v2007.09.13 - Convex Optimization

180 CHAPTER 2. CONVEX GEOMETRY2.13.11.0.1 Theorem. Dual cone intersection. [245,2.7]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 (405)i=1Proof. For X i ∈ R n×n , a complete matrix of linearly independentextreme directions (p.124) 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} (406)M⋃K i =i=1M⋃{X i c | c ≽ 0} (407)i=1The union of all K i can be equivalently expressed⎧⎡ ⎤⎪⎨[ X 1 X 2 · · · X M ] ⎢⎣⎪⎩aḅ.c⎥⎦ | a,b,... ,c ≽ 0 ⎫⎪ ⎬⎪ ⎭(408)Because extreme directions of the simplices K i are extreme directions of Kby assumption, then by the extremes theorem (2.8.1.1.1),K = { [ X 1 X 2 · · · X M ]d | d ≽ 0 } (409)Defining X ∆ = [X 1 X 2 · · · X M ] (with any redundant [sic] columns optionallyremoved from X), then K ∗ can be expressed, (315) (Cone Table S, p.167)⋄K ∗ = {y | X T y ≽ 0} =M⋂{y | Xi T y ≽ 0} =i=1M⋂K ∗ i (410)i=1