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Interior Point Methods in Semidefinite Programming with Applications to Combinatorial Optimization (1993)
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| Venue: | SIAM Journal on Optimization |
| Citations: | 543 - 12 self |
BibTeX
@ARTICLE{Alizadeh93interiorpoint,
author = {Farid Alizadeh},
title = {Interior Point Methods in Semidefinite Programming with Applications to Combinatorial Optimization},
journal = {SIAM Journal on Optimization},
year = {1993},
volume = {5},
pages = {13--51}
}
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Abstract
We study the semidefinite programming problem (SDP), i.e the problem of optimization of a linear function of a symmetric matrix subject to linear equality constraints and the additional condition that the matrix be positive semidefinite. First we review the classical cone duality as specialized to SDP. Next we present an interior point algorithm which converges to the optimal solution in polynomial time. The approach is a direct extension of Ye's projective method for linear programming. We also argue that most known interior point methods for linear programs can be transformed in a mechanical way to algorithms for SDP with proofs of convergence and polynomial time complexity also carrying over in a similar fashion. Finally we study the significance of these results in a variety of combinatorial optimization problems including the general 0-1 integer programs, the maximum clique and maximum stable set problems in perfect graphs, the maximum k- partite subgraph problem in graphs, and va...
Keyphrases
interior point method combinatorial optimization linear program semidefinite programming problem maximum clique optimal solution linear programming positive semidefinite polynomial time mechanical way additional condition symmetric matrix subject combinatorial optimization problem polynomial time complexity direct extension projective method maximum stable set problem perfect graph linear function equality constraint interior point algorithm maximum partite subgraph problem classical cone duality general 0-1 integer program similar fashion
