Revisiting second-order optimality conditions for equality-contrained minimization problem

The aim of this note is to give a geometric insight into the classical second order optimality conditions for equality-constrained minimization problem. We show that the Hessian's positivity of the Lagrangian function associated to the problem at a local minimum point $x^*$ corresponds to inequalities between the respective algebraic curvatures at point $x^*$ of the hypersurface $\mathcal{M}_{f, x^*}=\{ x \in \R^n \, | \, f(x) = f(x^*)\}$ defined by the objective function $f$ and the submanifold $\mathcal{M}_g = \{ x \in \R^n \, | \, g(x)= 0 \}$ defining the contraints. These inequalities highlight a geometric evidence on how, in order to guarantee the optimality, the submanifold $\mathcal{M}_g$ has to be locally included in the half space $\mathcal{M}_{f, x^*}^+ = \{ x \in \R^n \, | \, f(x) \geq f(x^*)\}$ limited by the hypersurface $\mathcal{M}_{f, x^*}.$ This presentation can be used for educational purposes and help to a better understanding of this property.