How to resolve total variation $F(u)=f(u)+\lambda\sum_i^N\int_{\Omega}|\nabla u_i(x)|dx$ Given $u(x)=[u_1(x)..u_N(x)]$, $0 \le u_i(x) \le 1, \sum_i^N u_i(x)=1$ and the cost function is:
$$F(u)=f(u(x))+\lambda\sum_i^N\int_{\Omega}|\nabla u_i(x)|dx$$
where $u_i(x)$ is a value that indicate the degree of $x$ and $f(u(x))$ is just simple function and convex function. I want to find optimal $arg \min_u F(u)$ subject to $0 \le u_i(x) \le 1, \sum_i u_i(x)=1$. As I saw that second term is a total variation (TV) of $u$. Could you have me to find $u$? This is what I done
The optimal solution of u is
$$\frac {\partial F(u)}{\partial u}=f'(u)+\lambda\int_{\Omega}...=0$$
$=>u=?$
 A: Since you seem to be unclear about the $f$ part of the energy / cost-function, I'll ignore it ($f \equiv 0$) in what follows. Also, since I imagine you're going to implement this on an actual computer, let's forget the integral, etc., and consider the discretization.
The TV semi-norm can then be written as
\begin{eqnarray}
\begin{split}
TV(x) &:= \|\nabla x\|_p = \begin{cases}\|\nabla x\|_1 := \sum_{j=1}^p\sum_{k=1}^d|(\nabla_k x)_j|, &\mbox{ in the anisotropic case,}\\
\|\nabla x\|_{2,1} := \sum_{j=1}^p\sqrt{\sum_{k=1}^d|(\nabla_k x)_j|^2}, &\mbox{ in the isotropic case}\end{cases}\\
&= \mathrm{max}\{\langle \nabla x, u\rangle | u \in \mathbb{R}^{dp}, \|u\|_{p^*} \le 1\},
\end{split}
\end{eqnarray}
where each linear operator $\nabla_k \in \mathbb{R}^{p \times p}$ is the discrete finite-difference gradient operator along the $k$th axis and $\nabla \in \mathbb{R}^{dp \times p}$ is formed by stacking them vertically. Imagine a $d$-dimensional photo with $p$ "pixels". Also note that in the last part of the formula, one has the dual norms $\|.\|_{1^*} = \|.\|_{\infty}$ and $\|.\|_{(2,1)^*} = \|.\|_{2,\infty}$.
It should be clear that the TV semi-norm is non-smooth. Thus your problem is a non-smooth convex optimization problem. Thanks to the formula above, we recognize your problem should be amenable to iterative methods using a scheme like Nesterov's smoothing (or see NESTA, for a "clearer" version of the algorithm for TV in particular).
Relevant literature also includes: TV in brain imaging (optimization of), benchmarking solvers for TV minimization and co., etc.
Hope this helps.
