What's the best way to optimize this energy function, and is it convex? I have an energy function

$E({\bf y})=||\,g({\bf Ay+c})-{\bf d}\,||^2_2 + ||\,{\bf y-e}\,||^2_2 + \alpha\,|{\bf y}|_1$

I need to minimize this with respect to $\bf y$, all other variables being constant. 

$g(\bf u)$ is a non-linear scalar function that acts on the elements of $\bf u$ independently. Typically this function is also monotonic and convex. In the simplest case, $g(\bf u) = u$, but other choices include $g({\bf u})=\log(1+\exp(\bf u))$.

My first question is whether this is a convex problem. I think that it is convex if $g(\bf u) = u$, but guessing it also might be convex if this function is nonlinear but has convex form, but I'm really not sure.

Then I would please like a suggestion as the quickest way to optimize for $\bf y$. I don't really know how to deal with the L1 term. The dimensionality of $\bf y$ is of the order 10,000 and $\bf A$ and $\bf B$ are typically sparse.


Turns out this is not convex - see comments.
Any suggestions on the best non-convex approach to use?
If $g(u)$ is convex, then you problem is convex as well: [EDIT by mcg: the linear/affine case is probably the only useful one for which this is convex.]

Other cases can be convex as well. There are several simple rules to be used to detect convexity. You may want to take a look to a classic book:

http://stanford.edu/~boyd/cvxbook/

The $|\cdot|_1$ is a simple regularization term that you an easily convert in a set of linear constraints: $\min |x|_1$ is equivalent to

$$
\min \sum t_i\\
t_i\geq |x_i| \quad i=1,\ldots,n
$$

and hence

$$
\min \sum t_i\\
t_i\geq x_i \quad i=1,\ldots,n
t_i\geq -x_i \quad i=1,\ldots,n
$$

More details are in the book I mentioned.

Than you can use any QP solver. If work on MATLAB and you do not want to work on the formulation, I would suggest you to use YALMIP or CVX.

You can find
If $g(u)$ is convex, then you problem is convex as well: [EDIT by mcg: the linear/affine case is probably the only useful one for which this is convex.]

Other cases can be convex as well. There are several simple rules to be used to detect convexity. You may want to take a look to a classic book:

http://stanford.edu/~boyd/cvxbook/

The $|\cdot|_1$ is a simple regularization term that you an easily convert in a set of linear constraints: $\min |x|_1$ is equivalent to

$$
\min \sum t_i\\
t_i\geq |x_i| \quad i=1,\ldots,n
$$

and hence

$$
\min \sum t_i\\
t_i\geq x_i \quad i=1,\ldots,n
t_i\geq -x_i \quad i=1,\ldots,n
$$

More details are in the book I mentioned.

Than you can use any QP solver. If work on MATLAB and you do not want to work on the formulation, I would suggest you to use YALMIP or CVX.

You can find