This is a question that it is not homework but I would like clear.
I have this Proximal interpretation, that is: the solution of the problem is a fixed point of the following mapping: $$ x^{\ast} \in \ \ \arg \min_{x \in X_{\text{adm}}} \{\ \frac{1}{2}||x-x^{\ast}||^{2} + \gamma \ \ f(x) \} \ , \ \gamma >0$$
with $$X_{\text{adm}} = \{\ x \in \mathbb{R^n} | x \geq 0 \ , \ A_{\text{eq}}x=b_{\text{eq}}, A_{\text{ineq}}x=b_{\text{ineq}} \}$$ These $A$'s can be interpreted as constrains and it is compact and convex subset of $\mathbb{R^n}$.
Here I suppose that $f(x)$ is convex and differentiable with the gradient satisfying the Lipschitz condition.
Does this proximal method always converge? , i.e , does the proximal interpretation diverge?
I think that yes it divenges but maybe someone can hit me with some example?
I want to understand this because I'm going to study the Proximal Gradient method. Thanks for your help and time.
The proximal minimization algorithm (iterating the mapping you wanted to describe) is the application to optimization of the so-called proximal point algorithm for finding zeroes of monotone operators.
Its convergence to a solution is ensured under basically no assumptions on the (nonzero) stepsize $\gamma$, as stated in Theorem 4 of Rockafellar, “Monotone operators and the proximal point algorithm”, 1976.
So there’s no counterexample showing divergence. You don’t even need smoothness of $f$ really.