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](http://arxiv.org/pdf/1203.3002v1.pdf)) 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.



Solving a QCQP problem with sparse regularization
There are several methods that one could apply to this problem. I don't have time to write out a full solution, but here's a quick idea. Replace $\Omega$ by $\hat\Omega = \Omega + \delta\_+$, where $\delta\_+$ is the indicator function for the nonnnegative orthant. Now, if you can reduce your problem to a proximal splitting method that works roughly as (not exactly this because of your equality constraints)


$$
 \beta^{k+1} \gets \mbox{prox}(\beta^k - \eta\_k\nabla L(\beta^k)),
$$
where $L$ is the differentiable part of the problem, and $\mbox{prox}$ is the *proximity-operator* that handles $\hat\Omega$, then you are done.


Fortunately, the proximity-operator of $\hat\Omega$ is merely the composition of the proximity operator of $\Omega$ with projection onto the nonnegative orthant. Thus, in some sense, you can easily use FISTA style methods.


