Derivative-free nonlinear optimization filter simplex

Aldina Correia ; João Matias ; Pedro Mestre ; Carlos Serodio

International Journal of Applied Mathematics and Computer Science, Tome 20 (2010), p. 679-688 / Harvested from The Polish Digital Mathematics Library

Résumé

The filter method is a technique for solving nonlinear programming problems. The filter algorithm has two phases in each iteration. The first one reduces a measure of infeasibility, while in the second the objective function value is reduced. In real optimization problems, usually the objective function is not differentiable or its derivatives are unknown. In these cases it becomes essential to use optimization methods where the calculation of the derivatives or the verification of their existence is not necessary: direct search methods or derivative-free methods are examples of such techniques. In this work we present a new direct search method, based on simplex methods, for general constrained optimization that combines the features of simplex and filter methods. This method neither computes nor approximates derivatives, penalty constants or Lagrange multipliers.

Détail

Publié le : 2010-01-01

Zbl 1211.90226

EUDML-ID : urn:eudml:doc:208017

Bibliographie

[000] Audet, C. (2004). Convergence results for pattern search algorithms are tight, Optimization and Engineering 2(5): 101-122. | Zbl 1085.90053

[001] Audet, C., Bchard, V. and Le Digabel, S. (2008). Nonsmooth optimization through mesh adaptive direct search and variable neighborhood search, Journal of Global Optimization 41(2): 299-318. | Zbl 1157.90535

[002] Audet, C. and Dennis Jr., J.E. (2007). A mads algorithm with a progressive barrier for derivative-free nonlinear programming, Technical Report G-2007-37, GERAD Reports, Polytechnic School of Montreal, Montreal.

[003] Audet, C. and Dennis Jr., J.E. (2006). Mesh adaptive direct search algorithms for constrained optimization, SIAM Journal on Optimization (17): 188-217. | Zbl 1112.90078

[004] Audet, C. and Dennis Jr., J. (2004). A pattern search filter method for nonlinear programming without derivatives, SIAM Journal on Optimization 5(14): 980-1010. | Zbl 1073.90066

[005] Audet, C. and Dennis Jr., J.E. (2002). Analysis of generalized pattern searches, SIAM Journal on Optimization 13(3): 889-903. | Zbl 1053.90118

[006] Audet, C., Dennis Jr., J.E. and Le Digabel, S. (2008). Globalization strategies for mesh adaptative direct search, Technical Report G-2008-74, GERAD Reports, Polytechnic School of Montreal, Montreal.

[007] Bertsekas, D.P. (1999). Nonlinear Programming, Athena Scientific, Belmont, MA. | Zbl 1015.90077

[008] Bongartz, I., Conn, A., Gould, N. and Toint, P. (1995). Cute: Constrained and unconstrained testing environment, ACM Transactions and Mathematical Software 21(1): 123-160. | Zbl 0886.65058

[009] Byrd, R.H., Nocedal, J. and Waltz, R.A. (2008). Steering exact penalty methods for nonlinear programming, Optimization Methods & Software 23(2): 197-213. | Zbl 1211.90224

[010] Correia, A., Matias, J., Mestre, P. and Serôdio, C. (2009). Derivative-free optimization and filter methods to solve nonlinear constrained problems, International Journal of Computer Mathematics 86(10): 1841-1851. | Zbl 1177.65085

[011] Fletcher, R. and Leyffer, S. (2002). Nonlinear programming without a penalty function, Mathematical Programming. Series A 91(2): 239-269. | Zbl 1049.90088

[012] Fletcher, R., Leyffer, S. and Toint, P. L. (2006). A brief history of filter method, Technical Report ANL/MCS-P1372-0906, Argonne National Laboratory, Mathematics and Computer Science Division, Argonne, IL.

[013] Karas, E.W., Ribeiro, A.A., Sagastizábal, C. and Solodov, M. (2006). A bundle-filter method for nonsmooth convex constrained optimization, Mathematical Programming 1(116): 297-320. | Zbl 1165.90024

[014] Nelder, J.A. and Mead, R. (1965). A simplex method for function minimization, The Computer Journal 7(4): 308-313. | Zbl 0229.65053