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On the conditions for the finite termination of ADMM and its applications to SOS polynomials feasibility problems

Title
On the conditions for the finite termination of ADMM and its applications to SOS polynomials feasibility problems
Authors
Komeiji H.Kim S.Yamashita M.
Ewha Authors
김선영
SCOPUS Author ID
김선영scopus
Issue Date
2019
Journal Title
Computational Optimization and Applications
ISSN
0926-6003JCR Link
Citation
Computational Optimization and Applications vol. 74, no. 2, pp. 317 - 344
Keywords
Alternating direction method of multipliersConditions for finite terminationSemidefinite programsSums of squares of polynomialsSums of squares of univariate trigonometric polynomials
Publisher
Springer New York LLC
Indexed
SCI; SCIE; SCOPUS WOS scopus
Document Type
Article
Abstract
We study finite termination properties of the alternating direction method of multipliers (ADMM) method applied to semidefinite programs (SDPs) generated from sums of squares (SOS) feasibility problems. Expressing a polynomial as SOS of lower degree by formulating the problem as SDPs is a key problem in many fields, and ADMM is frequently used to efficiently solve the SDPs whose size grows very rapidly with the degree and number of variables of the polynomial. We present conditions for the ADMM method to converges to an optimal solution in finite iterations and prove its finite termination under the conditions. In addition, for the problem of representing a univariate trigonometric polynomial as an SOS, we also provide similar conditions for the finite termination of the ADMM at an optimal solution. Numerical results demonstrate the finite termination if the conditions are satisfied and the size of the strictly feasible region is not too small. The size is determined by solving an SDP whose optimal value indicates how much the variable matrix of the original SDP can be diagonally increased, without violating the constraints of the original SDP. The finite termination discussed in this paper is a distinctive property of ADMM, and cannot be observed when implementing the interior-point methods. © 2019, Springer Science+Business Media, LLC, part of Springer Nature.
DOI
10.1007/s10589-019-00118-5
Appears in Collections:
자연과학대학 > 수학전공 > Journal papers
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