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Incorporating minimum Frobenius norm models in direct search. (English) Zbl 1190.90280
Summary: The goal of this paper is to show that the use of minimum Frobenius norm quadratic models can improve the performance of direct-search methods. The approach taken here is to maintain the structure of directional direct-search methods, organized around a search and a poll step, and to use the set of previously evaluated points generated during a direct-search run to build the models. The minimization of the models within a trust region provides an enhanced search step. Our numerical results show that such a procedure can lead to a significant improvement of direct search for smooth, piecewise smooth, and noisy problems.

90C56 Derivative-free methods and methods using generalized derivatives
90C30 Nonlinear programming
Full Text: DOI
[1] Conn, A.R., Scheinberg, K., Vicente, L.N.: Introduction to Derivative-Free Optimization. MPS-SIAM Series on Optimization. SIAM, Philadelphia (2009) · Zbl 1163.49001
[2] Custódio, A.L., Vicente, L.N.: Using sampling and simplex derivatives in pattern search methods. SIAM J. Optim. 18, 537–555 (2007) · Zbl 1144.65039 · doi:10.1137/050646706
[3] Custódio, A.L., Dennis, J.E. Jr., Vicente, L.N.: Using simplex gradients of nonsmooth functions in direct search methods. IMA J. Numer. Anal. 28, 770–784 (2008) · Zbl 1156.65059 · doi:10.1093/imanum/drn045
[4] DFO. http://www.coin-or.org/projects.html
[5] Fasano, G., Morales, J.L., Nocedal, J.: On the geometry phase in model-based algorithms for derivative-free optimization. Optim. Methods Softw. 24, 145–154 (2009) · Zbl 1154.90589 · doi:10.1080/10556780802409296
[6] Gray, G.A., Kolda, T.G.: Algorithm 856: APPSPACK 4.0: Asynchronous parallel pattern search for derivative-free optimization. ACM Trans. Math. Softw. 32, 485–507 (2006) · Zbl 1230.90196 · doi:10.1145/1163641.1163647
[7] Kolda, T.G., Lewis, R.M., Torczon, V.: Optimization by direct search: New perspectives on some classical and modern methods. SIAM Rev. 45, 385–482 (2003) · Zbl 1059.90146 · doi:10.1137/S003614450242889
[8] MATLAB, The MathWorks Inc. http://www.mathworks.com
[9] Moré, J.J., Wild, S.M.: Benchmarking derivative-free optimization algorithms. SIAM J. Optim. 20, 172–191 (2009). http://www.mcs.anl.gov/\(\sim\)more/dfo · Zbl 1187.90319 · doi:10.1137/080724083
[10] Moré, J.J., Sorensen, D.C., Hillstrom, K.E., Garbow, B.S.: The MINPACK project. In: Cowell, W.J. (ed.) Sources and Development of Mathematical Software, pp. 88–111. Prentice-Hall, New York (1984). http://www.netlib.org/minpack
[11] Powell, M.J.D.: Least Frobenius norm updating of quadratic models that satisfy interpolation conditions. Math. Program. 100, 183–215 (2004) · Zbl 1146.90526 · doi:10.1007/s10107-003-0490-7
[12] Powell, M.J.D.: Developments of NEWUOA for minimization without derivatives. IMA J. Numer. Anal. 28, 649–664 (2008) · Zbl 1154.65049 · doi:10.1093/imanum/drm047
[13] The Matrix Computation Toolbox. http://www.maths.manchester.ac.uk/\(\sim\)higham/mctoolbox
[14] Wild, S.M.: MNH: A derivative-free optimization algorithm using minimal norm Hessians. In: Tenth Copper Mountain Conference on Iterative Methods (April 2008)
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