v2010.10.26 - Convex Optimization

2.13. DUAL CONE & GENERALIZED INEQUALITY 209cone K ∗ is to first decompose K into simplicial parts K i so that K = ⋃ K i . 2.84Each component simplicial cone in K corresponds to some subset of nlinearly independent columns from X . The key idea, here, is how theextreme directions of the simplicial parts must remain extreme directions ofK . Finding the dual of K amounts to finding the dual of each simplicial part:2.13.11.0.1 Theorem. Dual cone intersection. [330,2.7]Suppose proper cone K ⊂ R n equals the union of M simplicial cones K i whoseextreme directions all coincide with those of K . Then proper dual cone K ∗is the intersection of M dual simplicial cones K ∗ i ; id est,M⋃M⋂K = K i ⇒ K ∗ = K ∗ i (470)i=1Proof. For X i ∈ R n×n , a complete matrix of linearly independentextreme directions (p.145) arranged columnar, corresponding simplicial K i(2.12.3.1.1) has vertex-descriptionNow suppose,K =K =i=1K i = {X i c | c ≽ 0} (471)M⋃K i =i=1M⋃{X i c | c ≽ 0} (472)i=1The union of all K i can be equivalently expressed⎧⎡ ⎤⎪⎨[ X 1 X 2 · · · X M ] ⎢⎣⎪⎩aḅ.c⎥⎦ | a , b ... c ≽ 0 ⎫⎪ ⎬⎪ ⎭(473)Because extreme directions of the simplices K i are extreme directions of Kby assumption, then2.84 That proposition presupposes, of course, that we know how to perform simplicialdecomposition efficiently; also called “triangulation”. [303] [173,3.1] [174,3.1] Existenceof multiple simplicial parts means expansion of x∈ K , like (411), can no longer be uniquebecause number N of extreme directions in K exceeds dimension n of the space.⋄