We consider the solution of nonlinear programs in the case where derivatives of the objective function and nonlinear constraints are unavailable. To solve such problems, we propose an adaptation of a method due to Conn, Gould, Sartenaer, and Toint that proceeds by approximately minimizing a succession of linearly constrained augmented Lagrangians. Our modification is to use a derivative-free generating set direct search algorithm to solve the linearly constrained subproblems. The stopping criterion proposed by Conn, Gould, Sartenaer and Toint for the approximate solution of the subproblems requires explicit knowledge of derivatives. Such information is presumed absent in the generating set search method we employ. Instead, we show that stationarity results for linearly constrained generating set search methods provide a derivative-free stopping criterion, based on a step-length control parameter, that is sufficient to preserve the convergence properties of the original augmented Lagrangian algorithm.
Nonlinear programming, augmented Lagrangian methods, constrained optimization, direct search, generating set search, generalized pattern search, derivative-free optimization
@techreport{SAND2006-5315,
author = {Tamara G. Kolda and Robert Michael Lewis and V. Torczon},
title = {A Generating Set Direct Search Augmented {Lagrangian} Algorithm for Optimization with a Combination of General and Linear Constraints},
number = {SAND2006-5315},
institution = {Sandia National Laboratories},
month = {August},
year = {2006},
doi = {10.2172/893121},
url = {http://www.osti.gov/scitech/biblio/893121},
urldate = {2014-04-17},
}