I have the following optimization problem:
$$ \begin{aligned} & \underset{\alpha, \gamma}{\text{minimize}} & & \end{aligned} \frac{1}{2} \|y - \sum\limits_{i=1}^{S}\gamma_{i}\cdot X_{i}\alpha_{i}\|_{2} + \sum\limits_{i=1}^{S}\frac{\lambda_{i}}{p}\|\alpha_{i}\|_{p}^{p} + \sum\limits_{i=1}^{S} \frac{\eta_{i}}{q}|\gamma|^{q} $$
Is this problem convex if $p, q, \lambda_{i}, \eta_i \geq 1$? If yes than what would make this problem non-convex?
The reason I ask is because this problem is mentioned in the following paper (Equation 2) and the authors claim that this is non-convex (They don't put any constraints on the various parameters though, the way I have.).
paper: www.site.uottawa.ca/~nat/Courses/csi5387_Winter2014/paper17.pdf
It is the quadratic term in $\gamma$ and $\alpha$ inside that first norm that makes it non-convex. It is convex separately in $\alpha$ and $\gamma$, just not jointly.