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### Статья

A small improvement in the structure of a material could potentially lower manufacturing costs. Thus, the free material design can be formulated as an optimization problem. However, due to its large scale, second-order methods cannot solve the free material design problem in a reasonable time. 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 $N \times N$. This is the first formulation of FMO without $A(E)^{-1}$. We apply the primal-dual subgradient method [Y. Nesterov, Math. Program., 120 (2009), pp. 221--259] 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 $\mathcal{O}(N^2)$ floating-point operations and an auxiliary vector storage of size $\mathcal{O}(N)$, compared with formulations having the inverse of $A(E)$ which requires $\mathcal{O}(N^3)$ arithmetic operations and an auxiliary vector storage of size $\mathcal{O}(N^2)$. 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. We also approximate a solution to the bounded Lagrangian dual problem. The problem is decomposed into small problems, each having only an unknown of $k\times k$ ($k=3$ or $6$) matrix, and can be solved in parallel. The iteration bound of our algorithm is optimal for a general subgradient scheme. Finally, we present promising numerical results.\