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A Subgradient Method for Free Material Design
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SYSNO ASEP 0507124 Document Type J - Journal Article R&D Document Type Journal Article Subsidiary J Článek ve WOS Title A Subgradient Method for Free Material Design Author(s) Kočvara, Michal (UTIA-B) RID, ORCID
Xia, Y. (CA)
Nesterov, Y. (BE)Number of authors 3 Source Title SIAM Journal on Optimization. - : SIAM Society for Industrial and Applied Mathematics - ISSN 1052-6234
Roč. 26, č. 4 (2016), s. 2314-2354Number of pages 41 s. Publication form Print - P Language eng - English Country US - United States Keywords fast gradient method ; Nesterov’s primal-dual subgradient method ; free material optimization Subject RIV BA - General Mathematics OECD category Pure mathematics Method of publishing Open access Institutional support UTIA-B - RVO:67985556 UT WOS 000391853600014 EID SCOPUS 85007240765 DOI 10.1137/15M1019660 Annotation 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.
Workplace Institute of Information Theory and Automation Contact Markéta Votavová, votavova@utia.cas.cz, Tel.: 266 052 201. Year of Publishing 2020 Electronic address https://epubs.siam.org/doi/10.1137/15M1019660
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