Advances in Optimization and Numerical Analysis
Volume 275 of the series Mathematics and Its Applications pp 5167
A Direct Search Optimization Method That Models the Objective and Constraint Functions by Linear Interpolation
 M. J. D. PowellAffiliated withDepartment of Applied Mathematics and Theoretical Physics, University of Cambridge
Abstract
An iterative algorithm is proposed for nonlinearly constrained optimization calculations when there are no derivatives. Each iteration forms linear approximations to the objective and constraint functions by interpolation at the vertices of a simplex and a trust region bound restricts each change to the variables. Thus a new vector of variables is calculated, which may replace one of the current vertices, either to improve the shape of the simplex or because it is the best vector that has been found so far, according to a merit function that gives attention to the greatest constraint violation. The trust region radius ρ is never increased, and it is reduced when the approximations of a wellconditioned simplex fail to yield an improvement to the variables, until ρ reaches a prescribed value that controls the final accuracy. Some convergence properties and several numerical results are given, but there are no more than 9 variables in these calculations because linear approximations can be highly inefficient. Nevertheless, the algorithm is easy to use for small numbers of variables.
Key words
Direct search Linear interpolation Nonlinear constraints Optimization without derivatives Title
 A Direct Search Optimization Method That Models the Objective and Constraint Functions by Linear Interpolation
 Book Title
 Advances in Optimization and Numerical Analysis
 Pages
 pp 5167
 Copyright
 1994
 DOI
 10.1007/9789401583305_4
 Print ISBN
 9789048143580
 Online ISBN
 9789401583305
 Series Title
 Mathematics and Its Applications
 Series Volume
 275
 Publisher
 Springer Netherlands
 Copyright Holder
 Springer Science+Business Media Dordrecht
 Additional Links
 Topics
 Keywords

 Direct search
 Linear interpolation
 Nonlinear constraints
 Optimization without derivatives
 Industry Sectors
 eBook Packages
 Editors

 Susana Gomez ^{(2)}
 JeanPierre Hennart ^{(2)}
 Editor Affiliations

 2. Instituto de Investigaciones en Mátematicas Aplicadas y Sistemas, Universidad Nacional Autónoma de México
 Authors

 M. J. D. Powell ^{(3)}
 Author Affiliations

 3. Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Silver Street, Cambridge, CB3 9EW, England
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