# Connecting optimization with spectral analysis of tri-diagonal matrices

## TL;DR

This paper establishes a novel connection between optimization and spectral analysis by approximating function extrema through eigenvalues of tri-diagonal matrices, linking optimization, orthogonal polynomials, and linear algebra.

## Contribution

It introduces a method to approximate global extrema of functions using eigenvalues of tri-diagonal matrices, bridging multiple mathematical fields.

## Key findings

- Extends the spectral analysis approach to optimization problems.
- Provides a hierarchy of matrices for increasingly accurate approximations.
- Links polynomial roots to optimization extrema.

## Abstract

We show that the global minimum (resp. maximum) of a continuous function on a compact set can be approximated from above (resp. from below) by computing the smallest (rest. largest) eigenvalue of a hierarchy of (r x r) tri-diagonal univariate moment matrices of increasing size. Equivalently it reduces to computing the smallest (resp. largest) root of a certain univariate degree-r orthonormal polynomial. This provides a strong connection between the fields of optimization, orthogonal polynomials, numerical analysis and linear algebra, via asymptotic spectral analysis of tri-diagonal symmetric matrices.