v2010.10.26 - Convex Optimization
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v2010.10.26 - Convex Optimization

v2010.10.26 - Convex Optimization v2010.10.26 - Convex Optimization

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766 APPENDIX E. PROJECTIONC ∞C 1C 2K ⊥ C 1 (0)K ⊥ C 2 (0)K ⊥ C ∞(0)Figure 176: Rough sketch of normal cone to set C ⊂ R 2 as C wanders towardinfinity. A point at which a normal cone is determined, here the origin, neednot belong to the set. Normal cone to C 1 is a ray. But as C moves outward,normal cone approaches a halfspace.closest to any point belonging to the translated normal cone KC ⊥ (a) + a ; e.g.,Theorem E.4.0.0.1. The normal cone to convex cone K at the originK ⊥ K(0) = −K ∗ (2089)is the negative dual cone. Any point belonging to −K ∗ , projected on K ,projects on the origin. More generally, [109,4.5]K ⊥ K(a) = −(K − a) ∗ (2090)K ⊥ K(a∈ K) = −K ∗ ∩ a ⊥ (2091)The normal cone to ⋂ C k at Pb in Figure 168 is ray {ξ(b −Pb) | ξ ≥0}illustrated in Figure 174. Applying Dykstra’s algorithm to that example,convergence to the desired result is achieved in two iterations as illustrated inFigure 173. Yet applying Dykstra’s algorithm to the example in Figure 167does not improve rate of convergence, unfortunately, because the givenpoint b and all the alternating projections already belong to the translatednormal cone at the vertex of intersection.

E.10. ALTERNATING PROJECTION 767E.10.3.3speculationDykstra’s algorithm always converges at least as quickly as classicalalternating projection, never slower [109], and it succeeds where alternatingprojection fails. Rate of convergence is wholly dependent on particulargeometry of a given problem. From these few examples we surmise, uniqueminimum-distance projection on blunt (not sharp or acute, informally)full-dimensional polyhedral cones may be found by Dykstra’s algorithm infew iterations. But total number of alternating projections, constitutingthose iterations, can never be less than number of convex sets.

E.10. ALTERNATING PROJECTION 767E.10.3.3speculationDykstra’s algorithm always converges at least as quickly as classicalalternating projection, never slower [109], and it succeeds where alternatingprojection fails. Rate of convergence is wholly dependent on particulargeometry of a given problem. From these few examples we surmise, uniqueminimum-distance projection on blunt (not sharp or acute, informally)full-dimensional polyhedral cones may be found by Dykstra’s algorithm infew iterations. But total number of alternating projections, constitutingthose iterations, can never be less than number of convex sets.

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