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Finding maximum rank moment matrices by facial reduction on primal form and Douglas-Rachford iteration

Content Provider	ACM Digital Library
Author	Wolkowicz, Henry Reid, Greg Wang, Fei
Abstract	Recent breakthroughs have been made in the use of semi-definite programming and its application to real polynomial solving. For example, the real radical of a zero dimensional ideal, can be determined by such approaches. Some progress has been made on the determination of the real radical in positive dimension by Ma, Wang and Zhi[5, 4]. Such work involves the determination of maximal rank semidefinite matrices. Existing methods are computationally expensive and have poorer accuracy on larger examples. In previous work we showed that regularity in the form of the Slater constraint qualification (strict feasibility) fails for the moment matrix in the SDP feasibility problem[6]. We used facial reduction to obtain a smaller regularized problem for which strict feasibility holds. However we did not have a theoretical guarantee that our methods, based on facial reduction and Douglas-Rachford iteration ensured the satisfaction of the maximum rank condition. Our work is motivated by the problems above. We discuss how to compute the moment matrix and its kernel using facial reduction techniques where the maximum rank property can be guaranteed by solving the dual problem. The facial reduction algorithms on the primal form is presented. We give examples that exhibit for the first time additional facial reductions beyond the first which are effective in practice with much better accuracy than SeDuMi(CVX).
Starting Page	35
Ending Page	37
Page Count	3
File Format	PDF
ISSN	19322240
DOI	10.1145/3096730.3096740
Journal	ACM Communications in Computer Algebra (ACCA)
Volume Number	51
Issue Number	1
Language	English
Publisher	Association for Computing Machinery (ACM)
Publisher Date	2015-02-05
Publisher Place	New York
Access Restriction	One Nation One Subscription (ONOS)
Content Type	Text
Resource Type	Article
Subject	Computational Theory and Mathematics Computational Mathematics

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