I've formulated some problem as follow and I want to check whether it is a convex optimization problem.
the optimization value is a vector, x and V is a fixed positive semidefinite matrix
$$ minimize \quad b^Tx $$ $$ subject \; to $$ $$ V \; \le \; (x^Tx)I+2xx^T $$
The inequality in the constraint means generalized inequality (not elementwise ineq)
Of course, the objective is linear thus, convex but I'm not sure about the constraint
Please tell me whether it is a convex problem or not.
Additionally, if it is, how can I optimize the vector x? the interior-point method with log determinant barrier?