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

TL;DR

This paper provides a geometric interpretation of second-order optimality conditions for equality-constrained minimization, linking Hessian positivity to curvature inequalities of related hypersurfaces.

Contribution

It introduces a geometric perspective on second-order conditions, connecting algebraic curvature inequalities to the optimality criteria in constrained optimization.

Findings

Hessian positivity corresponds to curvature inequalities.

Submanifold inclusion relates to optimality conditions.

Geometric insight aids educational understanding.

Abstract

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.