non convex optimization Hi there,
In my studies I come up with this nonconvex optimization problem
argmin |Ax|_2+lamda*|x|_1 subject to x'x=1
where cost function is nonsmooth but convex and the constrant in nonconvex.
I tries subgradient projection method for convex constraints but the global solution is not my desired solution.
My question is that I should solve this problem hurestically or there is a reliable method for this nonconvex optimization problem?
 A: You can have a look of these papers：
 1. Jonathan H. Manton, Optimization algorithms exploiting unitary constraints.
 2. Zaiwen Zai and Wotao Yin, A feasible method for optimization with orthogonality constraints.
Wish these studies can help you. 
A: Conceptually, for algorithm design, the following version of the problem might be amenable to a larger number of techniques:
\begin{equation*}
\min_x\quad\|Ax-b\|^2\quad\text{s.t.}\quad \|x\|_1 \le \gamma,\quad\|x\|=1.
\end{equation*}
There are two reasons behind this reformulation:


*

*The objective function is now differentiable, so without further ado you can invoke the Gradient-Projection method, which under reasonable assumptions can be guaranteed to converge.

*This formulation makes it easy to use Alternating-Projection approaches.


Of course, several other numerical ideas also apply. For example, to get a good solution, you could start with $\gamma$ very large so that the $\ell_1$ constraint essentially disappears; then solve the problem exactly, and then gradually tighten $\gamma$.
