Abstract.
In this paper we study primal-dual path-following algorithms for the second-order cone programming (SOCP) based on a family of directions that is a natural extension of the Monteiro-Zhang (MZ) family for semidefinite programming. We show that the polynomial iteration-complexity bounds of two well-known algorithms for linear programming, namely the short-step path-following algorithm of Kojima et al. and Monteiro and Adler, and the predictor-corrector algorithm of Mizuno et al., carry over to the context of SOCP, that is they have an O( logε-1) iteration-complexity to reduce the duality gap by a factor of ε, where n is the number of second-order cones. Since the MZ-type family studied in this paper includes an analogue of the Alizadeh, Haeberly and Overton pure Newton direction, we establish for the first time the polynomial convergence of primal-dual algorithms for SOCP based on this search direction.
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Received: June 5, 1998 / Accepted: September 8, 1999¶Published online April 20, 2000
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Monteiro, R., Tsuchiya, T. Polynomial convergence of primal-dual algorithms for the second-order cone program based on the MZ-family of directions. Math. Program. 88, 61–83 (2000). https://doi.org/10.1007/PL00011378
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DOI: https://doi.org/10.1007/PL00011378