v2010.10.26 - Convex Optimization

2.7. CONES 101Figure 39: Not a cone; ironically, the three-dimensional flared horn (withor without its interior) resembling mathematical symbol ≻ denoting strictcone membership and partial order.More familiar convex cones are Lorentz cone (confer Figure 46) 2.27{[ xK l =t]}∈ R n × R | ‖x‖ l ≤ t, l=2 (178)and polyhedral cone (2.12.1.0.1); e.g., any orthant generated by Cartesianhalf-axes (2.1.3). Esoteric examples of convex cones include the point at theorigin, any line through the origin, any ray having the origin as base suchas the nonnegative real line R + in subspace R , any halfspace partiallybounded by a hyperplane through the origin, the positive semidefinitecone S M + (191), the cone of Euclidean distance matrices EDM N (893)(6), completely positive semidefinite matrices {CC T |C ≥ 0} [40, p.71], anysubspace, and Euclidean vector space R n .2.27 a.k.a: second-order cone, quadratic cone, circular cone (2.9.2.8.1), unboundedice-cream cone united with its interior.