A Trust Region Framework for Managing the Use of Approximation Models in Optimization
01 Oct 1997-Structural Optimization (Institute for Computer Applications in Science and Engineering (ICASE))-Vol. 15, Iss: 1, pp 16-23
TL;DR: An analytically robust, globally convergent approach to managing the use of approximation models of varying fidelity in optimization, based on the trust region idea from nonlinear programming, which is shown to be provably convergent to a solution of the original high-fidelity problem.
Abstract: This paper presents an analytically robust, globally convergent approach to managing the use of approximation models of various fidelity in optimization. By robust global behavior we mean the mathematical assurance that the iterates produced by the optimization algorithm, started at an arbitrary initial iterate, will converge to a stationary point or local optimizer for the original problem. The approach we present is based on the trust region idea from nonlinear programming and is shown to be provably convergent to a solution of the original high-fidelity problem. The proposed method for managing approximations in engineering 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 approach 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 approximations. 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.
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TL;DR: This paper introduces the reader to a response surface methodology that is especially good at modeling the nonlinear, multimodal functions that often occur in engineering and shows how these approximating functions can be used to construct an efficient global optimization algorithm with a credible stopping rule.
Abstract: In many engineering optimization problems, the number of function evaluations is severely limited by time or cost. These problems pose a special challenge to the field of global optimization, since existing methods often require more function evaluations than can be comfortably afforded. One way to address this challenge is to fit response surfaces to data collected by evaluating the objective and constraint functions at a few points. These surfaces can then be used for visualization, tradeoff analysis, and optimization. In this paper, we introduce the reader to a response surface methodology that is especially good at modeling the nonlinear, multimodal functions that often occur in engineering. We then show how these approximating functions can be used to construct an efficient global optimization algorithm with a credible stopping rule. The key to using response surfaces for global optimization lies in balancing the need to exploit the approximating surface (by sampling where it is minimized) with the need to improve the approximation (by sampling where prediction error may be high). Striking this balance requires solving certain auxiliary problems which have previously been considered intractable, but we show how these computational obstacles can be overcome.
6,914 citations
TL;DR: The present state of the art of constructing surrogate models and their use in optimization strategies is reviewed and extensive use of pictorial examples are made to give guidance as to each method's strengths and weaknesses.
1,919 citations
01 Jan 2006
TL;DR: This work reviews the state-of-the-art metamodel-based techniques from a practitioner's perspective according to the role of meetamodeling in supporting design optimization, including model approximation, design space exploration, problem formulation, and solving various types of optimization problems.
Abstract: Computation-intensive design problems are becoming increasingly common in manufacturing industries. The computation burden is often caused by expensive analysis and simulation processes in order to reach a comparable level of accuracy as physical testing data. To address such a challenge, approximation or metamodeling techniques are often used. Metamodeling techniques have been developed from many different disciplines including statistics, mathematics, computer science, and various engineering disciplines. These metamodels are initially developed as “surrogates” of the expensive simulation process in order to improve the overall computation efficiency. They are then found to be a valuable tool to support a wide scope of activities in modern engineering design, especially design optimization. This work reviews the state-of-the-art metamodel-based techniques from a practitioner’s perspective according to the role of metamodeling in supporting design optimization, including model approximation, design space exploration, problem formulation, and solving various types of optimization problems. Challenges and future development of metamodeling in support of engineering design is also analyzed and discussed.Copyright © 2006 by ASME
1,503 citations
Cites background from "A Trust Region Framework for Managi..."
...Many researchers advocated the use of a sequential metamodeling approach using move limits [107] or trust regions [108, 109]....
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TL;DR: Model reduction aims to reduce the computational burden by generating reduced models that are faster and cheaper to simulate, yet accurately represent the original large-scale system behavior as mentioned in this paper. But model reduction of linear, nonparametric dynamical systems has reached a considerable level of maturity, as reflected by several survey papers and books.
Abstract: Numerical simulation of large-scale dynamical systems plays a fundamental role in studying a wide range of complex physical phenomena; however, the inherent large-scale nature of the models often leads to unmanageable demands on computational resources. Model reduction aims to reduce this computational burden by generating reduced models that are faster and cheaper to simulate, yet accurately represent the original large-scale system behavior. Model reduction of linear, nonparametric dynamical systems has reached a considerable level of maturity, as reflected by several survey papers and books. However, parametric model reduction has emerged only more recently as an important and vibrant research area, with several recent advances making a survey paper timely. Thus, this paper aims to provide a resource that draws together recent contributions in different communities to survey the state of the art in parametric model reduction methods. Parametric model reduction targets the broad class of problems for wh...
1,230 citations
TL;DR: For the first time, a mathematical motivation is presented and SM is placed into the context of classical optimization to achieve a satisfactory solution with a minimal number of computationally expensive "fine" model evaluations.
Abstract: We review the space-mapping (SM) technique and the SM-based surrogate (modeling) concept and their applications in engineering design optimization. For the first time, we present a mathematical motivation and place SM into the context of classical optimization. The aim of SM is to achieve a satisfactory solution with a minimal number of computationally expensive "fine" model evaluations. SM procedures iteratively update and optimize surrogates based on a fast physically based "coarse" model. Proposed approaches to SM-based optimization include the original algorithm, the Broyden-based aggressive SM algorithm, various trust-region approaches, neural SM, and implicit SM. Parameter extraction is an essential SM subproblem. It is used to align the surrogate (enhanced coarse model) with the fine model. Different approaches to enhance uniqueness are suggested, including the recent gradient parameter-extraction approach. Novel physical illustrations are presented, including the cheese-cutting and wedge-cutting problems. Significant practical applications are reviewed.
1,044 citations
Cites methods from "A Trust Region Framework for Managi..."
...Close to the solution, however, only small steps are needed, in which case, the classical optimization strategy based on local Taylor models is better....
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References
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Book•
01 Feb 1996
TL;DR: In this paper, Schnabel proposed a modular system of algorithms for unconstrained minimization and nonlinear equations, based on Newton's method for solving one equation in one unknown convergence of sequences of real numbers.
Abstract: Preface 1. Introduction. Problems to be considered Characteristics of 'real-world' problems Finite-precision arithmetic and measurement of error Exercises 2. Nonlinear Problems in One Variable. What is not possible Newton's method for solving one equation in one unknown Convergence of sequences of real numbers Convergence of Newton's method Globally convergent methods for solving one equation in one uknown Methods when derivatives are unavailable Minimization of a function of one variable Exercises 3. Numerical Linear Algebra Background. Vector and matrix norms and orthogonality Solving systems of linear equations-matrix factorizations Errors in solving linear systems Updating matrix factorizations Eigenvalues and positive definiteness Linear least squares Exercises 4. Multivariable Calculus Background Derivatives and multivariable models Multivariable finite-difference derivatives Necessary and sufficient conditions for unconstrained minimization Exercises 5. Newton's Method for Nonlinear Equations and Unconstrained Minimization. Newton's method for systems of nonlinear equations Local convergence of Newton's method The Kantorovich and contractive mapping theorems Finite-difference derivative methods for systems of nonlinear equations Newton's method for unconstrained minimization Finite difference derivative methods for unconstrained minimization Exercises 6. Globally Convergent Modifications of Newton's Method. The quasi-Newton framework Descent directions Line searches The model-trust region approach Global methods for systems of nonlinear equations Exercises 7. Stopping, Scaling, and Testing. Scaling Stopping criteria Testing Exercises 8. Secant Methods for Systems of Nonlinear Equations. Broyden's method Local convergence analysis of Broyden's method Implementation of quasi-Newton algorithms using Broyden's update Other secant updates for nonlinear equations Exercises 9. Secant Methods for Unconstrained Minimization. The symmetric secant update of Powell Symmetric positive definite secant updates Local convergence of positive definite secant methods Implementation of quasi-Newton algorithms using the positive definite secant update Another convergence result for the positive definite secant method Other secant updates for unconstrained minimization Exercises 10. Nonlinear Least Squares. The nonlinear least-squares problem Gauss-Newton-type methods Full Newton-type methods Other considerations in solving nonlinear least-squares problems Exercises 11. Methods for Problems with Special Structure. The sparse finite-difference Newton method Sparse secant methods Deriving least-change secant updates Analyzing least-change secant methods Exercises Appendix A. A Modular System of Algorithms for Unconstrained Minimization and Nonlinear Equations (by Robert Schnabel) Appendix B. Test Problems (by Robert Schnabel) References Author Index Subject Index.
6,831 citations
Book•
01 Mar 1983
TL;DR: Newton's Method for Nonlinear Equations and Unconstrained Minimization and methods for solving nonlinear least-squares problems with Special Structure.
Abstract: Preface 1. Introduction. Problems to be considered Characteristics of 'real-world' problems Finite-precision arithmetic and measurement of error Exercises 2. Nonlinear Problems in One Variable. What is not possible Newton's method for solving one equation in one unknown Convergence of sequences of real numbers Convergence of Newton's method Globally convergent methods for solving one equation in one uknown Methods when derivatives are unavailable Minimization of a function of one variable Exercises 3. Numerical Linear Algebra Background. Vector and matrix norms and orthogonality Solving systems of linear equations-matrix factorizations Errors in solving linear systems Updating matrix factorizations Eigenvalues and positive definiteness Linear least squares Exercises 4. Multivariable Calculus Background Derivatives and multivariable models Multivariable finite-difference derivatives Necessary and sufficient conditions for unconstrained minimization Exercises 5. Newton's Method for Nonlinear Equations and Unconstrained Minimization. Newton's method for systems of nonlinear equations Local convergence of Newton's method The Kantorovich and contractive mapping theorems Finite-difference derivative methods for systems of nonlinear equations Newton's method for unconstrained minimization Finite difference derivative methods for unconstrained minimization Exercises 6. Globally Convergent Modifications of Newton's Method. The quasi-Newton framework Descent directions Line searches The model-trust region approach Global methods for systems of nonlinear equations Exercises 7. Stopping, Scaling, and Testing. Scaling Stopping criteria Testing Exercises 8. Secant Methods for Systems of Nonlinear Equations. Broyden's method Local convergence analysis of Broyden's method Implementation of quasi-Newton algorithms using Broyden's update Other secant updates for nonlinear equations Exercises 9. Secant Methods for Unconstrained Minimization. The symmetric secant update of Powell Symmetric positive definite secant updates Local convergence of positive definite secant methods Implementation of quasi-Newton algorithms using the positive definite secant update Another convergence result for the positive definite secant method Other secant updates for unconstrained minimization Exercises 10. Nonlinear Least Squares. The nonlinear least-squares problem Gauss-Newton-type methods Full Newton-type methods Other considerations in solving nonlinear least-squares problems Exercises 11. Methods for Problems with Special Structure. The sparse finite-difference Newton method Sparse secant methods Deriving least-change secant updates Analyzing least-change secant methods Exercises Appendix A. A Modular System of Algorithms for Unconstrained Minimization and Nonlinear Equations (by Robert Schnabel) Appendix B. Test Problems (by Robert Schnabel) References Author Index Subject Index.
6,217 citations
Book•
01 Jan 1999
1,059 citations
"A Trust Region Framework for Managi..." refers background in this paper
...The trust radius hk is similar in purpose to a move limit (Vanderplaats 1984)....
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TL;DR: It is shown that, although the lack of comparative data established on reference test cases prevents an accurate assessment, there have been significant improvements in approximation concepts since the introduction of approximation concepts in the mid-seventies.
Abstract: This paper reviews the basic approximation concepts used in structural optimization. It also discusses some of the most recent developments in that area since the introduction of approximation concepts in the mid-seventies. The paper distinguishes between local, medium-range and global approximations; it covers function approximations and problem approximations. It shows that, although the lack of comparative data established on reference test cases prevents an accurate assessment, there have been significant improvements. The largest number of developments have been in the areas of local function approximations and use of intermediate variable and response quantities. It appears also that some new methodologies emerge which could greatly benefit from the introduction of new computer architectures.
581 citations
TL;DR: In this paper, an efficient automated minimum weight design procedure is presented which is applicable to sizing structural systems that can be idealized by truss, shear panel, and constant strain triangles.
Abstract: An efficient automated minimum weight design procedure is presented which is applicable to sizing structural systems that can be idealized by truss, shear panel, and constant strain triangles. Static stress and displacement constraints under alternative loading conditions are considered. The optimization algorithm is an adaptation of the method of inscribed hyperspheres and high efficiency is achieved by using several approximation concepts including temporary deletion of noncritical constraints, design variable linking, and Taylor series expansions for response variables in terms of design variables. Optimum designs for several planar and space truss examples problems are presented. The results reported support the contention that the innovative use of approximation concepts in structural synthesis can produce significant improvements in efficiency.
570 citations