Showing papers by "Robert M. Freund published in 2005"
TL;DR: In this paper, the modern theory of condition numbers for conic convex optimization is extended to the more general non-conic format (GP d ) s.t.Ax-b? C Y,x? C X, where P is any closed convex set, not necessarily a cone.
Abstract: The purpose of this paper is to extend, as much as possible, the modern theory of condition numbers for conic convex optimization:z *? min x c t x, s.t.Ax-b ? C Y ,x ? C X ,to the more general nonconic format:z *? min x c tx, ( GP d ) s.t.Ax - b ? C Y ,x ? P,whereP is any closed convex set, not necessarily a cone, which we call the ground-set. Although any convex problem can be transformed to conic form, such transformations are neither unique nor natural given the natural description of many problems, thereby diminishing the relevance of data-based condition number theory. Herein we extend the modern theory of condition numbers to the problem format ( GP d ). As a byproduct, we are able to state and prove natural extensions of many theorems from the conic-based theory of condition numbers to this broader problem format.
7 citations
TL;DR: Computational results indicate that this methodology holds the promise to markedly reduce the overall computation time for conic feasibility problems; for instance, a 46% decrease in average IPM iterations for 100 randomly generated poorly-behaved problem instances of dimension 1000 x 5000 is observed.
Abstract: We present a general theory for transforming a homogeneous conic system F: Ax = 0, x in C, x non-zero, to an equivalent system via projective transformation induced by the choice of a point in a related dual set. Such a projective transformation serves to pre-condition the conic system into a system that has both geometric and computational properties with certain guarantees. There must exist projective transformations that transform F to a system whose complexity is strongly-polynomial-time in m and the barrier parameter. We present a method for generating such a projective transformation based on sampling in the dual set, with associated probabilistic analysis. Finally, we present computational results that indicate that this methodology holds the promise to markedly reduce the overall computation time for conic feasibility problems; for instance we observe a 46% decrease in average IPM iterations for 100 randomly generated poorly-behaved problem instances of dimension 1000 x 5000.
3 citations