On constrained second derivatives

TL;DR

This paper investigates how to define constrained second derivatives in a general and unique way, demonstrating their usefulness in analyzing stationary points under constraints through theoretical development and a finite-dimensional example.

Contribution

It introduces a method for defining constrained second derivatives via successive constrained differentiations, enhancing stationary point analysis under constraints.

Findings

Constrained second derivatives can be properly defined through successive differentiation.

The method effectively characterizes stationary points under constraints.

An illustrative finite-dimensional example demonstrates the approach's applicability.

Abstract

The question of defining unique, generally applicable constrained second, and higher-order, derivatives is investigated. It is shown that second-order constrained derivatives obtained via two successive constrained differentiations provide a proper tool for the determination of the characters of stationary points under constraints. As an illustration, a finite-dimensional example is considered that has previously been used as an illustration for other stationary-point analysis methods.