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The Primal-Dual Second-Order Cone Approximations Algorithm for Symmetric Cone Programming

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Abstract

Given any open convex cone K, a logarithmically homogeneous, self-concordant barrier for K, and any positive real number r < 1, we associate, with each direction \(x \in K\), a second-order cone \(\Hat K_r(x)\) containing K. We show that K is the interior of the intersection of the second-order cones \(\Hat K_r(x)\), as x ranges over all directions in K. Using these second-order cones as approximations to cones of symmetric, positive definite matrices, we develop a new polynomial-time primal-dual interior-point algorithm for semidefinite programming. The algorithm is extended to symmetric cone programming via the relation between symmetric cones and Euclidean Jordan algebras.

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  1. Division of Mathematical Sciences, Nanyang Technological University, 1 Nanyang Walk, Blk 5 Level 3, Singapore 637616, Singapore

    Chek Beng Chua

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  1. Chek Beng Chua
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Correspondence to Chek Beng Chua.

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Chua, C. The Primal-Dual Second-Order Cone Approximations Algorithm for Symmetric Cone Programming. Found Comput Math 7, 271–302 (2007). https://doi.org/10.1007/s10208-004-0149-7

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  • DOI: https://doi.org/10.1007/s10208-004-0149-7

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