A Globally Convergent Filter Method for Nonlinear Programming

Abstract

In this paper we present a filter algorithm for nonlinear programming and prove its global convergence to stationary points. Each iteration is composed of a feasibility phase, which reduces a measure of infeasibility, and an optimality phase, which reduces the objective function in a tangential approximation of the feasible set. These two phases are totally independent, and the only coupling between them is provided by the filter. The method is independent of the internal algorithms used in each iteration, as long as these algorithms satisfy reasonable assumptions on their efficiency. Under standard hypotheses, we show two results: for a filter with minimum size, the algorithm generates a stationary accumulation point; for a slightly larger filter, all accumulation points are stationary.

A filter inexact-restoration method for nonlinear programming

“…Two relevant remarks were found -some problems, denoted by a), suffer from Marato's effect, which means that the solution is obtained in few iterations but the algorithm does not converge, and in two problems the failure is related to the existence of a stationary point x with h(x) > . The last remark had been already mentioned by Gonzaga et al [7] in the convergence proof. …”

“…A similar idea was suggested by Gonzaga et al [7] where no methods are specified for the IR phases. In each phase of the IR method a filter scheme with line search technique is used instead of a merit function.…”

“…The global convergence was studied by Gonzaga et al [7], where it is proved that this method is independent of the internal algorithms used in each iteration, since these algorithms satisfy some requirements in their efficiency. It is shown that, under some assumptions, for a filter with a minimal size, the algorithm generates a stationary accumulation point and that for bigger filters, all the accumulation points are stationary.…”

“…The high level algorithm is suggested by Gonzaga et al [7] but not yet implemented -the internal algorithms are not proposed. The filter, a new concept introduced by Fletcher and Leyffer [3], replaces the merit function avoiding the penalty parameter estimation and the difficulties related to the nondifferentiability.…”

A sequential quadratic programming algorithm that combines merit function and filter ideas

“…This approach was promptly followed by many authors, mainly in conjunction with SLP (sequential linear programming), SQP and interior-point type methods (see, for instance, [1,5,6,7,9,11,12,15,16,17,22,23,24,25]). …”