2017/04/21 by Roshchina, Vera, Tunçel, Levent
#49N15 #52A15 #52A20 #90C46 #FOS: Mathematics #Optimization and Control (math.OC)
paper · doi:10.48550/arxiv.1704.06368
We study the boundary structure of closed convex cones, with a focus on facially dual complete (nice) cones. These cones form a proper subset of facially exposed convex cones, and they behave well in the context of duality theory for convex optimization. Using the well-known and commonly used concept of tangent cones in nonlinear optimization, we introduce some new notions for exposure of faces of convex sets. Based on these new notions, we obtain a necessary condition and a sufficient condition for a cone to be facially dual complete. In our sufficient condition, we utilize a new notion called lexicographic tangent cones (these are a family of cones obtained from a recursive application of the tangent cone concept). Lexicographic tangent cones are related to Nesterov's lexicographic derivatives and to the notion of subtransversality in the context of variational analysis.