On Extending Some Primal--Dual Interior-Point Algorithms From Linear Programming to Semidefinite Programming

Authors:
Yin Zhang
Affiliations:
-
Venue:
SIAM Journal on Optimization
Year:
1998

Citing 0
Cited 9

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Solving semidefinite programming problems via alternating direction methods

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An improved prediction-correction method for monotone variational inequalities with separable operators

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A New Class of Large Neighborhood Path-Following Interior Point Algorithms for Semidefinite Optimization with $O(\sqrt{n}\log\frac{\mathrm{Tr}(X^0S^0)}{\epsilon})$ Iteration Complexity

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Superlinear Convergence of an Infeasible Predictor-Corrector Path-Following Interior Point Algorithm for a Semidefinite Linear Complementarity Problem Using the Helmberg-Kojima-Monteiro Direction

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A new O(nL) -iteration predictor-corrector algorithm with wide neighborhood for semidefinite programming

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Abstract

This work concerns primal--dual interior-point methods for semidefinite programming (SDP) that use a search direction originally proposed by Helmberg et al. [SIAM J. Optim., 6 (1996), pp. 342--361] and Kojima, Shindoh, and Hara [SIAM J. Optim., 7 (1997), pp. 86--125.] and recently rediscovered by Monteiro [SIAM J. Optim., 7 (1997), pp. 663--678] in a more explicit form. In analyzing these methods, a number of basic equalities and inequalities were developed in [Kojima, Shindoh, and Hara] and also in [Monteiro] through different means and in different forms.In this paper, we give a concise derivation of the key equalities and inequalities for complexity analysis along the exact line used in linear programming (LP), producing basic relationships that have compact forms almost identical to their counterparts in LP. We also introduce a new formulation of the central path and variable-metric measures of centrality. These results provide convenient tools for deriving polynomiality results for primal--dual algorithms extended from LP to SDP using the aforementioned and related search directions. We present examples of such extensions, including the long-step infeasible-interior-point algorithm of Zhang [SIAM J. Optim., 4 (1994), pp. 208--227].