On the constrained minimization of smooth Kurdyka—Łojasiewicz functions with the scaled gradient projection method

The scaled gradient projection (SGP) method is a first-order optimization method applicable to the constrained minimization of smooth functions and exploiting a scaling matrix multiplying the gradient and a variable steplength parameter to improve the convergence of the scheme. For a general nonconvex function, the limit points of the sequence generated by SGP have been proved to be stationary, while in the convex case and with some restrictions on the choice of the scaling matrix the sequence itself converges to a constrained minimum point. In this paper we extend these convergence results by showing that the SGP sequence converges to a limit point provided that the objective function satisfies the Kurdyka-Łojasiewicz property at each point of its domain and its gradient is Lipschitz continuous.

[1]  Ya-xiang,et al.  A NEW STEPSIZE FOR THE STEEPEST DESCENT METHOD , 2006 .

[2]  P. L. Combettes,et al.  Variable metric forward–backward splitting with applications to monotone inclusions in duality , 2012, 1206.6791.

[3]  I. Loris,et al.  On the convergence of variable metric line-search based proximal-gradient method under the Kurdyka-Lojasiewicz inequality , 2016 .

[4]  Émilie Chouzenoux,et al.  Variable Metric Forward–Backward Algorithm for Minimizing the Sum of a Differentiable Function and a Convex Function , 2013, Journal of Optimization Theory and Applications.

[5]  L. Zanni,et al.  Efficient deconvolution methods for astronomical imaging: algorithms and IDL-GPU codes , 2012, 1210.2258.

[6]  S. Bonettini,et al.  Accelerated gradient methods for the x-ray imaging of solar flares , 2013, 1311.5717.

[7]  L. Zanni,et al.  A scaled gradient projection method for constrained image deblurring , 2008 .

[8]  Bin Zhou,et al.  Gradient Methods with Adaptive Step-Sizes , 2006, Comput. Optim. Appl..

[9]  S. Łojasiewicz.  Sur la géométrie semi- et sous- analytique , 1993 .

[10]  S. Bonettini,et al.  Nonnegative image reconstruction from sparse Fourier data: a new deconvolution algorithm , 2010 .

[11]  M. Bertero,et al.  A convergent blind deconvolution method for post-adaptive-optics astronomical imaging , 2013, 1305.0421.

[12]  S. Bonettini,et al.  New convergence results for the scaled gradient projection method , 2014, 1406.6601.

[13]  Roger Fletcher,et al.  A limited memory steepest descent method , 2011, Mathematical Programming.

[14]  Marco Prato,et al.  A New Semiblind Deconvolution Approach for Fourier-Based Image Restoration: An Application in Astronomy , 2013, SIAM J. Imaging Sci..

[15]  K. Kurdyka.  On gradients of functions definable in o-minimal structures , 1998 .

[16]  Benar Fux Svaiter,et al.  Convergence of descent methods for semi-algebraic and tame problems: proximal algorithms, forward–backward splitting, and regularized Gauss–Seidel methods , 2013, Math. Program..

[17]  L. Zanni,et al.  Accelerating gradient projection methods for ℓ1-constrained signal recovery by steplength selection rules , 2009 .

[18]  J. Borwein,et al.  Two-Point Step Size Gradient Methods , 1988 .

[19]  Federica Porta,et al.  Variable Metric Inexact Line-Search-Based Methods for Nonsmooth Optimization , 2015, SIAM J. Optim..

[20]  Hédy Attouch,et al.  Proximal Alternating Minimization and Projection Methods for Nonconvex Problems: An Approach Based on the Kurdyka-Lojasiewicz Inequality , 2008, Math. Oper. Res..