Revisiting second-order optimality conditions for equality-constrained 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 M f,x * = {x ∈ R n | f (x) = f (x *)} defined by the objective function f and the submanifold M g = {x ∈ R n | g(x) = 0} defining the contraints. These inequalities highlight a geometric evidence on how, in order to guarantee the optimality, the submanifold M g has to be locally included in the half space M + f,x * = {x ∈ R n | f (x) ≥ f (x *)} limited by the hypersurface M f,x *. This presentation can be used for educational purposes and help to a better understanding of this property