function [gd,relres,iter]=gaboptdual(g,a,M,varargin)
%GABOPTDUAL  Compute dual window
%   Usage: gd=gaboptdual(g,a,M);
%          gd=gaboptdual(g,a,M, varagin);
%
%   Input parameters:
%     g      : Window function
%     a      : Time shift
%     M      : Number of Channels
%
%   Output parameters:
%     gd     : Dual window
%
%   `gaboptdual(g,a,M)` computes a window *gd* which is an
%   approximate dual window of the Gabor system defined by *g*, *a* and
%   *M*. 
%
%   This function solve a convex optimization problem that can be written
%   as:
%
%   .. gd  = argmin_x    || alpha x||_1 +  || beta Fx||_1  
%
%   ..                 + || omega (x -g_l) ||_2^2 + delta || x ||_S0
%
%   ..                 + gamma || nabla F x ||_2^2 + mu || nabla x ||_2^2
%
%   ..     such that  x is a dual windows of g
%
%   .. math:: \begin{split}  \text{gd}  = & \text{arg} \min_x     \| \alpha x \|_1 +  \| \beta \mathcal{F}x\|_1  \\ &  + \| \omega (x - g_l) \|_2^2  \\ &  \delta \| x \|_{S0}+ \mu \| \nabla x \|_2^2 +\gamma \| \nabla \mathcal{F} x \|_2^2 \\ &   \text{such that } x \text{ is a dual window of }g \end{split}
%
%   **Note**: This function require the unlocbox. You can download it at
%   `<http://unlocbox.sourceforge.net>`_
%
%   The function uses an iterative algorithm to compute the approximate
%   optimized dual. The algorithm can be controlled by the following flags:
%
%     'alpha',alpha  Weight in time. If it is a scalar, it represent the
%                  weights of the entire L1 function in time. If it is a 
%                  vector, it is the associated weight assotiated to each
%                  component of the L1 norm (length: Ldual).
%                  Default value is $\alpha=0$.
%                  **Warning**: this value should not be too big in order to
%                  avoid the the L1 norm proximal operator kill the signal.
%                  No L1-time constraint: $\alpha=0$
%
%     'beta',beta  Weight in frequency. If it is a scalar, it represent the
%                  weights of the entire L1 function in frequency. If it is a 
%                  vector, it is the associated weight assotiated to each
%                  component of the L1 norm in frequency. (length: Ldual).
%                  Default value is $\beta=0$.
%                  **Warning**: this value should not be too big in order to
%                  avoid the the L1 norm proximal operator kill the signal.
%                  No L1-frequency constraint: $\beta=0$
%
%     'omega',omega  Weight in time of the L2-norm. If it is a scalar, it represent the
%                  weights of the entire L2 function in time. If it is a 
%                  vector, it is the associated weight assotiated to each
%                  component of the L2 norm (length: Ldual).
%                  Default value is $\omega=0$.
%                  No L2-time constraint: $\omega=0$
%
%     'glike',g_l  $g_l$ is a windows in time. The algorithm try to shape
%                  the dual window like $g_l$. Normalization of $g_l$ is done
%                  automatically. To use option omega should be different
%                  from 0. By default $g_d=0$.
%
%     'mu',mu      Weight of the smooth constraint Default value is 1. 
%                  No smooth constraint: $\mu=0$
%   
%     'gamma',gamma  Weight of the smooth constraint in frequency. Default value is 1. 
%                    No smooth constraint: $\gamma=0$
%   
%     'delta',delta  Weight of the S0-norm. Default value is 0. 
%                    No S0-norm: $\delta=0$
%
%     'dual'       Look for a dual windows (default)
%
%     'painless'   Construct a starting guess using a painless-case
%                  approximation. This is the default
%
%     'zero'       Choose a starting guess of zero.
%
%     'rand'       Choose a random starting phase.
%
%     'tol',t      Stop if relative residual error is less than the 
%                  specified tolerance.  
%
%     'maxit',n    Do at most n iterations. default 200
%
%     'print'      Display the progress.
%
%     'debug'      Display all the progresses.
%
%     'quiet'      Don't print anything, this is the default.
%
%     'fast'       Fast algorithm, this is the default.
%
%     'slow'       Safer algorithm, you can try this if the fast algorithm
%                  is not working. Before using this, try to iterate more.
%
%     'printstep',p  If 'print' is specified, then print every p'th
%                    iteration. Default value is p=10;
%
%     'hardconstraint' Force the projection at the end (default)
%
%     'softconstaint' Do not force the projection at the end
%
%   See also: gabfirdual, gabdual, gabtight, gabopttight, gaboptdual

% Author: Nathanael Perraudin
% Date  : 18 Feb 2014 
    
if nargin<4
  error('%s: Too few input parameters.',upper(mfilename));
end;

[gd,relres,iter]=gabconvexopt(g,a,M,varargin{:});
