v2010.10.26 - Convex Optimization

E.10. ALTERNATING PROJECTION 761matrix closest to the positive semidefinite cone,⎡⎢⎣1 1 0.5454 01 1 1 0.54540.5454 1 1 10 0.5454 1 1⎤⎥⎦ (2083)and we find the positive semidefinite matrix closest to the affine subset A(2073):⎡⎤1.0521 0.9409 0.5454 0.0292⎢ 0.9409 1.0980 0.9451 0.5454⎥⎣ 0.5454 0.9451 1.0980 0.9409 ⎦ (2084)0.0292 0.5454 0.9409 1.0521These matrices (2083) and (2084) attain the Euclidean distance dist(A , S+).nConvergence is illustrated in Figure 172.E.10.3Optimization and projectionUnique projection on the nonempty intersection of arbitrary convex sets tofind the closest point therein is a convex optimization problem. The firstsuccessful application of alternating projection to this problem is attributedto Dykstra [127] [62] who in 1983 provided an elegant algorithm that prevailstoday. In 1988, Han [178] rediscovered the algorithm and provided aprimal−dual convergence proof. A synopsis of the history of alternatingprojection E.21 can be found in [64] where it becomes apparent that Dykstra’swork is seminal.E.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 173and Figure 174) associated with set k at iteration i . Initializey k0 = 0 ∀k=1... L and x 1,0 = b (2085)E.21 For a synopsis of alternating projection applied to distance geometry, see [353,3.1].