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A nonmonotone spectral projected gradient method for large-scale topology optimization problems
1. | Department of Material Science and Engineering, Sharif University of Technology, Tehran, P.O. Box 11365-9466, Iran |
2. | Department of Mathematics, Louisiana State University, Baton Rouge, LA, 70808, United States |
References:
[1] |
G. Allaire, "Shape Optimization by the Homogenization Method," Springer, 2002. |
[2] |
J. Barzilai and J. M. Borwein, Two-point step size gradient methods, IMA Journal of Numerical Analysis, 8 (1988), 141-148.
doi: 10.1093/imanum/8.1.141. |
[3] |
M. P. Bendsøe and O. Sigmund, "Topology Optimization: Theory, Methods, and Applications," Springer Verlag, 2003. |
[4] |
E. G. Birgin, J. M. Martínez and M. Raydan, Nonmonotone spectral projected gradient methods on convex sets, SIAM Journal on Optimization, 10 (2000), 1196-1211.
doi: 10.1137/S1052623497330963. |
[5] |
E. G. Birgin, J. M. Martínez, and M. Raydan, Algorithm 813: SPGsoftware for convex-constrained optimization, ACM Transactions on Mathematical Software, 27 (2001), 340-349.
doi: 10.1145/502800.502803. |
[6] |
R. P. Brent, An algorithm with guaranteed convergence for finding a zero of a function, The Computer Journal, 14 (1971), 422-425.
doi: 10.1093/comjnl/14.4.422. |
[7] |
L. Ceng, Q. H. Ansari and J. Yao, Extragradient-projection method for solving constrained convex minimization problems, Numerical Algebra, Control and Optimization, 1 (2011), 341-359. |
[8] |
A. R. Conn, N. I. M. Gould and Ph. L. Toint, "Trust-Region Methods," SIAM, Philadelphia, PA, USA, 2000.
doi: 10.1137/1.9780898719857. |
[9] |
Y. H. Dai, Alternate step gradient method, Optimization, 52 (2003), 395-415.
doi: 10.1080/02331930310001611547. |
[10] |
Y. H. Dai, W. W. Hager, K. Schittkowski and H. Zhang, The cyclic Barzilai-Borwein method for unconstrained optimization, IMA Journal of Numerical Analysis, 26 (2006), 604-627.
doi: 10.1093/imanum/drl006. |
[11] |
A. Donoso, Numerical simulations in 3D heat conduction: Minimizing the quadratic mean temperature gradient by an optimality criteria method, SIAM Journal on Scientific Computing, 28 (2006), 929-941.
doi: 10.1137/060650453. |
[12] |
R. Fletcher, On the Barzilai-Borwein method, Optimization and Control with Applications, 96 (2005), 235-256.
doi: 10.1007/0-387-24255-4_10. |
[13] |
C. Fleury, CONLIN: An efficient dual optimizer based on convex approximation concepts, Structural and Multidisciplinary Optimization, 1 (1989), 81-89. |
[14] |
G. E. Forsythe, On the asymptotic directions of the s-dimensional optimum gradient method, Numerische Mathematik, 11 (1968), 57-76.
doi: 10.1007/BF02165472. |
[15] |
P. E. Gill, W. Murray, and M. A. Saunders, SNOPT: An SQP algorithm for large-scale constrained optimization, SIAM J. Optim., 12 (2002), 979-1006.
doi: 10.1137/S1052623499350013. |
[16] |
L. Grippo, F. Lampariello and S. Lucidi, A nonmonotone line search technique for Newton's method, SIAM Journal on Numerical Analysis, (1986), 707-716.
doi: 10.1137/0723046. |
[17] |
W. W. Hager and H. Zhang, A new active set algorithm for box constrained optimization, SIAM Journal on Optimization, 17 (2006), 526-557.
doi: 10.1137/050635225. |
[18] |
J. Nocedal and S. J. Wright, "Numerical Optimization," Springer, 2006. |
[19] |
W. H. Press, S. A. Teukolsky, W. T. Vetterling and B. P. Flannery, "Numerical Recipes 3rd edition: The Art of Scientific Computing,", 2007., ().
|
[20] |
M. Raydan, The Barzilai and Borwein gradient method for the large scale unconstrained minimization problem, SIAM Journal on Optimization, 7 (1997), 26-33.
doi: 10.1137/S1052623494266365. |
[21] |
J. B. Rosen, The gradient projection method for nonlinear programming. Part I. Linear constraints, Journal of the Society for Industrial and Applied Mathematics, (1960), 181-217.
doi: 10.1137/0108011. |
[22] |
K. Svanberg, The method of moving asymptotes- A new method for structural optimization, International Journal for Numerical Methods in Engineering, 24 (1987), 359-373.
doi: 10.1002/nme.1620240207. |
[23] |
K. Svanberg, A class of globally convergent optimization methods based on conservative convex separable approximations, SIAM J. Optim., 12 (2002), 555-573.
doi: 10.1137/S1052623499362822. |
[24] |
Y. X. Yuan, Recent advances in numerical methods for nonlinear equations and nonlinear least squares, Numerical Algebra Control and Optimization, 1 (2011), 15-34.
doi: 10.3934/naco.2011.1.15. |
[25] |
C. Zillober, A globally convergent version of the method of moving asymptotes, Structural and Multidisciplinary Optimization, 6 (1993), 166-174. |
[26] |
C. Zillober, SCPIP: an efficient software tool for the solution of structural optimization problems, Struct. Multidisc. Optim., 24 (2002), 362-371.
doi: 10.1007/s00158-002-0248-5. |
show all references
References:
[1] |
G. Allaire, "Shape Optimization by the Homogenization Method," Springer, 2002. |
[2] |
J. Barzilai and J. M. Borwein, Two-point step size gradient methods, IMA Journal of Numerical Analysis, 8 (1988), 141-148.
doi: 10.1093/imanum/8.1.141. |
[3] |
M. P. Bendsøe and O. Sigmund, "Topology Optimization: Theory, Methods, and Applications," Springer Verlag, 2003. |
[4] |
E. G. Birgin, J. M. Martínez and M. Raydan, Nonmonotone spectral projected gradient methods on convex sets, SIAM Journal on Optimization, 10 (2000), 1196-1211.
doi: 10.1137/S1052623497330963. |
[5] |
E. G. Birgin, J. M. Martínez, and M. Raydan, Algorithm 813: SPGsoftware for convex-constrained optimization, ACM Transactions on Mathematical Software, 27 (2001), 340-349.
doi: 10.1145/502800.502803. |
[6] |
R. P. Brent, An algorithm with guaranteed convergence for finding a zero of a function, The Computer Journal, 14 (1971), 422-425.
doi: 10.1093/comjnl/14.4.422. |
[7] |
L. Ceng, Q. H. Ansari and J. Yao, Extragradient-projection method for solving constrained convex minimization problems, Numerical Algebra, Control and Optimization, 1 (2011), 341-359. |
[8] |
A. R. Conn, N. I. M. Gould and Ph. L. Toint, "Trust-Region Methods," SIAM, Philadelphia, PA, USA, 2000.
doi: 10.1137/1.9780898719857. |
[9] |
Y. H. Dai, Alternate step gradient method, Optimization, 52 (2003), 395-415.
doi: 10.1080/02331930310001611547. |
[10] |
Y. H. Dai, W. W. Hager, K. Schittkowski and H. Zhang, The cyclic Barzilai-Borwein method for unconstrained optimization, IMA Journal of Numerical Analysis, 26 (2006), 604-627.
doi: 10.1093/imanum/drl006. |
[11] |
A. Donoso, Numerical simulations in 3D heat conduction: Minimizing the quadratic mean temperature gradient by an optimality criteria method, SIAM Journal on Scientific Computing, 28 (2006), 929-941.
doi: 10.1137/060650453. |
[12] |
R. Fletcher, On the Barzilai-Borwein method, Optimization and Control with Applications, 96 (2005), 235-256.
doi: 10.1007/0-387-24255-4_10. |
[13] |
C. Fleury, CONLIN: An efficient dual optimizer based on convex approximation concepts, Structural and Multidisciplinary Optimization, 1 (1989), 81-89. |
[14] |
G. E. Forsythe, On the asymptotic directions of the s-dimensional optimum gradient method, Numerische Mathematik, 11 (1968), 57-76.
doi: 10.1007/BF02165472. |
[15] |
P. E. Gill, W. Murray, and M. A. Saunders, SNOPT: An SQP algorithm for large-scale constrained optimization, SIAM J. Optim., 12 (2002), 979-1006.
doi: 10.1137/S1052623499350013. |
[16] |
L. Grippo, F. Lampariello and S. Lucidi, A nonmonotone line search technique for Newton's method, SIAM Journal on Numerical Analysis, (1986), 707-716.
doi: 10.1137/0723046. |
[17] |
W. W. Hager and H. Zhang, A new active set algorithm for box constrained optimization, SIAM Journal on Optimization, 17 (2006), 526-557.
doi: 10.1137/050635225. |
[18] |
J. Nocedal and S. J. Wright, "Numerical Optimization," Springer, 2006. |
[19] |
W. H. Press, S. A. Teukolsky, W. T. Vetterling and B. P. Flannery, "Numerical Recipes 3rd edition: The Art of Scientific Computing,", 2007., ().
|
[20] |
M. Raydan, The Barzilai and Borwein gradient method for the large scale unconstrained minimization problem, SIAM Journal on Optimization, 7 (1997), 26-33.
doi: 10.1137/S1052623494266365. |
[21] |
J. B. Rosen, The gradient projection method for nonlinear programming. Part I. Linear constraints, Journal of the Society for Industrial and Applied Mathematics, (1960), 181-217.
doi: 10.1137/0108011. |
[22] |
K. Svanberg, The method of moving asymptotes- A new method for structural optimization, International Journal for Numerical Methods in Engineering, 24 (1987), 359-373.
doi: 10.1002/nme.1620240207. |
[23] |
K. Svanberg, A class of globally convergent optimization methods based on conservative convex separable approximations, SIAM J. Optim., 12 (2002), 555-573.
doi: 10.1137/S1052623499362822. |
[24] |
Y. X. Yuan, Recent advances in numerical methods for nonlinear equations and nonlinear least squares, Numerical Algebra Control and Optimization, 1 (2011), 15-34.
doi: 10.3934/naco.2011.1.15. |
[25] |
C. Zillober, A globally convergent version of the method of moving asymptotes, Structural and Multidisciplinary Optimization, 6 (1993), 166-174. |
[26] |
C. Zillober, SCPIP: an efficient software tool for the solution of structural optimization problems, Struct. Multidisc. Optim., 24 (2002), 362-371.
doi: 10.1007/s00158-002-0248-5. |
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