Derivative Free Multilevel Optimization

Yıl 2015, Cilt: 5 Sayı: 1, 46 - 60, 01.06.2015

B. Karasözen

Öz

Optimization problems with different levels arise by discretization of ordinary and partial differential equations. We present a trust-region based derivative-freemultilevel optimization algorithm. The performance of the algorithm is shown on a shapeoptimization problem and global convergence to the Şrst order critical point is proved

Anahtar Kelimeler

Derivative-free optimization, multilevel optimization, shape optimization, trust-region methods

Kaynakça

• Berghen, F.V. and Bersini, H., (2005), CONDOR, a new parallel, constrained extension of Powell’s UOBYQA algorithm: Experimental results and comparison with the DFO algorithm, Journal of Computational and Applied Mathematics, 181, pp. 157-175.
• Borzi, A. and Schulz, V., (2009), Multigrid methods for PDE optimization, SIAM Review, 51, pp. 361-395.
• Conn, A.R. and Toint, Ph. L., (1996), An algorithm using quadratic interpolation for unconstrained derivative free optimization, in ”Nonlinear Optimization and Applications”, G. Di Pillo and F. Gi- anessi, eds, Plenum Publishing, New York, pp. 27-47.
• Conn, A.R., Scheinberg, K. and Toint, Ph. L, (1997), Recent progress in unconstrained nonlinear optimization without derivatives, Mathematical Programming, 79, pp. 397-414.
• Conn, A. R., Scheinberg, K. and Toint, Ph. L., (1997), On the convergence of derivative-free meth- ods for unconstrained optimization, In Approximation Theory and Optimization: Tribute to M.J.D. Powell, editors: A. Iserles and M. Buhmann, Cambridge University Press, Cambridge, pp. 83-108.
• Conn, A.R., Sheinberg, K. and Vicente, L.N., (2009), Introduction to Derivative-Free Optimization, SIAM Series on Optimization.
• Gratton, S., Sartenaer, A. and Toint, Ph. L, (2010), Numerical Experience with a recursive trust- region method for multilevel nonlinear optimization, Optimization Methods and Software, 25, pp. 359-386.
• Gratton, S., Sartenaer, A. and Toint, Ph. L, (2006), Second-order convergence properties of trustre- gion methods using incomplete curvature information, with an application to multigrid optimization, Journal of Computational and Applied Mathematics, 24, pp. 676-692.
• Gratton, S., Sartenaer, A. and Toint, Ph. L, (2008), Recursive Trust-Region Methods for Multiscale Nonlinear Optimization, SIAM Journal on Optimization, 19, pp. 414-444.
• Karas¨ozen, B., (2007), Survey of Trust-Region Derivative Free Optimization Methods, Journal of Industrial and Management Optimization, 3, pp. 321-334.
• Lewis, R.M., and Nash, S.G., (2005), Model problems for the multigrid optimization of systems governed by differential equations, SIAM J. Sci. Comput., 26, pp. 18111837.
• Mor´e, J.J. and Sorenson, D.C., (1979), On the use of directions of negative curvature in a modiŞed Newton Method. Mathematical Programming, 16, pp. 1-20.
• Nash, S.G., (2000), A multigrid approach to discretized optimization problems, Optim. Methods Softw., 14, pp. 99116.
• Powell, M.J.D., (2002), UOBYQA: Unconstrained Optimization By Quadratic Approximation, Math- ematical Programming, 92, pp. 555-583.
• Terlaky, T., (2004), Algorithms for Continuous Optimization DFO-trust region Interpolation Algo- rithm. Computer and Software, McMaster University, Hamilton, Canada.
• Toint, Ph. L., Conn, A.R. and Gould, N.I.M., (2009), Trust-Region Methods, SIAM.
• Toint, Ph. L., Tomanos, D. and Mendon¸ca, M. W., (2009), A multilevel algorithm for solving the trust-region subproblem. Optimization Methods and Software, 24, pp. 299-311.
• B¨ulent Karas¨ozen for the photography and short autobiography, see TWMS J. App. Eng. Math., V.1, N.2.