Let f = 3*x^2 + 3*y^2 + 2*x*y. Hessian of f = [6 2;2 6], so f is a convex quadratic. The good news is that it can be entered in CVX as 0.5*[x y]*[6 2;2 6]*[x;y]. The bad news is that because a_i > 0, the inequality goes in the wrong direction to be convex.
Because the optimization problem is non-convex,a point satisfying the Karush-Kuhn Tucker (KKT) first order optimality conditions could be a global minimum, a local minimum which is not a global minimum, a saddle point, or a local or global maximum.
There are global solvers available for non-convex problems, but they do not always succeed in finding the globally optimal solution within available run-time and memory.
The only hope for getting this problem entered into CVX and submitted by it to a solver is if CVX’s MIDCP capability http://cvxr.com/cvx/doc/intro.html#mixed-integer-problems can be used to handle the non-convexity via introduction of binary (or integer)variables, I have no reason to believe that can be done in this case, but I am not sure that it can’t be, and haven;t pout in the effort to try to do that. If you find such a formulation, please post it here for the benefit of forum readers.