BibTex RIS Cite

Derivative Free Multilevel Optimization

Year 2015, Volume: 5 Issue: 1, 46 - 60, 01.06.2015

Abstract

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

References

  • 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.
Year 2015, Volume: 5 Issue: 1, 46 - 60, 01.06.2015

Abstract

References

  • 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.
There are 18 citations in total.

Details

Primary Language English
Journal Section Research Article
Authors

B. Karasözen This is me

Publication Date June 1, 2015
Published in Issue Year 2015 Volume: 5 Issue: 1

Cite