v2010.10.26 - Convex Optimization

2.13. DUAL CONE & GENERALIZED INEQUALITY 179To find its smallest face F(K ∋c) containing a given point c∈K , by thediscretized membership theorem 2.13.4.2.1, it is necessary and sufficient tofind generators for the smallest face. We may do so one generator at atime: 2.67 Consider generator Γ i . If there exists a vector z ∈ K ∗ orthogonal toc but not to Γ i , then Γ i cannot belong to the smallest face of K containingc . Such a vector z can be realized by a linear feasibility problem:find z ∈ R nsubject to c T z = 0X T z ≽ 0Γ T i z = 1(374)If there exists a solution z for which Γ T i z=1, thenΓ i ̸⊥ K ∗ ∩ c ⊥ = {z ∈ R n | X T z ≽0, c T z=0} (375)so Γ i /∈ F(K ∋c) ; solution z is a certificate of null membership. If thisproblem is infeasible for generator Γ i ∈ K , conversely, then Γ i ∈ F(K ∋c)by (372) and (363) because Γ i ⊥ K ∗ ∩ c ⊥ ; in that case, Γ i is a generator ofF(K ∋c).Since the constant in constraint Γ T i z =1 is arbitrary positively, then bytheorem of the alternative there is correspondence between (374) and (348)admitting the alternative linear problem: for a given point c∈Kfinda∈R N , µ∈Ra , µsubject to µc − Γ i = Xaa ≽ 0(376)Now if this problem is feasible (bounded) for generator Γ i ∈ K , then (374)is infeasible and Γ i ∈ F(K ∋c) is a generator of the smallest face thatcontains c .2.13.4.3.2 Exercise. Finding smallest face of pointed closed convex cone.Show that formula (372) and algorithms (374) and (376) apply more broadly;id est, a full-dimensional cone K is an unnecessary condition. 2.68 2.67 When finding a smallest face, generators of K in matrix X may not be diminished innumber (by discarding columns) until all generators of the smallest face have been found.2.68 Hint: A hyperplane, with normal in K ∗ , containing cone K is admissible.