Mathematical Programming

, Volume 100, Issue 3, pp 613–662

The volumetric barrier for convex quadratic constraints

Article

DOI: 10.1007/s10107-003-0513-4

Cite this article as:
Anstreicher, K. Math. Program., Ser. A (2004) 100: 613. doi:10.1007/s10107-003-0513-4
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Abstract.

Let https://static-content.springer.com/image/art%3A10.1007%2Fs10107-003-0513-4/MediaObjects/s10107-003-0513-4flb1.gif where https://static-content.springer.com/image/art%3A10.1007%2Fs10107-003-0513-4/MediaObjects/s10107-003-0513-4flb2.gif and https://static-content.springer.com/image/art%3A10.1007%2Fs10107-003-0513-4/MediaObjects/s10107-003-0513-4flb3.gifi is an n×n positive semidefinite matrix. We prove that the volumetric and combined volumetric-logarithmic barriers for https://static-content.springer.com/image/art%3A10.1007%2Fs10107-003-0513-4/MediaObjects/s10107-003-0513-4flb3.gif are https://static-content.springer.com/image/art%3A10.1007%2Fs10107-003-0513-4/MediaObjects/s10107-003-0513-4flb4.gif and https://static-content.springer.com/image/art%3A10.1007%2Fs10107-003-0513-4/MediaObjects/s10107-003-0513-4flb4.gif self-concordant, respectively. Our analysis uses the semidefinite programming (SDP) representation for the convex quadratic constraints defining https://static-content.springer.com/image/art%3A10.1007%2Fs10107-003-0513-4/MediaObjects/s10107-003-0513-4flb3.gif, and our earlier results on the volumetric barrier for SDP. The self-concordance results actually hold for a class of SDP problems more general than those corresponding to the SDP representation of https://static-content.springer.com/image/art%3A10.1007%2Fs10107-003-0513-4/MediaObjects/s10107-003-0513-4flb3.gif.

Keywords

Volumetric barrierConvex quadratic constraintsSemidefinite programming

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  1. 1.Department of Management SciencesUniversity of IowaUSA