A Subgradient Method for Free Material Design

A small improvement in the structure of the material could save the manufactory a lot of money. The free material design can be formulated as an optimization problem. However, due to its large scale, secondorder methods cannot solve the free material design problem in reasonable size. We formulate the free material optimization (FMO) problem into a saddle-point form in which the inverse of the stiffness matrix A(E) in the constraint is eliminated. The size of A(E) is generally large, denoted as N ×N . This is the first formulation of FMO without A(E)−1. We apply the primal-dual subgradient method [17] to solve the restricted saddle-point formula. This is the first gradient-type method for FMO. Each iteration of our algorithm takes a total of O(N) floating-point operations and an auxiliary vector storage of size O(N), compared with formulations having the inverse of A(E) which requires O(N) arithmetic operations and an auxiliary vector storage of size O(N). To solve the problem, we developed a closed-form solution to a semidefinite least squares problem and an efficient parameter update scheme for the gradient method, which are included in the appendix. We also approximate a solution to the bounded Lagrangian dual problem. The problem is decomposed into small problems each only having an unknown of k × k (k = 3 or 6) matrix, and can be solved in parallel. The iteration bound of our algorithm is optimal for general subgradient scheme. Finally we present promising numerical results. keywords fast gradient method, Nesterov’s primal-dual subgradient method, free material optimization, large-scale problems, first-order method, saddle-point, Lagrangian, complexity, duality, constrained least squares. AMS 90C90, 90C06, 90C25, 90C30, 90C47, 9008