Connecting optimization with spectral analysis of tri-diagonal matrices
Optimization and Control
2020-03-17 v4 Numerical Analysis
Numerical Analysis
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.
Cite
@article{arxiv.1907.09784,
title = {Connecting optimization with spectral analysis of tri-diagonal matrices},
author = {Jean Lasserre},
journal= {arXiv preprint arXiv:1907.09784},
year = {2020}
}