Preconditioning of a Generalized Forward-Backward Splitting and Application to Optimization on Graphs
SIAM JOURNAL ON IMAGING SCIENCES(2015)
摘要
We present a preconditioning of a generalized forward-backward splitting algorithm for finding a zero of a sum of maximally monotone operators Sigma(n)(i=1) A(i) + B with B cocoercive, involving only the computation of B and of the resolvent of each A(i) separately. This allows us in particular to minimize functionals of the form Sigma(n)(i=1) g(i) + f with f smooth, using only the gradient of f and the proximity operator of each gi separately. By adapting the underlying metric, such preconditioning can serve two practical purposes: first, it might accelerate the convergence or, second, it might simplify the computation of the resolvent of A(i) for some i. In addition, in many cases of interest, our preconditioning strategy allows the economy of storage and computation concerning some auxiliary variables. In particular, we show how this approach can handle large-scale, nonsmooth, convex optimization problems structured on graphs, which arise in many image processing or learning applications, and that it compares favorably to alternatives in the literature.
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关键词
preconditioning,forward-backward splitting,monotone operator splitting,nonsmooth convex optimization,graph learning,total variation
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