Initialize dual variables to \nabla f(x)

[foges]

I'm not sure if there is an ideal way to guess the dual variables, but assuming x0 and y0 are primal optimal, then the relation

x0 = prox_g(x0 - \tilde x) y0 = prox_f(y0 - \tilde y)

holds, which means

\nabla g(x0) + rho \tilde x = 0 <=> \tilde x = -(1/rho) \nabla g(x0) and \tilde y = -(1/rho) \nabla f(y0)

[foges]

Done for initialization of x. Need to do the same for initialization of lambda.