文件列表:

__MACOSX\toolbox_optim\._.dropbox (82, 2010-12-29)
__MACOSX\toolbox_optim\._compute_dual_prox.m (82, 2010-12-28)
__MACOSX\toolbox_optim\._Icon (362425, 2011-01-02)
__MACOSX\toolbox_optim\._perform_admm.m (82, 2010-12-29)
__MACOSX\toolbox_optim\._perform_dr.m (82, 2011-01-02)
__MACOSX\toolbox_optim\._perform_fb.m (82, 2010-12-29)
__MACOSX\toolbox_optim\._perform_fb_strongly.m (82, 2010-12-29)
__MACOSX\toolbox_optim\tests\old\._test_fbstrongly_analysis.m (82, 2010-12-28)
__MACOSX\toolbox_optim\tests\._publish_all.m (82, 2010-12-29)
__MACOSX\toolbox_optim\tests\._test_l1_constraint.m (82, 2010-12-29)
__MACOSX\toolbox_optim\tests\._test_l1_lagrangian.m (82, 2010-12-29)
__MACOSX\toolbox_optim\tests\._test_tv_constraint.m (82, 2010-12-29)
__MACOSX\toolbox_optim\tests\._test_tv_lagrangian.m (82, 2010-12-29)
__MACOSX\toolbox_optim\toolbox\._.DS_Store (82, 2010-12-29)
__MACOSX\toolbox_optim\toolbox\._compute_correlation_error.m (82, 2010-11-13)
__MACOSX\toolbox_optim\toolbox\._compute_operator_norm.m (186, 2010-12-29)
__MACOSX\toolbox_optim\toolbox\._perform_l1ball_projection.m (82, 2010-12-27)
__MACOSX\toolbox_optim\toolbox\._perform_soft_thresholding.m (82, 2010-12-28)
__MACOSX\toolbox_optim\toolbox\._s2v.m (82, 2010-12-29)
__MACOSX\._toolbox_optim (82, 2011-01-03)
toolbox_optim\.dropbox (9, 2010-12-29)
toolbox_optim\compute_dual_prox.m (517, 2010-12-28)
toolbox_optim\Icon (0, 2011-01-02)
toolbox_optim\perform_admm.m (2037, 2010-12-29)
toolbox_optim\perform_dr.m (1211, 2011-01-02)
toolbox_optim\perform_fb.m (2197, 2010-12-29)
toolbox_optim\perform_fb_strongly.m (1352, 2010-12-29)
toolbox_optim\tests\html\test_l1_constraint.html (5416, 2010-12-29)
toolbox_optim\tests\html\test_l1_constraint.png (3619, 2010-12-29)
toolbox_optim\tests\html\test_l1_constraint_01.png (3624, 2010-12-29)
toolbox_optim\tests\html\test_l1_constraint_02.png (5770, 2010-12-29)
toolbox_optim\tests\html\test_l1_lagrangian.html (4719, 2010-12-29)
toolbox_optim\tests\html\test_l1_lagrangian.png (2887, 2010-12-29)
toolbox_optim\tests\html\test_l1_lagrangian_01.png (6565, 2010-12-29)
toolbox_optim\tests\html\test_tv_constraint.html (5782, 2010-12-29)
toolbox_optim\tests\html\test_tv_constraint.png (1920, 2010-12-29)
toolbox_optim\tests\html\test_tv_constraint_01.png (4315, 2010-12-29)
toolbox_optim\tests\html\test_tv_lagrangian.html (5801, 2010-12-29)
... ...

Toolbox for non-smooth convex optimization using first order schemes. by Gabriel Peyré, www.numerical-tours.com ****** DISCLAIMER ****** This toolbox contains the implementation of what I consider to be fundamental algorithms for non-smooth convex optimization of structured functions. These algorithms might not be the fasted (although they certainly are quite efficient), but they all have a simple implementation in term of black boxes (gradient and proximal mappings, given as callbacks). However, you should have some knowledge about what is a gradient operator and a proximal mapping in order to be able to use this toolbox on your own problems. I suggest you have a look at the "suggested readings" for some more information about all this. ****** CONTENT Of THE TOOLBOX ****** The algorithms are implemented in the functions: * perform_admm.m * perform_dr.m * perform_fb_strongly.m * perform_fb.m A useful helper function is compute_dual_prox.m to switch between proximal mapping of the primal and dual functions. The directory tests/ contains some simple examples of application to TV and L1 minimization. You can use them as starting point for your own problems. The directory tests/html/ contains the HTML files with the results. ****** PROBLEMS CONSIDERED ****** These schemes are used for the minimization of convex problems that can be split as either min_x F(x)+G(x) (*) or min_x F(K(x)) + G(x) (**) where F and G are convex functions and K is a linear operator. The two important quality of a function F is * Being smooth, i.e. one can compute its gradient nabla F. * Being proximable, i.e. one can compute its proximal mapping Prox_{lambda*F}(x) = argmin_{y} 1/2*norm(x-y)^2 + lambda*F(x) Basic examples of proximable function are function that can be written as F(x) = sum_i f(x_i) where x_i is a 1-D or 2-D component of x, and f is a low dimensional function. In this case Prox_{lambda*F}(x) = ( Prox_{lambda*f}(x_i) )_i * Being strongly convex (larger than a quadratic function). Some algorithms make use of the dual function, defined as F^*(y) = max_x -F(x) ****** ALGORITHMS IMPLEMENTED ****** The algorithms implemented are: * Forward-backward (FB) splitting schemes (includes classical FB and accelerated schemes such as Nesterov and FISTA). ==> Works for problem (*), with F smooth and G proximable. * Preconditonned ADMM scheme ==> Works for problem (**), with F and G proximable. * Douglas-Rachord (DR) splitting scheme ==> Works for problem (*), with F and G proximable. * FB on the dual ==> Works for (**) with F proximable, G strongly convex so that one can compute the gradient of G^*. Both gradients and proximal mappings should be given as callback functions. ****** SUGGESTED READINGS ****** To learn about first order schemes: * Proximal Splitting Methods in Signal Processing, Patrick L. Combettes and Jean-Christophe Pesquet * Gradient-Based Algorithms with Applications to Signal Recovery Problems, Amir Beck and Marc Teboulle To learn more about the connexion with signal and image processing: * Blind Source Separation: the Sparsity Revolution, Bobin J. a Starck J.-L. a Moudden Y. a Fadili M.J * A wavelet tour of signal processing, 3rd edition, S. Mallat. * Numerical Tours of Signal Processing, www.numerical-tours.com Copyright (c) 2010 Gabriel Peyre

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