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Title: Generalized pattern search algorithms with adaptive precision function evaluations

Abstract

In the literature on generalized pattern search algorithms, convergence to a stationary point of a once continuously differentiable cost function is established under the assumption that the cost function can be evaluated exactly. However, there is a large class of engineering problems where the numerical evaluation of the cost function involves the solution of systems of differential algebraic equations. Since the termination criteria of the numerical solvers often depend on the design parameters, computer code for solving these systems usually defines a numerical approximation to the cost function that is discontinuous with respect to the design parameters. Standard generalized pattern search algorithms have been applied heuristically to such problems, but no convergence properties have been stated. In this paper we extend a class of generalized pattern search algorithms to a form that uses adaptive precision approximations to the cost function. These numerical approximations need not define a continuous function. Our algorithms can be used for solving linearly constrained problems with cost functions that are at least locally Lipschitz continuous. Assuming that the cost function is smooth, we prove that our algorithms converge to a stationary point. Under the weaker assumption that the cost function is only locally Lipschitz continuous, wemore » show that our algorithms converge to points at which the Clarke generalized directional derivatives are nonnegative in predefined directions. An important feature of our adaptive precision scheme is the use of coarse approximations in the early iterations, with the approximation precision controlled by a test. Such an approach leads to substantial time savings in minimizing computationally expensive functions.« less

Authors:
;
Publication Date:
Research Org.:
Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States)
Sponsoring Org.:
U.S. DOE. Assistant Secretary for Energy Efficiency and Renewable Building System Design (US)
OSTI Identifier:
813385
Report Number(s):
LBNL-52629
R&D Project: 474610; TRN: US200316%%175
DOE Contract Number:  
AC03-76SF00098
Resource Type:
Technical Report
Resource Relation:
Other Information: PBD: 14 May 2003
Country of Publication:
United States
Language:
English
Subject:
32 ENERGY CONSERVATION, CONSUMPTION, AND UTILIZATION; ACCURACY; ALGORITHMS; COMPUTER CODES; CONVERGENCE; DESIGN; EVALUATION

Citation Formats

Polak, Elijah, and Wetter, Michael. Generalized pattern search algorithms with adaptive precision function evaluations. United States: N. p., 2003. Web. doi:10.2172/813385.
Polak, Elijah, & Wetter, Michael. Generalized pattern search algorithms with adaptive precision function evaluations. United States. https://doi.org/10.2172/813385
Polak, Elijah, and Wetter, Michael. 2003. "Generalized pattern search algorithms with adaptive precision function evaluations". United States. https://doi.org/10.2172/813385. https://www.osti.gov/servlets/purl/813385.
@article{osti_813385,
title = {Generalized pattern search algorithms with adaptive precision function evaluations},
author = {Polak, Elijah and Wetter, Michael},
abstractNote = {In the literature on generalized pattern search algorithms, convergence to a stationary point of a once continuously differentiable cost function is established under the assumption that the cost function can be evaluated exactly. However, there is a large class of engineering problems where the numerical evaluation of the cost function involves the solution of systems of differential algebraic equations. Since the termination criteria of the numerical solvers often depend on the design parameters, computer code for solving these systems usually defines a numerical approximation to the cost function that is discontinuous with respect to the design parameters. Standard generalized pattern search algorithms have been applied heuristically to such problems, but no convergence properties have been stated. In this paper we extend a class of generalized pattern search algorithms to a form that uses adaptive precision approximations to the cost function. These numerical approximations need not define a continuous function. Our algorithms can be used for solving linearly constrained problems with cost functions that are at least locally Lipschitz continuous. Assuming that the cost function is smooth, we prove that our algorithms converge to a stationary point. Under the weaker assumption that the cost function is only locally Lipschitz continuous, we show that our algorithms converge to points at which the Clarke generalized directional derivatives are nonnegative in predefined directions. An important feature of our adaptive precision scheme is the use of coarse approximations in the early iterations, with the approximation precision controlled by a test. Such an approach leads to substantial time savings in minimizing computationally expensive functions.},
doi = {10.2172/813385},
url = {https://www.osti.gov/biblio/813385}, journal = {},
number = ,
volume = ,
place = {United States},
year = {Wed May 14 00:00:00 EDT 2003},
month = {Wed May 14 00:00:00 EDT 2003}
}