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Speeding up a memetic algorithm for the max-bisection problem
A wedge trust region method with self-correcting geometry for derivative-free optimization
1. | School of Mathematical Sciences, Jiangsu Key Labratory for NSLSCS, Nanjing Normal University, Nanjing 210023, China, China |
2. | PPGEPS, Pontifical Catholic University of Parana (PUCPR), Curitiba, Parana, Brazil |
3. | Department of Mathematics, Universidade Federal do Parana (UFPR), Curitiba, Parana, Brazil |
References:
[1] |
P. G. Ciarlet and P. A. Raviart, General Lagrange and Hermite interpolation in Rn with applications to finite element methods,, Archive for Rational Mechanics and Analysis, 46 (1972), 178.
|
[2] |
A. R. Conn, N. I. M. Gould and Ph. L. Toint, Trust-Region Methods,, SIAM, (2000).
doi: 10.1137/1.9780898719857. |
[3] |
A. R. Conn, K. Scheinberg and Ph. L. Toint, A derivative free optimization algorithm in practice,, In the proceeding of the 7th AIAA/USAF/NASA/ISSMO Symposium on Multidisciplinary Analysis and Optimization, (1998). Google Scholar |
[4] |
A. R. Conn, K. Scheinberg and L. N. Vicente, Introduction to Derivative-Free Optimization,, MPS-SIAM Series on Optimization, (2008).
doi: 10.1137/1.9780898718768. |
[5] |
Y. H. Dai, W. W. Hager, K. Schittkowski and H. C. Zhang, The cyclic Barzilai-Borwein method for unconstrained optimization,, IMA J. Numerical Analysis, 26 (2006), 604.
doi: 10.1093/imanum/drl006. |
[6] |
E. D. Dolan and J. J. Moré, Benchmarking optimization software with performance profiles,, Mathematical Programming, 91 (2002), 201.
doi: 10.1007/s101070100263. |
[7] |
K. R. Fowler, J. P. Reese, C. E. Kees, J. E. Dennis, C. T. Kelley, C. T. Miller, C. Audet, A. J. Booker, G. Couture, R. W. Darwin, M. W. Farthing, D. E. Finkel, J. M. Gablonsky, G. Gray and T. G. Kolda, Comparison of derivative-free optimization methods for groundwater supply and hydraulic capture community problems,, Advances in Water Resources, 31 (2008), 743. Google Scholar |
[8] |
S. Gratton, Ph. L. Toint and A. Tröltzsch, An active-set trust-region method for derivative-free nonlinear bound-constrained optimization,, Optimization Methods and Software, 26 (2011), 873.
doi: 10.1080/10556788.2010.549231. |
[9] |
G. Gray, T. Kolda, K. Sale and M. Young, Optimizing an empirical scoring function for transmembrane protein structure determination,, INFORMS Journal on Computing, 16 (2004), 406.
doi: 10.1287/ijoc.1040.0102. |
[10] |
R. Hooke and T. A. Jeeves, Direct search solution of numerical and statistical problems,, Journal of the Association for Computing Machinery, 8 (1961), 212. Google Scholar |
[11] |
X. W. Liu and Y. Yuan, A robust algorithm for optimization with general equality and inequality constraints,, SIAM J. Scientific Computing, 22 (2000), 517.
doi: 10.1137/S1064827598334861. |
[12] |
M. Marazzi, Nonlinear Optimization with and without Derivatives,, PhD thesis, (2001). Google Scholar |
[13] |
M. Marazzi and J. Nocedal, Wedge trust region methods for derivative free optimization,, Mathematical Programming, 91 (2002), 289.
doi: 10.1007/s101070100264. |
[14] |
J. J. Moré, B. S. Garbow and K. E. Hillstrom, Testing unconstrained optimization software,, ACM Transactions on Mathematical Software, 4 (1981), 136.
doi: 10.1145/355934.355936. |
[15] |
J. J. Moré and D. C. Sorensen, Computing a trust region step,, SIAM Journal on Scientific and Statistical Computing, 4 (1983), 553.
doi: 10.1137/0904038. |
[16] |
J. A. Nelder and R. Mead, A simplex method for function minimization,, The Computer Journal, 7 (1965), 308. Google Scholar |
[17] |
J. Nocedal and S. J. Wright, Numerical Optimization,, Springer, (1999).
doi: 10.1007/b98874. |
[18] |
R. Oeuvray, Trust-Region Methods Based on Radial Basis Functions with Application to Biomedical Imaging,, PhD thesis, (2005). Google Scholar |
[19] |
M. J. D. Powell, A new algorithm for unconstrained optimization,, In Nonlinear Programming (eds. J. B. Rosen, (1970), 31.
|
[20] |
M. J. D. Powell, On the global convergence of trust region algorithms for unconstrained optimization,, Mathematical Programming, 29 (1984), 297.
doi: 10.1007/BF02591998. |
[21] |
M. J. D. Powell, A direct search optimization method that models the objective by quadratic interpolation,, In presentation at the 5th Stockholm Optimization Days, (1994). Google Scholar |
[22] |
M. J. D. Powell, A quadratic model trust region method for unconstained minimization without derivatives,, presentation at the International Conference on Nonlinear Programming and Variational Inequalities, (1998). Google Scholar |
[23] |
M. J. D. Powell, On the Lagrange functions of quadratic models that are defined by interpolation,, Optimization Methods and Software, 16 (2001), 289.
doi: 10.1080/10556780108805839. |
[24] |
M. J. D. Powell, UOBYQA: Unconstrained optimization by quadratic approximation,, Mathematical Programming, 92 (2002), 555.
doi: 10.1007/s101070100290. |
[25] |
M. J. D. Powell, Least Frobenius norm updating of quadratic models that satisfy interpolation conditions,, Mathematical Programming, 100 (2004), 183.
doi: 10.1007/s10107-003-0490-7. |
[26] |
M. J .D. Powell, The NEWUOA software for unconstrained optimization without derivatives,, In Large-Scale Nonlinear Optimization (eds. P. Pardalos, (2006), 255.
doi: 10.1007/0-387-30065-1_16. |
[27] |
M. J. D. Powell, On nonlinear optimization since 1959,, In The Birth of Numerical Analysis (eds. A. Bultheel and R. Cools), (2010), 141.
|
[28] |
M. J. D. Powell and Y. Yuan, A trust region algorithm for equality constrained optimization,, Mathematical Programming, 49 (1991), 189.
doi: 10.1007/BF01588787. |
[29] |
K. Scheinberg and Ph. L. Toint, Self-correcting geometry in model-based algorithms for derivative-free unconstrained optimization,, SIAM Journal on Optimization, 20 (2010), 3512.
doi: 10.1137/090748536. |
[30] |
W. Sun, Q. K. Du and J. R. Chen, Computational Methods,, Science Press, (2007). Google Scholar |
[31] |
W. Sun, J. Yuan and Y. Yuan, Conic trust region method for linearly constrained optimization,, Journal of Computational Mathematics, 21 (2003), 295.
|
[32] |
W. Sun and Y. Yuan, A conic trust-region method for nonlinearly constrained optimization,, Anals of Operations Research, 103 (2001), 175.
doi: 10.1023/A:1012955122229. |
[33] |
W. Sun and Y. Yuan, Optimization Theory and Methods: Nonlinear Programming,, Springer Optimization and Its Applications, (2006).
|
[34] |
S. M. Wild, R. G. Regis and C. A. Shoemaker, ORBIT: optimization by radial basis function interpolation in trust-regions,, SIAM Journal on Scientific Computing, 30 (2008), 3197.
doi: 10.1137/070691814. |
[35] |
D. Winfield, Function and Functional Optimization by Interpolation in Data Tables,, PhD thesis, (1969). Google Scholar |
[36] |
D. Winfield, Function minimization by interpolation in a data table,, J. Inst. Math. Appl., 12 (1973), 339.
|
[37] |
D. Xue and W. Sun, On convergence analysis of a derivative-free trust region algorithm for constrained optimization with separable structure,, Science China Mathematics, 57 (2014), 1287.
doi: 10.1007/s11425-013-4677-y. |
[38] |
Y. Yuan, On a subproblem of trust region algorithms for constrained optimization,, Mathematical Programming, 47 (1990), 53.
doi: 10.1007/BF01580852. |
[39] |
H. C. Zhang, A. R. Conn and K. Scheinberg, A derivative-free algorithm for least-squares minimization,, SIAM J. Optimization, 20 (2010), 3555.
doi: 10.1137/09075531X. |
[40] |
L. Zhao and W. Sun, Nonmonotone retrospective conic trust region method for unconstrained optimization,, Numerical Algebra, 3 (2013), 309.
doi: 10.3934/naco.2013.3.309. |
show all references
References:
[1] |
P. G. Ciarlet and P. A. Raviart, General Lagrange and Hermite interpolation in Rn with applications to finite element methods,, Archive for Rational Mechanics and Analysis, 46 (1972), 178.
|
[2] |
A. R. Conn, N. I. M. Gould and Ph. L. Toint, Trust-Region Methods,, SIAM, (2000).
doi: 10.1137/1.9780898719857. |
[3] |
A. R. Conn, K. Scheinberg and Ph. L. Toint, A derivative free optimization algorithm in practice,, In the proceeding of the 7th AIAA/USAF/NASA/ISSMO Symposium on Multidisciplinary Analysis and Optimization, (1998). Google Scholar |
[4] |
A. R. Conn, K. Scheinberg and L. N. Vicente, Introduction to Derivative-Free Optimization,, MPS-SIAM Series on Optimization, (2008).
doi: 10.1137/1.9780898718768. |
[5] |
Y. H. Dai, W. W. Hager, K. Schittkowski and H. C. Zhang, The cyclic Barzilai-Borwein method for unconstrained optimization,, IMA J. Numerical Analysis, 26 (2006), 604.
doi: 10.1093/imanum/drl006. |
[6] |
E. D. Dolan and J. J. Moré, Benchmarking optimization software with performance profiles,, Mathematical Programming, 91 (2002), 201.
doi: 10.1007/s101070100263. |
[7] |
K. R. Fowler, J. P. Reese, C. E. Kees, J. E. Dennis, C. T. Kelley, C. T. Miller, C. Audet, A. J. Booker, G. Couture, R. W. Darwin, M. W. Farthing, D. E. Finkel, J. M. Gablonsky, G. Gray and T. G. Kolda, Comparison of derivative-free optimization methods for groundwater supply and hydraulic capture community problems,, Advances in Water Resources, 31 (2008), 743. Google Scholar |
[8] |
S. Gratton, Ph. L. Toint and A. Tröltzsch, An active-set trust-region method for derivative-free nonlinear bound-constrained optimization,, Optimization Methods and Software, 26 (2011), 873.
doi: 10.1080/10556788.2010.549231. |
[9] |
G. Gray, T. Kolda, K. Sale and M. Young, Optimizing an empirical scoring function for transmembrane protein structure determination,, INFORMS Journal on Computing, 16 (2004), 406.
doi: 10.1287/ijoc.1040.0102. |
[10] |
R. Hooke and T. A. Jeeves, Direct search solution of numerical and statistical problems,, Journal of the Association for Computing Machinery, 8 (1961), 212. Google Scholar |
[11] |
X. W. Liu and Y. Yuan, A robust algorithm for optimization with general equality and inequality constraints,, SIAM J. Scientific Computing, 22 (2000), 517.
doi: 10.1137/S1064827598334861. |
[12] |
M. Marazzi, Nonlinear Optimization with and without Derivatives,, PhD thesis, (2001). Google Scholar |
[13] |
M. Marazzi and J. Nocedal, Wedge trust region methods for derivative free optimization,, Mathematical Programming, 91 (2002), 289.
doi: 10.1007/s101070100264. |
[14] |
J. J. Moré, B. S. Garbow and K. E. Hillstrom, Testing unconstrained optimization software,, ACM Transactions on Mathematical Software, 4 (1981), 136.
doi: 10.1145/355934.355936. |
[15] |
J. J. Moré and D. C. Sorensen, Computing a trust region step,, SIAM Journal on Scientific and Statistical Computing, 4 (1983), 553.
doi: 10.1137/0904038. |
[16] |
J. A. Nelder and R. Mead, A simplex method for function minimization,, The Computer Journal, 7 (1965), 308. Google Scholar |
[17] |
J. Nocedal and S. J. Wright, Numerical Optimization,, Springer, (1999).
doi: 10.1007/b98874. |
[18] |
R. Oeuvray, Trust-Region Methods Based on Radial Basis Functions with Application to Biomedical Imaging,, PhD thesis, (2005). Google Scholar |
[19] |
M. J. D. Powell, A new algorithm for unconstrained optimization,, In Nonlinear Programming (eds. J. B. Rosen, (1970), 31.
|
[20] |
M. J. D. Powell, On the global convergence of trust region algorithms for unconstrained optimization,, Mathematical Programming, 29 (1984), 297.
doi: 10.1007/BF02591998. |
[21] |
M. J. D. Powell, A direct search optimization method that models the objective by quadratic interpolation,, In presentation at the 5th Stockholm Optimization Days, (1994). Google Scholar |
[22] |
M. J. D. Powell, A quadratic model trust region method for unconstained minimization without derivatives,, presentation at the International Conference on Nonlinear Programming and Variational Inequalities, (1998). Google Scholar |
[23] |
M. J. D. Powell, On the Lagrange functions of quadratic models that are defined by interpolation,, Optimization Methods and Software, 16 (2001), 289.
doi: 10.1080/10556780108805839. |
[24] |
M. J. D. Powell, UOBYQA: Unconstrained optimization by quadratic approximation,, Mathematical Programming, 92 (2002), 555.
doi: 10.1007/s101070100290. |
[25] |
M. J. D. Powell, Least Frobenius norm updating of quadratic models that satisfy interpolation conditions,, Mathematical Programming, 100 (2004), 183.
doi: 10.1007/s10107-003-0490-7. |
[26] |
M. J .D. Powell, The NEWUOA software for unconstrained optimization without derivatives,, In Large-Scale Nonlinear Optimization (eds. P. Pardalos, (2006), 255.
doi: 10.1007/0-387-30065-1_16. |
[27] |
M. J. D. Powell, On nonlinear optimization since 1959,, In The Birth of Numerical Analysis (eds. A. Bultheel and R. Cools), (2010), 141.
|
[28] |
M. J. D. Powell and Y. Yuan, A trust region algorithm for equality constrained optimization,, Mathematical Programming, 49 (1991), 189.
doi: 10.1007/BF01588787. |
[29] |
K. Scheinberg and Ph. L. Toint, Self-correcting geometry in model-based algorithms for derivative-free unconstrained optimization,, SIAM Journal on Optimization, 20 (2010), 3512.
doi: 10.1137/090748536. |
[30] |
W. Sun, Q. K. Du and J. R. Chen, Computational Methods,, Science Press, (2007). Google Scholar |
[31] |
W. Sun, J. Yuan and Y. Yuan, Conic trust region method for linearly constrained optimization,, Journal of Computational Mathematics, 21 (2003), 295.
|
[32] |
W. Sun and Y. Yuan, A conic trust-region method for nonlinearly constrained optimization,, Anals of Operations Research, 103 (2001), 175.
doi: 10.1023/A:1012955122229. |
[33] |
W. Sun and Y. Yuan, Optimization Theory and Methods: Nonlinear Programming,, Springer Optimization and Its Applications, (2006).
|
[34] |
S. M. Wild, R. G. Regis and C. A. Shoemaker, ORBIT: optimization by radial basis function interpolation in trust-regions,, SIAM Journal on Scientific Computing, 30 (2008), 3197.
doi: 10.1137/070691814. |
[35] |
D. Winfield, Function and Functional Optimization by Interpolation in Data Tables,, PhD thesis, (1969). Google Scholar |
[36] |
D. Winfield, Function minimization by interpolation in a data table,, J. Inst. Math. Appl., 12 (1973), 339.
|
[37] |
D. Xue and W. Sun, On convergence analysis of a derivative-free trust region algorithm for constrained optimization with separable structure,, Science China Mathematics, 57 (2014), 1287.
doi: 10.1007/s11425-013-4677-y. |
[38] |
Y. Yuan, On a subproblem of trust region algorithms for constrained optimization,, Mathematical Programming, 47 (1990), 53.
doi: 10.1007/BF01580852. |
[39] |
H. C. Zhang, A. R. Conn and K. Scheinberg, A derivative-free algorithm for least-squares minimization,, SIAM J. Optimization, 20 (2010), 3555.
doi: 10.1137/09075531X. |
[40] |
L. Zhao and W. Sun, Nonmonotone retrospective conic trust region method for unconstrained optimization,, Numerical Algebra, 3 (2013), 309.
doi: 10.3934/naco.2013.3.309. |
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