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On the connection of facially exposed, and nice cones

Gabor Pataki(gabor***at***unc.edu)

Abstract: A closed convex cone K is called nice, if the set K^* + F^\perp is closed for all F faces of K, where K^* is the dual cone of K, and F^\perp is the orthogonal complement of the linear span of F. The niceness property plays a role in the facial reduction algorithm of Borwein and Wolkowicz, and the question whether the linear image of a nice cone is closed also has a simple answer. We prove several characterizations of nice cones and show a strong connection with facial exposedness. We prove that a nice cone must be facially exposed; in reverse, facial exposedness with an added technical condition implies niceness. We conjecture that nice, and facially exposed cones are actually the same, and give some supporting evidence.

Keywords: facially exposed cones, nice cones

Category 1: Convex and Nonsmooth Optimization

Category 2: Linear, Cone and Semidefinite Programming (Semi-definite Programming )

Category 3: Linear, Cone and Semidefinite Programming (Semi-definite Programming )

Citation:

Download: [PDF]

Entry Submitted: 09/23/2011
Entry Accepted: 09/23/2011
Entry Last Modified: 09/23/2011

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