries to produce an acceptable point keeping the efficiency of the rejected step. The resulting point is tested by trust region criteria. Under the usual hypotheses, the algorithm inherits the quadratic convergence properties of the feasibility and optimality phases. This article includes a comparison between classical Sequential Quadratic Programming (SQP) and Inexact Restoration (IR) iterations, showing that both methods share the same asymptotic convergence properties.

Fenchel-Moreau conjugation for lower semi-continuous functions

We introduce a modification of Fenchels conjugation which is a particular case of Moreaus conjugation. We obtain good properties such as convexity of the conjugate function even though the function is not convex. We also introduce the concept of conjugate dual space as a class of continuous operators, while in the Fenchel conjugation the conjugate dual space is the classical topological dual space. Finally, we present some examples for illustrating the difference between the Fenchel–Moreau conjugation and our modification.

Global convergence of a general filter algorithm based on an efficiency condition of the step

In this work we discuss global convergence of a general filter algorithm that depends neither on the definition of the forbidden region, which can be given by the original or slanting filter rule, nor on the way in which the step is computed. This algorithm basically consists of calculating a point not forbidden by the filter from the current point. Assuming that this step must be efficient, in the sense that near a feasible non-stationary point the decrease in the objective function is relatively large, we prove the global convergence of the algorithm. We also discuss that such a condition is satisfied if the step is computed by the SQP or Inexact Restoration methods. For SQP we present a general proof of this result that is valid for both the original and the slanting filter criterion. In order to compare the performance of the general filter algorithm according to the method used to calculate the step and the filter rule regarded, we present numerical experiments performed with problems from CUTEr collection.

Uma nova taxa de convergência para o Método do Gradiente aplicado á minimização de funções quadráticas

Realizamos neste trabalho um estudo sobre a velocidade de convergencia do Metodo do Gradiente aplicado a minimizacao de funcoes quadraticas convexas com busca exata. Destacamos a ocorrencia de uma nova taxa de convergencia para a sequencia gerada pelo algoritmo, diferente das apresentadas na literatura, a qual demonstramos para o caso 2 x 2.