I have the following optimization problem:
$min$ $x_1^2+x_2^2+x_3^2$
$s.t.$
$x_1+2x_2+3x_3\geq 4$
$x_3\leq 1$
It is asked without solving the KKT system, prove the problem has a unique optimal solution and it satisfies the KKT conditions.
Objective is quadratic $x^TQx$, and $Q$ is equal to the identity matrix which is positive definite. Therefore, the objective function is strictly convex. As the constraint is linear, it is convex too. I thought the answer should be related to those facts, but couldn't find it.