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

Department of Statistics and Operations Research, University of North Carolina, CB #3260, Chapel Hill, North Carolina 27599
gabor{at}unc.edu, http://www.unc.edu/pataki

When is the linear image of a closed convex cone closed? We present very simple and intuitive necessary conditions that (1) unify, and generalize seemingly disparate, classical sufficientconditions such as 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); and (3) provide similarly attractive conditions for an equivalent problem: the closedness of the sum of two closed convex cones.

Key Words: closedness; linear image; closed convex cone; sum of closed convex cones; duality; common root of Slater’s condition and polyhedrality
History: Received: March 27, 2003; revision received: June 8, 2006;