# American Institute of Mathematical Sciences

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

## 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

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.
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). [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. [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. 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. 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. 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, 1 (2011), 341. [8] A. R. Conn, N. I. M. Gould and Ph. L. Toint, "Trust-Region Methods,", SIAM, (2000). doi: 10.1137/1.9780898719857. [9] Y. H. Dai, Alternate step gradient method,, Optimization, 52 (2003), 395. 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. 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. doi: 10.1137/060650453. [12] R. Fletcher, On the Barzilai-Borwein method,, Optimization and Control with Applications, 96 (2005), 235. 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. [14] G. E. Forsythe, On the asymptotic directions of the s-dimensional optimum gradient method,, Numerische Mathematik, 11 (1968), 57. 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. 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. 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. 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. 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. 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. 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. 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. 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. [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.

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##### 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. 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. 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. 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. 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, 1 (2011), 341. [8] A. R. Conn, N. I. M. Gould and Ph. L. Toint, "Trust-Region Methods,", SIAM, (2000). doi: 10.1137/1.9780898719857. [9] Y. H. Dai, Alternate step gradient method,, Optimization, 52 (2003), 395. 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. 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. doi: 10.1137/060650453. [12] R. Fletcher, On the Barzilai-Borwein method,, Optimization and Control with Applications, 96 (2005), 235. 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. [14] G. E. Forsythe, On the asymptotic directions of the s-dimensional optimum gradient method,, Numerische Mathematik, 11 (1968), 57. 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. 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. 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. 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. 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. 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. 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. 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. 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. [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.
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