An Analytic Center Cutting Plane Method for Semidefinite Feasibility Problems

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 in the cone of symmetric positive semidefinite matrices. Assume that is defined by an oracle, which for any given m...

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Vydané v:Mathematics of operations research Ročník 27; číslo 2; s. 332 - 346
Hlavní autori: Sun, Jie, Toh, Kim-Chuan, Zhao, Gongyun
Médium: Journal Article
Jazyk:English
Vydavateľské údaje: Linthicum INFORMS 01.05.2002
Institute for Operations Research and the Management Sciences
Predmet:
Algorithms
Analysis
Analytic center
Analytics
cutting plane methods
Feasibility studies
Geometric planes
Harmonic functions
Management science
Mathematical theorems
Mathematical vectors
Mathematics
Matrices
Newton approximation methods
Operations research
Oracles
semidefinite programming
United States
ISSN:0364-765X, 1526-5471
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Shrnutí: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 in the cone of symmetric positive semidefinite matrices. Assume that is defined by an oracle, which for any given m x m symmetric positive semidefinite matrix either confirms that or returns a cut, i.e., a symmetric matrix A such that is in the half-space { Y : A | Y A | }. 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 * ( m 3 / 2 ) on the total number of cuts to be used, where is the maximum radius of a ball contained by .
Bibliografia:ObjectType-Article-1
SourceType-Scholarly Journals-1
content type line 14
ISSN:0364-765X
1526-5471
DOI:10.1287/moor.27.2.332.327