I want to solve the follong QCQP problem:

$$ \mbox{Minimize}\quad\beta^TA\beta+\mu\Omega(\beta)$$ $$ \mbox{s.t.}\quad\beta^TB\beta=1 \quad\mbox{and}\quad\beta\ge0 $$

where $A$ and $B$ are both positive definite, and $\Omega(\cdot)$ is a sparse(thus non-smooth, but still convex) norm, like $\ell_1$ norm or $\ell_1/\ell_2$ norm for a certain group set.

My tentative plan is to combine *FISTA algorithm* and *Homotopy methods* (a relevant paper I've found) for $f(\beta)=\beta^TA\beta++\lambda(\beta^TB\beta-1)+\mu\Omega(\beta)$, which can handle both $\beta$ and the Lagrange multiplier $\lambda$, but I did not find an algorithm considering the constraint $\beta\ge 0$(in the entry-wise sense). So can I improve the algorithm to embrace the conic constraits? Thank you.