Leading Principal Minors of Bordered Hessian in Constrained Max Problems

[Kolmin]

I am struggling a bit with the second order conditions of a constrained maximization problem with $n$ variables and $k$ constraints (with $k>n$).

In the equality constraints case we have to check if the $(n-k)$ leading principal minors of the bordered Hessian alternate in sign, starting from the sign given by $(-1)^{n}$.

In the inequality constraints case we have to check if the $[n-(e+k)]$ leading principal minors of the bordered Hessian alternate in sign, starting from the sign given by $(-1)^{n}$ (with $e$ equals to the number of binding constraints and $k$ equals to the number of not-binding constraints).

Fair enough, but how to I behave when I have $n-k<0$ or $n-(e+k)<0$ (e.g. 2 variables in the objective function and 4 equality constraints)?
Do I have to focus only on the number I get in order to know which minors I have to check, without focusing on the sign?


Thanks a lot.

 

[Kolmin]

Sorry again with my stupid questions, but could somebody give me an hint on why this problem is probably that dumb?

Is it maybe that you cannot have more constraints than variables?
But still, if the constraints equals the number of the varibales, what should I look for in the second order conditions?

Thanks. : )