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".