American Institute of Mathematical Sciences

Advanced
2012, 2(2): 395-412. doi: 10.3934/naco.2012.2.395

A nonmonotone spectral projected gradient method for large-scale topology optimization problems

Rouhollah Tavakoli 1, and  Hongchao Zhang 2,

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

Received  October 2011 Revised  March 2012 Published  May 2012

An efficient gradient-based method to solve the volume constrained topology optimization problems is presented. Each iterate of this algorithm is obtained by the projection of a Barzilai-Borwein step onto the feasible set consisting of box and one linear constraints (volume constraint). To ensure the global convergence, an adaptive nonmonotone line search is performed along the direction that is given by the current and projection point. The adaptive cyclic reuse of the Barzilai-Borwein step is applied as the initial stepsize. The minimum memory requirement, the guaranteed convergence property, and almost only one function and gradient evaluations per iteration make this new method very attractive within common alternative methods to solve large-scale optimal design problems. Efficiency and feasibility of the presented method are supported by numerical experiments.
Keywords: volume constraint., distributed parameter identification, method of moving asymptotic (MMA), nonmonotone line search, large-scale topology optimization, Barzilai-Borwein step-size.
Mathematics Subject Classification: Primary: 90C06, 90C26; Secondary: 65Y2.
Citation: Rouhollah Tavakoli, Hongchao Zhang. A nonmonotone spectral projected gradient method for large-scale topology optimization problems. Numerical Algebra, Control & Optimization, 2012, 2 (2) : 395-412. doi: 10.3934/naco.2012.2.395
References:
[1]

G. Allaire, "Shape Optimization by the Homogenization Method,", Springer, (2002).   Google Scholar

[2]

J. Barzilai and J. M. Borwein, Two-point step size gradient methods,, IMA Journal of Numerical Analysis, 8 (1988), 141.  doi: 10.1093/imanum/8.1.141.  Google Scholar

[3]

M. P. Bendsøe and O. Sigmund, "Topology Optimization: Theory, Methods, and Applications,", Springer Verlag, (2003).   Google Scholar

[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.  doi: 10.1137/S1052623497330963.  Google Scholar

[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.  doi: 10.1145/502800.502803.  Google Scholar

[6]

R. P. Brent, An algorithm with guaranteed convergence for finding a zero of a function,, The Computer Journal, 14 (1971), 422.  doi: 10.1093/comjnl/14.4.422.  Google Scholar

[7]

L. Ceng, Q. H. Ansari and J. Yao, Extragradient-projection method for solving constrained convex minimization problems,, Numerical Algebra, 1 (2011), 341.   Google Scholar

[8]

A. R. Conn, N. I. M. Gould and Ph. L. Toint, "Trust-Region Methods,", SIAM, (2000).  doi: 10.1137/1.9780898719857.  Google Scholar

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Y. H. Dai, Alternate step gradient method,, Optimization, 52 (2003), 395.  doi: 10.1080/02331930310001611547.  Google Scholar

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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.  doi: 10.1093/imanum/drl006.  Google Scholar

[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.  doi: 10.1137/060650453.  Google Scholar

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R. Fletcher, On the Barzilai-Borwein method,, Optimization and Control with Applications, 96 (2005), 235.  doi: 10.1007/0-387-24255-4_10.  Google Scholar

[13]

C. Fleury, CONLIN: An efficient dual optimizer based on convex approximation concepts,, Structural and Multidisciplinary Optimization, 1 (1989), 81.   Google Scholar

[14]

G. E. Forsythe, On the asymptotic directions of the s-dimensional optimum gradient method,, Numerische Mathematik, 11 (1968), 57.  doi: 10.1007/BF02165472.  Google Scholar

[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.  doi: 10.1137/S1052623499350013.  Google Scholar

[16]

L. Grippo, F. Lampariello and S. Lucidi, A nonmonotone line search technique for Newton's method,, SIAM Journal on Numerical Analysis, (1986), 707.  doi: 10.1137/0723046.  Google Scholar

[17]

W. W. Hager and H. Zhang, A new active set algorithm for box constrained optimization,, SIAM Journal on Optimization, 17 (2006), 526.  doi: 10.1137/050635225.  Google Scholar

[18]

J. Nocedal and S. J. Wright, "Numerical Optimization,", Springer, (2006).   Google Scholar

[19]

W. H. Press, S. A. Teukolsky, W. T. Vetterling and B. P. Flannery, "Numerical Recipes 3rd edition: The Art of Scientific Computing,", 2007., ().   Google Scholar

[20]

M. Raydan, The Barzilai and Borwein gradient method for the large scale unconstrained minimization problem,, SIAM Journal on Optimization, 7 (1997), 26.  doi: 10.1137/S1052623494266365.  Google Scholar

[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.  doi: 10.1137/0108011.  Google Scholar

[22]

K. Svanberg, The method of moving asymptotes- A new method for structural optimization,, International Journal for Numerical Methods in Engineering, 24 (1987), 359.  doi: 10.1002/nme.1620240207.  Google Scholar

[23]

K. Svanberg, A class of globally convergent optimization methods based on conservative convex separable approximations,, SIAM J. Optim., 12 (2002), 555.  doi: 10.1137/S1052623499362822.  Google Scholar

[24]

Y. X. Yuan, Recent advances in numerical methods for nonlinear equations and nonlinear least squares,, Numerical Algebra Control and Optimization, 1 (2011), 15.  doi: 10.3934/naco.2011.1.15.  Google Scholar

[25]

C. Zillober, A globally convergent version of the method of moving asymptotes,, Structural and Multidisciplinary Optimization, 6 (1993), 166.   Google Scholar

[26]

C. Zillober, SCPIP: an efficient software tool for the solution of structural optimization problems,, Struct. Multidisc. Optim., 24 (2002), 362.  doi: 10.1007/s00158-002-0248-5.  Google Scholar

show all references

References:
[1]

G. Allaire, "Shape Optimization by the Homogenization Method,", Springer, (2002).   Google Scholar

[2]

J. Barzilai and J. M. Borwein, Two-point step size gradient methods,, IMA Journal of Numerical Analysis, 8 (1988), 141.  doi: 10.1093/imanum/8.1.141.  Google Scholar

[3]

M. P. Bendsøe and O. Sigmund, "Topology Optimization: Theory, Methods, and Applications,", Springer Verlag, (2003).   Google Scholar

[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.  doi: 10.1137/S1052623497330963.  Google Scholar

[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.  doi: 10.1145/502800.502803.  Google Scholar

[6]

R. P. Brent, An algorithm with guaranteed convergence for finding a zero of a function,, The Computer Journal, 14 (1971), 422.  doi: 10.1093/comjnl/14.4.422.  Google Scholar

[7]

L. Ceng, Q. H. Ansari and J. Yao, Extragradient-projection method for solving constrained convex minimization problems,, Numerical Algebra, 1 (2011), 341.   Google Scholar

[8]

A. R. Conn, N. I. M. Gould and Ph. L. Toint, "Trust-Region Methods,", SIAM, (2000).  doi: 10.1137/1.9780898719857.  Google Scholar

[9]

Y. H. Dai, Alternate step gradient method,, Optimization, 52 (2003), 395.  doi: 10.1080/02331930310001611547.  Google Scholar

[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.  doi: 10.1093/imanum/drl006.  Google Scholar

[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.  doi: 10.1137/060650453.  Google Scholar

[12]

R. Fletcher, On the Barzilai-Borwein method,, Optimization and Control with Applications, 96 (2005), 235.  doi: 10.1007/0-387-24255-4_10.  Google Scholar

[13]

C. Fleury, CONLIN: An efficient dual optimizer based on convex approximation concepts,, Structural and Multidisciplinary Optimization, 1 (1989), 81.   Google Scholar

[14]

G. E. Forsythe, On the asymptotic directions of the s-dimensional optimum gradient method,, Numerische Mathematik, 11 (1968), 57.  doi: 10.1007/BF02165472.  Google Scholar

[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.  doi: 10.1137/S1052623499350013.  Google Scholar

[16]

L. Grippo, F. Lampariello and S. Lucidi, A nonmonotone line search technique for Newton's method,, SIAM Journal on Numerical Analysis, (1986), 707.  doi: 10.1137/0723046.  Google Scholar

[17]

W. W. Hager and H. Zhang, A new active set algorithm for box constrained optimization,, SIAM Journal on Optimization, 17 (2006), 526.  doi: 10.1137/050635225.  Google Scholar

[18]

J. Nocedal and S. J. Wright, "Numerical Optimization,", Springer, (2006).   Google Scholar

[19]

W. H. Press, S. A. Teukolsky, W. T. Vetterling and B. P. Flannery, "Numerical Recipes 3rd edition: The Art of Scientific Computing,", 2007., ().   Google Scholar

[20]

M. Raydan, The Barzilai and Borwein gradient method for the large scale unconstrained minimization problem,, SIAM Journal on Optimization, 7 (1997), 26.  doi: 10.1137/S1052623494266365.  Google Scholar

[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.  doi: 10.1137/0108011.  Google Scholar

[22]

K. Svanberg, The method of moving asymptotes- A new method for structural optimization,, International Journal for Numerical Methods in Engineering, 24 (1987), 359.  doi: 10.1002/nme.1620240207.  Google Scholar

[23]

K. Svanberg, A class of globally convergent optimization methods based on conservative convex separable approximations,, SIAM J. Optim., 12 (2002), 555.  doi: 10.1137/S1052623499362822.  Google Scholar

[24]

Y. X. Yuan, Recent advances in numerical methods for nonlinear equations and nonlinear least squares,, Numerical Algebra Control and Optimization, 1 (2011), 15.  doi: 10.3934/naco.2011.1.15.  Google Scholar

[25]

C. Zillober, A globally convergent version of the method of moving asymptotes,, Structural and Multidisciplinary Optimization, 6 (1993), 166.   Google Scholar

[26]

C. Zillober, SCPIP: an efficient software tool for the solution of structural optimization problems,, Struct. Multidisc. Optim., 24 (2002), 362.  doi: 10.1007/s00158-002-0248-5.  Google Scholar

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