Article

Mathematical Programming

, Volume 100, Issue 3, pp 613-662

The volumetric barrier for convex quadratic constraints

  • Kurt M. AnstreicherAffiliated withDepartment of Management Sciences, University of Iowa Email author 

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Abstract.

Let http://static-content.springer.com/image/art%3A10.1007%2Fs10107-003-0513-4/MediaObjects/s10107-003-0513-4flb1.gif where http://static-content.springer.com/image/art%3A10.1007%2Fs10107-003-0513-4/MediaObjects/s10107-003-0513-4flb2.gif and http://static-content.springer.com/image/art%3A10.1007%2Fs10107-003-0513-4/MediaObjects/s10107-003-0513-4flb3.gif i is an n×n positive semidefinite matrix. We prove that the volumetric and combined volumetric-logarithmic barriers for http://static-content.springer.com/image/art%3A10.1007%2Fs10107-003-0513-4/MediaObjects/s10107-003-0513-4flb3.gif are http://static-content.springer.com/image/art%3A10.1007%2Fs10107-003-0513-4/MediaObjects/s10107-003-0513-4flb4.gif and http://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 http://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 http://static-content.springer.com/image/art%3A10.1007%2Fs10107-003-0513-4/MediaObjects/s10107-003-0513-4flb3.gif .

Keywords

Volumetric barrier Convex quadratic constraints Semidefinite programming