I.3.4: Generalized Block Relaxation
In some cases, even the supposedly simple minimizations within blocks may not have very simple solutions. In that case, we often use generalized block relaxation, which is defined by p maps As mapping X into (subsets of) X. We have As(x)∈{x1}⊗⋯⊗{xs−1}⊗Fs(x)⊗{xs+1}⊗⋯⊗{xp}, where z∈Fs(x) implies f(x1,⋯,xs−1,z,xs+1,⋯,xp)≤f(x1,⋯,xs−1,xs,xs+1,⋯,xp).
In ordinary block relaxation Fs(x)=argminz∈Xsf(x1,⋯,xs−1,z,xs+1,⋯,xp), but in generalized block relaxation we could update xs by taking one or more steps of a stable and convergent iterative algorithm for minimizing f(x1,⋯,xs−1,z,xs+1,⋯,xp) over z∈Xs.