v2010.10.26 - Convex Optimization

E.10. ALTERNATING PROJECTION 763K ⊥ H 1 ∩ H 2(0)K ⊥ H 1 ∩ H 2(Pb) + PbH 10H 2K H 1 ∩ H 2PbbFigure 174: 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 173. The normal cone at the origin K ⊥ H 1 ∩ H 2(0) is simply −K ∗ .Assuming a nonempty intersection, then the iterates converge to the uniqueminimum-distance projection of point b on that intersection; [109,9.24]Pb = limi→∞x 1i (2087)In the case that all C k are affine, then calculation of y ki is superfluousand the algorithm becomes identical to alternating projection. [109,9.26][148,1] Dykstra’s algorithm is so simple, elegant, and represents such a tinyincrement in computational intensity over alternating projection, it is nearlyalways arguably cost effective.E.10.3.2Normal coneGlunt [155,4] observes that the overall effect of Dykstra’s iterativeprocedure is to drive t toward the translated normal cone to ⋂ C k atthe solution Pb (translated to Pb). The normal cone gets its namefrom its graphical construction; which is, loosely speaking, to draw theoutward-normals at Pb (Definition E.9.1.0.1) to all the convex sets C ktouching Pb . Relative interior of the normal cone subtends these normalvectors.