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On the Closedness of the Linear Image of a Closed Convex Cone

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

Abstract: When is the linear image of a closed convex cone closed? We present very simple, and intuitive necessary conditions, which 1) unify, and generalize seemingly disparate, classical sufficient conditions: polyhedrality of the cone, and ``Slater'' type conditions; 2) are necessary and sufficient, when the dual cone belongs to a class, that we call nice cones. Nice cones subsume all cones amenable to treatment by efficient optimization algorithms: for instance, polyhedral, semidefinite, and $p$-cones. 3) provide similarly attractive conditions for an equivalent problem: the closedness of the sum of two closed convex cones.

Keywords: closedness; linear image; closed convex cone; sum of closed convex cones; duality; common root of Slater's condition and polyhedrality

Category 1: Convex and Nonsmooth Optimization (Convex Optimization )

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

Category 3: Linear, Cone and Semidefinite Programming (Second-Order Cone Programming )


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Entry Submitted: 12/27/2006
Entry Accepted: 12/27/2006
Entry Last Modified: 12/27/2006

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