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Product-form Cholesky factorization in interior point methods for second-order cone programming

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Abstract.

Second-order cone programming (SOCP) problems are typically solved by interior point methods. As in linear programming (LP), interior point methods can, in theory, solve SOCPs in polynomial time and can, in practice, exploit sparsity in the problem data. Specifically, when cones of large dimension are present, the density that results in the normal equations that are solved at each iteration can be remedied in a manner similar to the treatment of dense columns in an LP. Here we propose a product-form Cholesky factorization (PFCF) approach, and show that it is more numerically stable than the alternative Sherman-Morrison-Woodbury approach. We derive several PFCF variants and compare their theoretical perfomance. Finally, we prove that the elements of L in the Cholesky factorizations LDL T that arise in interior point methods for SOCP are uniformly bounded as the duality gap tends to zero as long as the iterates remain is some conic neighborhood of the cental path.

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Authors and Affiliations

  1. Dept. of IEOR, Columbia University, New York, NY, USA

    D. Goldfarb

  2. IBM T.J. Watson Research Center, Yorktown Heights, NY, USA

    K. Scheinberg

Authors
  1. D. Goldfarb
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  2. K. Scheinberg
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Correspondence to D. Goldfarb.

Additional information

Mathematics Subject Classification (1991): 90C25, 90C51, 15A23

Research supported in part by NSF Grants CDA 97-26385, DMS 01-04282, ONR Grant N000140310514 and DOE Grant GE-FG01-92ER-25126

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Goldfarb, D., Scheinberg, K. Product-form Cholesky factorization in interior point methods for second-order cone programming. Math. Program. 103, 153–179 (2005). https://doi.org/10.1007/s10107-004-0556-1

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