swMATH ID: 4829
Software Authors: Kelley, C.T.
Description: Iterative methods for optimization This book gives an introduction to optimization methods for unconstrained and bound constrained minimization problems. The style of the book is probably best described by the following quote from the book’s preface: ‘dots{} we treat a small number of methods in depth, giving less detailed description of only a few [dots{}]. We aim for clarity and brevity rather than complete generality and confine our scope to algorithms that are easy to implement (by the reader!) and understand.’ par This book is partitioned into two parts. The first part, occupying approximately 100 pages, is devoted to the optimization of smooth functions. The methods studied in this first part rely on the availability and accuracy of first order, and sometimes also second order derivatives of the objective function. The first part contains five chapters. The first chapter provides basic concepts. It also introduces a parameter identification problem and a discretized optimal control problem, both of which are used to demonstrate all methods discussed in the first part. Chapter 2 studies the local convergence of Newton’s method, inexact Newton methods, and the Gauss-Newton method for the solution of nonlinear least squares problems. Both, overdetermined and underdetermined nonlinear least squares problems are considered. Chapter 3 is devoted to line-search and trust-region methods, which are used to globalize convergence, i.e., remove the restriction that the starting point of the optimization iteration is sufficiently close to a solution. par The BFGS method is studied in chapter 4. A local convergence analysis is provided and implementation details are discussed. Other quasi-Newton methods are sketched. The last chapter of the first part, chapter 5, studies projection methods for the solution of bound constrained problems. All chapters conclude with a demonstration of the methods discussed in the respective chapter using the parameter identification problem and the discretized optimal control problem introduced in chapter 1, and with a set of exercises. par The second part of the book, which is approximately 50 pages long, deals with the optimization of noisy functions. Such optimization problems arise, e.g., when the evaluation of the objective function involves computer simulations. In such cases the noise often introduces artificial minimizers. Gradient information, even if available, cannot expected to be reliable. This second part contains three chapters. The first chapter provides a discussion of noisy functions, basics concepts, and three simple examples that are later used to demonstrate the behavior of optimization algorithms. Chapter 7 introduces implicit filtering, a technique due to the author and his group. Implicit filtering methods use finite difference approximations of the gradient, which are adjusted to the noise level in the function. Direct search algorithms, including the Nelder-Mead, multidirectional search, and the Hooke-Jeves algorithms are discussed in Chapter 8. Again, the latter two chapters conclude with a numerical demonstration of the methods discussed in the respective chapter, and with a set of exercises. par The treatment of both, optimization methods for smooth and for noisy functions is a unique feature of this book. Matlab implementations of all algorithms discussed in this book are easily accessible from the author’s or the publisher’s web-page.
Homepage: http://www4.ncsu.edu/~ctk/optimization_codes.html
Dependencies: Matlab
Keywords: unconstrained optimization; bound constraints; Newton’s method; Gauss-Newton method; quasi-Newton methods; line-search; trust-region; nonlinear least squares; projection methods; noisy functions; direct search methods; implicit filtering
Related Software: minpack; Matlab; L-BFGS; IMFIL; NITSOL; DFO; CUTEr; NEWUOA; PETSc; TRON; UOBYQA; mctoolbox; NewtonLib; SifDec; fminsearch; LAPACK; SDBOX; tn; L-BFGS-B; DFBOX_IMPR
Cited in: 649 Publications
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Cited by 1,142 Authors

24 Kelley, Carl T.
10 Bellavia, Stefania
10 Kevrekidis, Ioannis George
9 Banks, Harvey Thomas
9 Herty, Michael Matthias
8 Martínez, José Mario
8 Morini, Benedetta
8 Volkwein, Stefan
7 Antil, Harbir
7 Bardsley, Johnathan M.
7 Chacón, Luis
7 Fukushima, Masao
7 Haber, Eldad
7 Hedar, Abdel-Rahman
7 Olufsen, Mette Sofie
7 Siettos, Constantinos I.
6 Nagy, James Gerard
6 Ottesen, Johnny Tom
6 Quyen, Tran Nhan Tam
6 Vicente, Luis Nunes
5 Bai, Zhongzhi
5 Coutinho, Alvaro L. G. A.
5 Heinkenschloss, Matthias
5 Hinze, Michael
5 Kalita, Jiten C.
5 Klar, Axel
5 Pinnau, René
5 Seaïd, Mohammed
5 Yotov, Ivan
4 Bergamaschi, Luca
4 Blanco, Pablo Javier
4 Cătinaş, Emil
4 Chen, Ke
4 Galligani, Emanuele
4 Gear, Charles William
4 Karátson, János
4 Kavanagh, Kathleen Rose
4 Lapenta, Giovanni
4 Macconi, Maria
4 Martínez, Ángeles
4 Martins, Marcos A. D.
4 Miller, Cass T.
4 Sabo, Kristian
4 Scitovski, Rudolf
4 Tran, Hien T.
4 Wheeler, Mary Fanett
3 Ascher, Uri M.
3 Banda, Mapundi Kondwani
3 Birgin, Ernesto G.
3 Bru, Rafael
3 Buscaglia, Gustavo Carlos
3 Diniz-Ehrhardt, Maria Aparecida
3 Duintjer Tebbens, Jurjen
3 Dussault, Jean-Pierre
3 Elman, Howard C.
3 Fan, Jinyan
3 Gablonsky, Joerg M.
3 Gomes-Ruggiero, Márcia Ap.
3 Gondzio, Jacek
3 Gugat, Martin
3 Hare, Warren L.
3 Hoppe, Ronald H. W.
3 Huang, Jingfang
3 Jarry-Bolduc, Gabriel
3 Kees, Christopher E.
3 Klibanov, Michael V.
3 Krejić, Nataša
3 Lee, Taeyoung
3 Leiva, Jorge S.
3 Li, Zhaoyan
3 McClamroch, N. Harris
3 Mehlsen, Jesper
3 Regis, Rommel G.
3 Reynolds, Daniel R.
3 Sachs, Ekkehard W.
3 Sciandrone, Marco
3 Tůma, Miroslav
3 Xiao, Yunhai
3 Xin, Jianguo
3 Yuan, Ya-xiang
3 Zhang, Daoping
2 Ahookhosh, Masoud
2 Al-Baali, Mehiddin
2 Al-Jeiroudi, Ghussoun
2 Alves, José L. D.
2 Amini, Keyvan
2 An, Dong
2 Atkinson, Kendall Eugene
2 Bachmann, Freya
2 Baldick, Ross
2 Banerjee, Malay
2 Blauth, Sebastian
2 Brady, Renee
2 Caldwell, Timothy M.
2 Caliari, Marco
2 Calvetti, Daniela
2 Cardona, Alberto
2 Chan, Swee-Huat
2 Chan, Tony Fan-Cheong
2 Chen, Xiaojun
...and 1,042 more Authors
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Cited in 209 Serials

28 Journal of Computational and Applied Mathematics
28 Computational Optimization and Applications
26 Journal of Computational Physics
22 Optimization Methods & Software
19 SIAM Journal on Scientific Computing
18 SIAM Journal on Optimization
17 Applied Mathematics and Computation
16 Computer Methods in Applied Mechanics and Engineering
15 Inverse Problems
13 Journal of Optimization Theory and Applications
12 Optimization and Engineering
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4 International Journal for Numerical Methods in Fluids
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4 M\(^3\)AS. Mathematical Models & Methods in Applied Sciences
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4 Mathematical Programming. Series A. Series B
4 Journal of Inverse and Ill-Posed Problems
4 SIAM Journal on Applied Dynamical Systems
4 Multiscale Modeling & Simulation
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3 IMA Journal of Numerical Analysis
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3 Bulletin of Mathematical Biology
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3 SIAM Journal on Control and Optimization
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3 Numerical Methods for Partial Differential Equations
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3 SIAM Journal on Matrix Analysis and Applications
3 International Journal of Computer Mathematics
3 SIAM Review
3 Communications in Numerical Methods in Engineering
3 CEJOR. Central European Journal of Operations Research
3 Acta Numerica
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2 International Journal for Numerical and Analytical Methods in Geomechanics
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2 ACM Transactions on Mathematical Software
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2 Journal of Parallel and Distributed Computing
2 IEEE Transactions on Circuits and Systems. I: Fundamental Theory and Applications
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2 Advances in Design and Control
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Cited in 46 Fields

376 Numerical analysis (65-XX)
189 Operations research, mathematical programming (90-XX)
116 Partial differential equations (35-XX)
89 Calculus of variations and optimal control; optimization (49-XX)
89 Fluid mechanics (76-XX)
60 Biology and other natural sciences (92-XX)
46 Computer science (68-XX)
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24 Optics, electromagnetic theory (78-XX)
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