A Subgradient Method for Free Material Design

0507124 - ÚTIA 2020 RIV US eng J - Journal Article
Kočvara, Michal - Xia, Y. - Nesterov, Y.
A Subgradient Method for Free Material Design.
SIAM Journal on Optimization. Roč. 26, č. 4 (2016), s. 2314-2354. ISSN 1052-6234. E-ISSN 1095-7189
Institutional support: RVO:67985556
Keywords : fast gradient method * Nesterov’s primal-dual subgradient method * free material optimization
OECD category: Pure mathematics
Impact factor: 1.968, year: 2016
Method of publishing: Open access
http://library.utia.cas.cz/separaty/2019/MTR/kocvara-0507124.pdf https://epubs.siam.org/doi/10.1137/15M1019660

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, second order 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. We apply the primal-dual subgradient method 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^2) 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^3) arithmetic operations and an auxiliary vector storage of size 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, 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.

Permanent Link: http://hdl.handle.net/11104/0298529