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 $x^{\ast }$ corresponds to inequalities between the respective algebraic curvatures at point $x^{\ast }$ of the hypersurface ${\mathcal{M}}_{f,x^{\ast }}=\{x\in {\mathbb{R}}^n\mathrm{\mid }f(x)=f(x^{\ast })\}$ defined by the objective function $f$ and the submanifold ${\mathcal{M}}_g=\{x\in {\mathbb{R}}^n\mathrm{\mid }g(x)=0\}$ 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 ${\mathcal{M}}_{f,x^{\ast }}^{+}=\{x\in {\mathbb{R}}^n\mathrm{\mid }f(x)\ge f(x^{\ast })\}$ limited by the hypersurface ${\mathcal{M}}_{f,x^{\ast }}\mathrm{.}$ This presentation can be used for educational purposes and help to a better understanding of this property.