Suppose I want to find all the interior local minimizers of the following problem: $\underset{x,y}{min}f(x,y)$ subject to $x,y>0, g(x,y)\geq 0$.
By setting the first order conditions $\frac{\partial f(x,y)}{\partial x}=0$ and $\frac{\partial f(x,y)}{\partial y}=0$, I found $n$ candidates $(x^*_1,y^*_1)$,$(x^*_2,y^*_2)$ ... $(x^*_n,y^*_n)$.
Is it true that all I need to do now is to see which ones out of the $n$ candidates make the Hessian $H\equiv \begin{bmatrix}f_{xx},f_{xy}\\f_{xy}, f_{yy}\end{bmatrix}$ positive semidefinite?
Also, I think I do not need to worry about the KKT first order conditions here, because the KKT conditions aim to find candidates for both interior and corner solutions. Is that correct?