# Identifying saddle point in constrained optimization

Suppose we are minimizing $$f(x)$$. The first order necessary condition of $$x^*$$ being local minmum is: $$\nabla f(x^*)= \mathbf{0}.$$ For sufficiency, we check if also $$\nabla^2f(x^*) \succ 0$$, i.e., the Hessian is positive definite at $$x^*$$. I know that if $$\nabla^2f(x^*) \prec 0$$ then this is a local maximizer. Moreover, if it is indefinite, $$x^*$$ is a saddle point. Finally, the sufficiency condition fails to classify if $$\nabla^2 f(x^*) \succeq 0$$ or $$\nabla^2f(x^*)\preceq 0$$ but not strictly positive/negative definite. So semi- definiteness makes it not possible to conclude.

Now, I want to see such a situation in constrained optimization. Assume we are minimizing $$f(x)$$ over some $$h(x) \leq 0, \ g(x) =0$$. The necessary condition of $$x^*$$ being a local minimum is (given that this point is a regular point) satisfying the KKT conditions. It is known that a regular point $$x^*$$ satisfying KKT conditions is local minimizer if it also satisfies: $$s^\top \nabla_{xx}^2 L(x^*, \lambda^*, \rho^*)s >0$$ for all $$s \neq 0$$ such that: $$\begin{bmatrix} \frac{\partial g(x^*)}{\partial x} \\ \frac{\partial h_a(x^*)}{\partial x} \end{bmatrix} s = \mathbf{0}.$$ Here, $$h_a$$ denotes the set of active inequality constraints at $$x^*$$. Also $$\lambda^*, \rho^*$$ are the optimal Lagrange multipliers corresponding to the inequality and equality constraints (but unnecessary detail here).

Similarly, for all $$s$$ satisfying the second condition if we have $$s^\top \nabla_{xx}^2 L(x^*, \lambda^*, \rho^*)s <0$$ then we have a local maximizer. My question is, is there any way to conclude 'saddle point' or 'not enough evidence to classify' as in the unconstrained case?

Can I say if for some of the $$s$$ values we have $$s^\top \nabla_{xx}^2 L(x^*, \lambda^*, \rho^*)s >0$$ and for some others we have $$s^\top \nabla_{xx}^2 L(x^*, \lambda^*, \rho^*)s = 0$$ is this non-concludable? And similarly, if we have some $$s^\top \nabla_{xx}^2 L(x^*, \lambda^*, \rho^*)s > 0$$ and some $$s^\top \nabla_{xx}^2 L(x^*, \lambda^*, \rho^*)s < 0$$, is this definitely something like a constrained saddle point?