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Journal of Optimization
Volume 2015, Article ID 790451, 16 pages
http://dx.doi.org/10.1155/2015/790451
Research Article

Constraint Consensus Methods for Finding Strictly Feasible Points of Linear Matrix Inequalities

Department of Mathematics and Statistics, Northern Arizona University, Flagstaff, AZ 86011-5717, USA

Received 25 July 2014; Revised 31 October 2014; Accepted 6 November 2014

Academic Editor: Manlio Gaudioso

Copyright © 2015 Shafiu Jibrin and James W. Swift. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We give algorithms for solving the strict feasibility problem for linear matrix inequalities. These algorithms are based on John Chinneck’s constraint consensus methods, in particular, the method of his original paper and the modified DBmax constraint consensus method from his paper with Ibrahim. Our algorithms start with one of these methods as “Phase 1.” Constraint consensus methods work for any differentiable constraints, but we take advantage of the structure of linear matrix inequalities. In particular, for linear matrix inequalities, the crossing points of each constraint boundary with the consensus ray can be calculated. In this way we check for strictly feasible points in “Phase 2” of our algorithms. We present four different algorithms, depending on whether the original (basic) or DBmax constraint consensus vector is used in Phase 1 and, independently, in Phase 2. We present results of numerical experiments that compare the four algorithms. The evidence suggests that one of our algorithms is the best, although none of them are guaranteed to find a strictly feasible point after a given number of iterations. We also give results of numerical experiments indicating that our best method compares favorably to a new variant of the method of alternating projections.