A Note on the Calculation of Step-Lengths in Interior-Point Methods for Semidefinite Programming

Authors:
Kim-Chuan Toh
Affiliations:
Department of Mathematics, National University of Singapore, 10 Kent Ridge Crescent, Singapore 119620. mattohkc@math.nus.edu.sg
Venue:
Computational Optimization and Applications
Year:
2002

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Abstract

In each iteration of an interior-point method for semidefinite programming, the maximum step-length that can be taken by the iterate while maintaining the positive semidefiniteness constraint needs to be estimated. In this note, we show how the maximum step-length can be estimated via the Lanczos iteration, a standard iterative method for estimating the extremal eigenvalues of a matrix. We also give a posteriori error bounds for the estimate. Numerical results on the performance of the proposed method against two commonly used methods for calculating step-lengths (backtracking via Cholesky factorizations and exact eigenvalues computations) are included.