principal "pseudo eigenvector" of a real symmetric positive-semidefinite matrix Let $A$ be a real symmetric positive-semidefinite matrix and suppose that $c>0$ is a sufficiently small number. I wonder if it is possible to solve the non-convex optimization 
$$\arg\max_u\ u^\mathrm{T}Au\\ \mathrm{subject\ to\ }\left\Vert u\right\Vert_2\leq 1\\ \quad\quad\quad\quad\left\Vert u\right\Vert_1\leq c,$$ efficiently.
For solving the optimization, I couldn't get farther than writing KKT conditions which do not help much in specifying the multipliers.
Given that without the $\ell_1$-norm constraint (i.e. $c\to\infty$), the problem reduces to finding the principal eigenvector of $A$ that can be solved efficiently (e.g., using power iteration method), we can think of the solution to the optimization above as "psuedo eigenvector".
 A: I am not sure about the objective function part. But for the constraints, May be this will help. Write $u=u^{+}-u^{-}$, where $u^{+}$ denotes positive part in $u$ and $u^{-}$ denotes the negative part. Now with this, the above problem will become 
\begin{align}
\arg\max_{u^{+},u^{-}}\ {u^{+}}^{T}Au^{+}+{u^{-}}^{T}Au^{-}-2{u^{-}}^{T}Au^{+} \\
subject ~~to& [1,1\ldots,1]^{T}({u^{+}}+{u^{-}}) <= c \\
& (u^{+}-u^{-})^{T}(u^{+}-u^{-})<=1 \\
&  u^{+}>=0, u^{-}>=0
\end{align} 
You can make the objective function linear by bringing in a variable $t$ and hence adding a non-convex quadratic constraint. All other linear constraints can be reformulated as quadratic constraints. I think then you can reformulate it as a semi-definite program. In details
\begin{align}
\min_{t}~~  -t \\
subject~to~ &u^{H}Au \geq t \\
& [1,1\ldots,1]^{T}({u^{+}}+{u^{-}}) <= c \\
& (u^{+}-u^{-})^{T}(u^{+}-u^{-})<=1 \\
&  u^{+}>=0, u^{-}>=0,t>=0
\end{align}
Note that this is equivalent to your original problem. Now the variables $u^{+},u^{-},t$ can be combined to form a single vector $x$ and I believe you can write all of them as a optimization problem with a linear objective and non-convex quadratic constraints. If it is possible, then Semi-definite Relaxation (SDR) is a very famous technique to deal with such non convex problems. Also in your case, SDR should give exact solutions. In details, you problem should look something like 
\begin{align}
\min_{x}~~  a^{T}x \\
subject~to~ &x^{H}F_{1}x \geq c_1 \\
&  x^{H}F_{2}x \geq c_2\\
& x^{H}F_{3}x \geq c_3 \\
&  x>=0
\end{align}
If you can reformulate your problem this way which I think is possible, then semi-definite relaxation will work. 
