Classifying extrema using intervals

TL;DR

The paper addresses classifying stationary points of a smooth function $f:\mathbb{R}^n\to\mathbb{R}$ by a verified interval-based method that solves $\nabla f=0$ and then tests local behavior without relying on Hessian definiteness. It critiques the standard Hessian/eigenvalue approach for numerical rounding errors and degeneracy, offering a robust alternative. The proposed interval algorithm builds surrounding interval boxes around each candidate along all coordinate directions, evaluates $f$ on these boxes, and uses interval ranges to determine minima, maxima, or saddles with guaranteed correctness. This interval-based method scales linearly with dimension and provides reliable classifications by comprehensively sampling directions and enforcing enclosure guarantees, addressing the key pitfalls of traditional methods.

Abstract

We present a straightforward and verified method of deciding whether the n-dimensional point x (n>=1), such that \nabla f(x)=0, is the local minimizer, maximizer or just a saddle point of a real-valued function f. The method scales linearly with dimensionality of the problem and never produces false results.