v2010.10.26 - Convex Optimization

2.13. DUAL CONE & GENERALIZED INEQUALITY 163prototypical linear program and its dual. Were K a positive semidefinitecone, then problem (p) has the form of prototypical semidefinite program(649) with (d) its dual. It is sometimes possible to solve a primal problem byway of its dual; advantageous when the dual problem is easier to solve thanthe primal problem, for example, because it can be solved analytically, or hassome special structure that can be exploited. [61,5.5.5] (4.2.3.1) 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 [330,2.7](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 (306)where each dual is determined with respect to a cone’s ambient space.(conjugation) [307,14] [109,4.5] [330, p.52] When K is any convexcone, dual of the dual cone equals closure of the original cone;K ∗∗ = K (307)is the intersection of all halfspaces about the origin that contain K .Because K ∗∗∗ = K ∗ always holds,K ∗ = (K) ∗ (308)When convex cone K is closed, then dual of the dual cone is the originalcone; K ∗∗ = K ⇔ K is a closed convex cone: [330, p.53, p.95]K = {x ∈ R n | 〈y , x〉 ≥ 0 ∀y ∈ K ∗ } (309)If any cone K is full-dimensional, then K ∗ is pointed;K full-dimensional ⇒ K ∗ pointed (310)If the closure of any convex cone K is pointed, conversely, then K ∗ isfull-dimensional;K pointed ⇒ K ∗ full-dimensional (311)Given that a cone K ⊂ R n is closed and convex, K is pointed if and onlyif K ∗ − K ∗ = R n ; id est, iff K ∗ is full-dimensional. [55,3.3 exer.20]