v2007.09.13 - Convex Optimization

640 APPENDIX E. PROJECTIONE.10.3.1Dykstra’s algorithmAssume we are given some point b ∈ R n and closed convex sets{C k ⊂ R n | k=1... L}. Let x ki ∈ R n and y ki ∈ R n respectively denote aprimal and dual vector (whose meaning can be deduced from Figure 127and Figure 128) associated with set k at iteration i . Initializey k0 = 0 ∀k=1... L and x 1,0 = b (1848)Denoting by P k t the unique minimum-distance projection of t on C k , andfor convenience x L+1,i ∆ = x 1,i−1 , calculation of the iterates x 1i proceeds: E.21for i=1, 2,...until convergence {for k=L... 1 {t = x k+1,i − y k,i−1x ki = P k ty ki = P k t − t}}(1849)Assuming a nonempty intersection, then the iterates converge to the uniqueminimum-distance projection of point b on that intersection; [73,9.24]Pb = limi→∞x 1i (1850)In the case all the C k are affine, then calculation of y ki is superfluousand the algorithm becomes identical to alternating projection. [73,9.26][98,1] 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 [105,4] observes that the overall effect of Dykstra’s iterative procedureis to drive t toward the translated normal cone to ⋂ C k at the solutionPb (translated to Pb). The normal cone gets its name from its graphicalconstruction; which is, loosely speaking, to draw the outward-normals at Pb(Definition E.9.1.0.1) to all the convex sets C k touching Pb . The relativeinterior of the normal cone subtends these normal vectors.E.21 We reverse order of projection (k=L...1) in the algorithm for continuity of exposition.