# 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
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##### References:
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