I'm solving an optimization problem with KKT method and I've found set of solutions for the problem that meat KKT conditions but now I'm trying to use second order optimality condition for my answer to see if it's a minimum or not but it seems that there is a problem which I can't figure it out. the optimization problem is to minimize:
$ Minimize\space\space\space -5(x_{1} - 3)^2 + 12(x_{2} + 5)^2 - 6x_{1}x_{2} $
$ s.t.\space\space0 \leq x_{1} \leq 10 \space\space and\space\space 0 \leq x_{2} \leq 5$
by reformulation:
$ g_{1}(x):\space -x_{1} \leq 0 $
$ g_{2}(x):\space x_{1} - 10 \leq 0 $
$ g_{3}(x):\space -x_{2} \leq 0 $
$ g_{4}(x):\space x_{2} - 5 \leq 0 $
I solved and found the points meat KKT conditions that are:
$ P_{1} = (0, 0), \space\space\space\space P_{2} = (3, 0), \space\space P_{3} = (10, 0)$
But when I try to use second order optimality condition to determine if these points are minimum or not for example for $ P_{3} $ by:
$ \nabla g(x)d = 0 $
In which $\nabla g(x) $ is the matrix of active constraints gradients. here for $ P_{3} $ , $\space g_{2}(x) $ and $\space g_{3}(x) $ are active so:
$ \nabla g(x)d = 0 \to \begin{pmatrix} 1 & 0 \\ 0 & -1 \\ \end{pmatrix} \begin{pmatrix} d_{1} \\ d_{2} \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \end{pmatrix} \to d = \begin{pmatrix} d_{1} \\ d_{2} \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \end{pmatrix} $
so when I try to determine the sign of
$ d^TH(x)d $
I end up with 0 that I don't know if it is minimum or maximum. so my problem is $ d = 0 $ and I can't figure out the sign of $ d^TH(x)d $ . How can I overcome this problem.