v2007.09.17 - Convex Optimization

2.13. DUAL CONE & GENERALIZED INEQUALITY 179Volumes have been written about this problem, most notably by Cottle [59].The problem is not convex if both vectors w and z are variable. But if one ofthem is fixed, then the problem becomes convex with a very simple geometricinterpretation: Define the affine subsetA ∆ = {y ∈ R n | Ay=w − q} (403)For w T z to vanish, there must be a complementary relationship between thenonzero entries of vectors w and z ; id est, w i z i =0 ∀i. Given w ≽0, thenz belongs to the convex set of feasible solutions:z ∈ −K ⊥ R n + (w ∈ Rn +) ∩ A = R n + ∩ w ⊥ ∩ A (404)where KR ⊥ (w) is the normal cone to n Rn + + at w (393). If this intersection isnonempty, then the problem is solvable.2.13.11 Proper nonsimplicial K , dual, X fat full-rankAssume we are given a set of N conically independent generators 2.61 (2.10)of an arbitrary polyhedral proper cone K in R n arranged columnar inX ∈ R n×N such that N > n (fat) and rankX = n . Having found formula(361) to determine the dual of a simplicial cone, the easiest way to find avertex-description of the proper dual cone K ∗ is to first decompose K intosimplicial parts K i so that K = ⋃ K i . 2.62 Each component simplicial conein K corresponds to some subset of n linearly independent columns from X .The key idea, here, is how the extreme directions of the simplicial parts mustremain extreme directions of K . Finding the dual of K amounts to findingthe dual of each simplicial part:2.61 We can always remove conically dependent columns from X to construct K or todetermine K ∗ . (F.2)2.62 That proposition presupposes, of course, that we know how to perform simplicialdecomposition efficiently; also called “triangulation”. [226] [123,3.1] [124,3.1] Existenceof multiple simplicial parts means expansion of x∈ K like (352) can no longer be uniquebecause N the number of extreme directions in K exceeds n the dimension of the space.