function [x,R] = fb_dual(x, K, KS, GradFS, ProxGS, be, options)
% fb_dual - Forward-Backward algorithm applied to the Fenchel-Rockafellar dual (**) for strongly convex problems
%
%    [x,R] = fb_dual(x, K, KS, GradFS, ProxGS, be, options);
%
%   Minimization of 
%       min_x F(x)+G(K*x)                   			(*)
%   where F is strongly convex, and F is a proper closed convex and simple function.
%
%   Under appropriate domain equilification conditions, we have strong duality and (* is such that
%       min_x F(x)+G(K*x) = -min_u F^*(-K^* u) + G^*(u)		(**)
%   and the extremality Fenchel condition allows to recover the primal solution from a dual one, i.e.
%	x = grad(F^*)(-K^* u)
%
%
%
%   INPUTS:
%   GradFS(x) is grad(F^*)(x) (is Lipschitz since F is strongly convex)
%   ProxGS(u,mu) is prox_{mu*G^*}(u)
%   K is the handle to the linear operator.
%   KS is the handle to the adjoint linear operator.
%   L is the lipshitz constant of K*GradFS*KS
%   options.niter is the number of iterations.
%   options.verb is for the diaplay of iterations.
%   options.mu is the FB descent step-size.
%   options.report(x) is a function to fill in R.
%
%   OUTPUTS:
%   x=grad(F^*)(-K^* u) is the final solution.
%   R(i) = options.report(x) at iteration i.
%

if isnumeric(K)
    K = @(x)K*x;
end  
if isnumeric(KS)
    KS = @(x)KS*x;
end   

report_old = getoptions(options, 'report', @(x)0);
options.report = @(u)report_old(GradFS(-KS(u)));

NewGrad = @(u)-K(GradFS(-KS(u)));
u0 = K(x);
[u, R] = fb(u0, ProxGS, NewGrad, be, options);
x = GradFS(-KS(u));