On the closedness of the linear image of a closed convex cone

On the closedness of the linear image of a closed convex cone

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Article ID: iaor200954094
Country: United States
Volume: 32
Issue: 2
Start Page Number: 395
End Page Number: 412
Publication Date: May 2007
Journal: Mathematics of Operations Research
Authors:
Keywords: convex hulls
Abstract:

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.

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