A primal-dual interior point method for large-scale free material optimization

Free Material Optimization (FMO) is a branch of structural optimization in which the design variable is the elastic material tensor that is allowed to vary over the design domain. The requirements are that the material tensor is symmetric positive semidefinite with bounded trace. The resulting optimization problem is a nonlinear semidefinite program with many small matrix inequalities for which a special-purpose optimization method should be developed. The objective of this article is to propose an efficient primal-dual interior point method for FMO that can robustly and accurately solve large-scale problems. Several equivalent formulations of FMO problems are discussed and recommendations on the best choice based on the results from our numerical experiments are presented. Furthermore, the choice of search direction is also investigated numerically and a recommendation is given. The number of iterations the interior point method requires is modest and increases only marginally with problem size. The computed optimal solutions obtain a higher precision than other available special-purpose methods for FMO. The efficiency and robustness of the method is demonstrated by numerical experiments on a set of large-scale FMO problems.