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Alexandrove-amf - S tructural Opt-23(~ Springer-Verlag 1998...

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Structural Optimization 15, 16-23 (~) Springer-Verlag 1998 A trust-region framework for managing the use of approxima- tion models in optimization N.M. Alexandrov Multidisciplinary Optimization Branch, MS 159, NASA Langley Research Center, Hampton, VA 23681, USA J.E. Dennis, Jr.* Department of Computational and Applied Mathematics, Rice University, Houston, TX 77005, USA R.M. Lewis** ICASE, MS 403, NASA Langley Research Center, Hampton, VA 23681, USA V. Torczont Department of Computer Science, College of William ~ Mary, Williamsburg, VA 23187, USA Abstract This paper presents an analytically robust, globally convergent approach to managing the use of approximation mod- els of varying fidelity in optimization. By robust global behaviour we mean the mathematical assurance that the iterates produced by the optimization algorithm, started at an arbitrary initial iter- ate, will converge to a stationary point or local optimizer for the original problem. The approach presented is based on the trust region idea from nonlinear programming and is shown to be prov- ably convergent to a solution of the original high-fidelity problem. The proposed method for managing approximations in engineer- ing optimization suggests ways to decide when the fidelity, and thus the cost, of the approximations might be fruitfully increased or decreased in the course of the optimization iterations. The ap- proach is quite general. We make no assumptions on the structure of the original problem, in particular, no assumptions of convexity and separability, and place only mild requirements on the approx- imations. The approximations used in the framework can be of any nature appropriate to an application; for instance, they can be represented by analyses, simulations, or simple algebraic models. This paper introduces the approach and outlines the convergence analysis. 1 Introduction In this paper we present an approach to managing the use of approximation models in optimization that is based on the * This reseal~ch was supported by the Dept. of Energy grant DE- FG03-95ER25257 and Air Force Office of Scientific Research grant F49620-95-1-0210 ** This research was supported by the National Aeronautics and Space Administration under NASA Contract No. NAS1-19480 while the author was in residence at the Institute for Computer Applications in Science and Engineering (ICASE), NASA Langley Research Center, Hampton, VA 23681, USA t This research was supported by the Air Force Office of Scien- tific Research ~rant F49620-95-1-0210 and by the National Aeronautics and Space Administration under NASA Contract No. NAS1-19480 while the author was in residence at the Institute for Computer Applications in Science and Engineering (ICASE), NASA Langley Research Center, Hampton, VA 23681, USA trust region approach from nonlinear programming (Dennis and Schnabel 1983; Mo% 1983). The approach presented inherits the mathematical robustness and global and local convergence properties of the classical trust region methods.
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