v2007.09.13 - Convex Optimization

E.10. ALTERNATING PROJECTION 643When set S is a convex cone K , then the normal cone to K at the originK ⊥ K(0) = −K ∗ (1853)is the negative dual cone. Any point belonging to −K ∗ , projected on K ,projects on the origin. More generally, [73,4.5]K ⊥ K(a) = −(K − a) ∗ (1854)KK(a∈ ⊥ K) = −K ∗ ∩ a ⊥ (1855)⋂The normal cone to Ck at Pb in Figure 122 is the ray{ξ(b −Pb) | ξ ≥0} illustrated in Figure 128. Applying Dykstra’s algorithm tothat example, convergence to the desired result is achieved in two iterations asillustrated in Figure 127. Yet applying Dykstra’s algorithm to the examplein Figure 121 does not improve rate of convergence, unfortunately, becausethe given point b and all the alternating projections already belong to thetranslated normal cone at the vertex of intersection.E.10.3.3speculationFrom these few examples we surmise, unique minimum-distance projection onblunt polyhedral cones having nonempty interior may be found by Dykstra’salgorithm in few iterations.