v2009.01.01 - Convex Optimization

704 APPENDIX E. PROJECTION
When set S is a convex cone K , then the normal cone to K at the origin
K ⊥ K(0) = −K ∗ (1966)
is the negative dual cone. Any point belonging to −K ∗ , projected on K ,
projects on the origin. More generally, [92,4.5]
K ⊥ K(a) = −(K − a) ∗ (1967)
K ⊥ K(a∈ K) = −K ∗ ∩ a ⊥ (1968)
The normal cone to ⋂ C k at Pb in Figure 143 is ray {ξ(b −Pb) | ξ ≥0}
illustrated in Figure 149. Applying Dykstra’s algorithm to that example,
convergence to the desired result is achieved in two iterations as illustrated in
Figure 148. Yet applying Dykstra’s algorithm to the example in Figure 142
does not improve rate of convergence, unfortunately, because the given
point b and all the alternating projections already belong to the translated
normal cone at the vertex of intersection.
E.10.3.3
speculation
From these few examples we surmise, unique minimum-distance projection on
blunt polyhedral cones having nonempty interior may be found by Dykstra’s
algorithm in few iterations.