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

Gábor Pataki
2007 Mathematics of Operations Research  
When is the linear image of a closed convex cone closed? Despite being fundamental in convex analysis, this problem has received suprisingly little attention. We present very simple, and intuitive necessary conditions, which • unify, and generalize seemingly disparate, classical sufficient conditions: polyhedrality of the cone, and "Slater" type conditions; • are necessary and sufficient, when the dual cone belongs to a class, that we call nice cones. Nice cones subsume all cones amenable to
more » ... ones amenable to treatment by efficient optimization algorithms: for instance, polyhedral, semidefinite, and p-cones. • provide similarly attractive conditions for an equivalent problem: the closedness of the sum of two closed convex cones. Our problem frequently appears in a different guise: given closed, convex cones K 1 and K 2 , ( ) When is K * 1 + K * 2 closed? A necessary and/or sufficient condition for either one of ( ) and ( ) yields such a condition for the other, as explained in Section 5. Literature review The first reference that we are aware of, which implies the sufficiency of (IMG-RI) is Theorem 2 in [13] . (The proof in [13] only works in the case when K is full-dimensional -for the general case, one needs to modify it.) The
doi:10.1287/moor.1060.0242 fatcat:wbw4v4trzvgszgxrcslxx2lsca