A non-convex QCQP

Question

I have a symmetric and positive semidefinite matrix $M$. Is it possible to solve the following problem as a quadratic problem?

$$\begin{array}{ll} \text{maximize} & w^T M w\\ \text{subject to} & w^T w \le 1\\ & A w \ge 0\end{array}$$

where $A$ is some matrix. If not, what is the right tool (maybe even software) to solve this kind of problem? We know also that $w^T M w$ is bounded above by some number.

Answer 1

This is a non-convex problem (as you are maximizing a convex function), so it may be difficult. Cplex should be able to do it, though, if the number of variables is not too big.

Answer 2

This is not a hard problem, both of your constraints (the sphere and the linear plane) are differentiable and convex. You can use the "fmincon" function of MATLAB. Remember to use the negative of the objective since fmincon minimizes by default