Looks like you're trying to maximise a quadratic on a ball intersected with an affine space and an ugly half-space.
Maximising a quadratic on a ball is known as the "trust region subproblem." Surprisingly, it admits an efficient solution. I've never used it, but LSTRS by Marielba Rojas et al. apparently implements a TRS solver.
You'll need to deal with the inequality constraint, though. I'd suggest finding the smallest feasible Lagrange multiplier that results in the optimal solution to your TRS actually lying in the desired halfplane. Binary search is a pretty low-tech way to do this.