Two improvements of the fast iterative shrinkage/thresholding algorithm: overrelaxation and weak convergence (通信方式) Two improvements of the fast iterative shrinkage/thresholding algorithm : Overrelaxation and weak convergence
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Abstract
In this paper, we present two improved schemes of FISTA (an iterative gradientbased algorithm, whose convergence rate of the objective function is Ο(1/k^2} in term of the iteration counter k) for the minimization of the sum of a smooth and a nonsmooth convex function. Such minimization problems arise naturally in signal and image processing. Our two schemes overcome two limitations of FISTA: (i) The stepsize in the for wardback ward splitting step in FISTA is bounded by a constant value determined by the Lipschitz constant of the gradient of the smooth function and (ii) no results on weak convergence of FISTA to a solution had been presented to our best knowledge. The first scheme admits variable stepsizes in broader ranges than FISTA while keeping the same convergence rate Ο(1/k^2) of the objective function. The second scheme, whose convergence rate of the objective function is not guaranteed as Ο(1/k^2), converges weakly to a solution by using a sequence generated by FISTA. A numerical example demonstrates the effectiveness of each improvement by showing that the proposed schemes outperform conventional algorithms in terms of speed of convergence.
Journal

 IEICE technical report

IEICE technical report 110(441), 143148, 20110224
The Institute of Electronics, Information and Communication Engineers