v2007.09.13 - Convex Optimization

2.13. DUAL CONE & GENERALIZED INEQUALITY 141whose optimal objective always has the saddle value 0 (regardless of theparticular convex cone K and other problem parameters). [267,3.2] Thusdetermination of convergence for either primal or dual problem is facilitated.2.13.1.1 Key properties of dual coneFor any cone, (−K) ∗ = −K ∗For any cones K 1 and K 2 , K 1 ⊆ K 2 ⇒ K ∗ 1 ⊇ K ∗ 2 [245,2.7](Cartesian product) For closed convex cones K 1 and K 2 , theirCartesian product K = K 1 × K 2 is a closed convex cone, andK ∗ = K ∗ 1 × K ∗ 2 (266)(conjugation) [228,14] [73,4.5] When K is any convex cone, the dualof the dual cone is the closure of the original cone; K ∗∗ = K . BecauseK ∗∗∗ = K ∗ K ∗ = (K) ∗ (267)When K is closed and convex, then the dual of the dual cone is theoriginal cone; K ∗∗ = K .If any cone K has nonempty interior, then K ∗ is pointed;K nonempty interior ⇒ K ∗ pointed (268)Conversely, if the closure of any convex cone K is pointed, then K ∗ hasnonempty interior;K pointed ⇒ K ∗ nonempty interior (269)Given that a cone K ⊂ R n is closed and convex, K is pointed ifand only if K ∗ − K ∗ = R n ; id est, iff K ∗has nonempty interior.[41,3.3, exer.20](vector sum) [228, thm.3.8] For convex cones K 1 and K 2K 1 + K 2 = conv(K 1 ∪ K 2 ) (270)