Abstract.
The authors of this paper recently introduced a transformation [4] that converts a class of semidefinite programs (SDPs) into nonlinear optimization problems free of matrix-valued constraints and variables. This transformation enables the application of nonlinear optimization techniques to the solution of certain SDPs that are too large for conventional interior-point methods to handle efficiently. Based on the transformation, we proposed a globally convergent, first-order (i.e., gradient-based) log-barrier algorithm for solving a class of linear SDPs. In this paper, we discuss an efficient implementation of the proposed algorithm and report computational results on semidefinite relaxations of three types of combinatorial optimization problems. Our results demonstrate that the proposed algorithm is indeed capable of solving large-scale SDPs and is particularly effective for problems with a large number of constraints.
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Received: June 22, 2001 / Accepted: January 20, 2002 Published online: December 9, 2002
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ID="†"Computational results reported in this paper were obtained on an SGI Origin2000 computer at Rice University acquired in part with support from NSF Grant DMS-9872009.
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ID="⋆"This author was supported in part by NSF Grants CCR-9902010, INT-9910084 and CCR-0203426
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ID="⋆⋆"This author was supported in part by NSF Grants CCR-9902010, INT-9910084 and CCR-0203113
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ID="⋆⋆⋆"This author was supported in part by DOE Grant DE-FG03-97ER25331, DOE/LANL Contract 03891-99-23 and NSF Grant DMS-9973339.
Key Words. semidefinite program – semidefinite relaxation – nonlinear programming – interior-point methods – limited memory quasi-Newton methods.
Mathematics Subject Classification (1991): 90C06, 90C27, 90C30.
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Burer, S., Monteiro, R. & Zhang, Y. A computational study of a gradient-based log-barrier algorithm for a class of large-scale SDPs. Math. Program., Ser. B 95, 359–379 (2003). https://doi.org/10.1007/s10107-002-0353-7
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DOI: https://doi.org/10.1007/s10107-002-0353-7