v2007.11.26 - Convex Optimization

E.10. ALTERNATING PROJECTION 657K ⊥ H 1 ∩ H 2(0)K ⊥ H 1 ∩ H 2(Pb) + PbH 10H 2K ∆ = H 1 ∩ H 2PbbFigure 134: Two examples (truncated): Normal cone to H 1 ∩ H 2 at theorigin, and at point Pb on the boundary. H 1 and H 2 are the same halfspacesfrom Figure 133. The normal cone at the origin K ⊥ H 1 ∩ H 2(0) is simply −K ∗ .E.10.3.2.1 Definition. Normal cone. [198] [30, p.261] [148,A.5.2][41,2.1] [231,3] The normal cone to any set S ⊆ R n at any particularpoint a∈ R n is defined as the closed coneK ⊥ S (a) ∆ = {z ∈ R n | z T (y −a)≤0 ∀y ∈ S} = −(S − a) ∗ (1882)an intersection of halfspaces about the origin in R n hence convex regardlessof the convexity of S ; the negative dual cone to the translate S − a . △Examples of normal cone construction are illustrated in Figure 134: Thenormal cone at the origin is the vector sum (2.1.8) of two normal cones;[41,3.3, exer.10] for H 1 ∩ int H 2 ≠ ∅K ⊥ H 1 ∩ H 2(0) = K ⊥ H 1(0) + K ⊥ H 2(0) (1883)This formula applies more generally to other points in the intersection.The normal cone to any affine set A at α∈ A , for example, is theorthogonal complement of A − α . Projection of any point in the translatednormal cone KC ⊥ (a∈ C) + a on convex set C is identical to a ; in other words,point a is that point in C closest to any point belonging to the translatednormal cone KC ⊥ (a) + a ; e.g., Theorem E.4.0.0.1.