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Title: An analytic center cutting plane method for semidefinite feasibility problems
Authors: Sun, J. 
Toh, K.-C. 
Zhao, G. 
Keywords: Analytic center
Cutting plane methods
Semidefinite programming
Issue Date: 2002
Citation: Sun, J.,Toh, K.-C.,Zhao, G. (2002). An analytic center cutting plane method for semidefinite feasibility problems. Mathematics of Operations Research 27 (2) : 332-346. ScholarBank@NUS Repository.
Abstract: Semidefinite feasibility problems arise in many areas of operations research. The abstract form of these problems can be described as finding a point in a nonempty bounded convex body Γ in the cone of symmetric positive semidefinite matrices. Assume that Γ is defined by an oracle, which for any given m × m symmetric positive semidefinite matrix Ŷ either confirms that Ŷ ε Γ or returns a cut, i.e., a symmetric matrix A such that Γ is in the half-space {Y : A · Y ≤ A · Ŷ}. We study an analytic center cutting plane algorithm for this problem. At each iteration, the algorithm computes an approximate analytic center of a working set defined by the cutting plane system generated in the previous iterations. If this approximate analytic center is a solution, then the algorithm terminates; otherwise the new cutting plane returned by the oracle is added into the system. As the number of iterations increases, the working set shrinks and the algorithm eventually finds a solution to the problem. All iterates generated by the algorithm are positive definite matrices. The algorithm has a worst-case complexity of O* (m3/ε2) on the total number of cuts to be used, where ε is the maximum radius of a ball contained by Γ.
Source Title: Mathematics of Operations Research
ISSN: 0364765X
Appears in Collections:Staff Publications

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