LDLˆT Direction in interior point method for semidefinite programming

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1 MITSUBISHI ELECTRIC RESEARCH LABORATORIES LDLˆT Direction in interior point method for semidefinite programming Raghunathan, A.U.; Biegler, L.T. TR March 2018 Abstract We present an interior point method for semidefinite programming where the semidefinite constraints on a matrix X are formulated as nonnegative constraints on d[1](x),..., d[n] (X) obtained from the LDLT factorization X = LDiag(d[1](X),..., d[n] (X))LT. The approach was first proposed by Fletcher who also provided analytic expressions for the derivatives of the factors in terms of X and the approach was subsequently utilized in an interior point algorithm by Benson and Vanderbei. However, the evaluation of first and second derivatives of d[i] (X) has been a bottleneck in such an algorithm. In this paper, we: (i) derive formulae for the first and second derivatives of d[i] (X) that are efficient and numerically stable to compute, (ii) show that the LDLT based search direction can be viewed in the standard framework of interior point methods for semidefinite programs with comparable computational cost per iteration, (iii) characterize the central path, and (iv) analyze the numerical conditioning of the linear system arising in the algorithm. We provide detailed numerical results on 79 SDP instances from the SDPLIB test set. SIAM Journal on Optimization This work may not be copied or reproduced in whole or in part for any commercial purpose. Permission to copy in whole or in part without payment of fee is granted for nonprofit educational and research purposes provided that all such whole or partial copies include the following: a notice that such copying is by permission of Mitsubishi Electric Research Laboratories, Inc.; an acknowledgment of the authors and individual contributions to the work; and all applicable portions of the copyright notice. Copying, reproduction, or republishing for any other purpose shall require a license with payment of fee to Mitsubishi Electric Research Laboratories, Inc. All rights reserved. Copyright c Mitsubishi Electric Research Laboratories, Inc., Broadway, Cambridge, Massachusetts 02139

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