A Geometric Analysis of Renegar's Condition Number, and its interplay with Conic Curvature

Alexandre Belloni(bellonimit.edu)
Robert M. Freund(rfreundmit.edu)

Abstract: For a conic linear system of the form Ax \in K, K a convex cone, several condition measures have been extensively studied in the last dozen years. Herein we show that Renegar's condition number is bounded from above and below by certain purely geometric quantities associated with A and K, and highlights the role of the singular values of A and their relationship with the condition number. Moreover, by using the notion of conic curvature, we show how Renegar's condition number can be used to provide both lower and upper bounds on the width of the set of feasible solutions. This complements the literature where only lower bounds have heretofore been developed.

Keywords: Conic Programming; Condition Numbers; Conic Curvature

Category 1: Linear, Cone and Semidefinite Programming

Citation: MIT Operations Research Center Working Paper OR-zzz-07

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Entry Submitted: 04/16/2007
Entry Accepted: 04/16/2007
Entry Last Modified: 04/16/2007

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