Linear Optimization of Polynomial Rational Functions: Applications for Positivity Analysis

Abstract

In this paper, we provide tight linear lower bounding functions for multivariate polynomials given over boxes. These functions are obtained by the expansion of polynomials into Bernstein basis and using the linear least squares function. Convergence properties for the absolute difference between the given polynomials and their lower bounds are shown with respect to raising the degree and the width of boxes and subdivision. Subsequently, we provide a new method for constructing an affine lower bounding function for a multivariate continuous rational function based on the Bernstein control points, the convex hull of a non-positive polynomial s, and degree elevation. Numerical comparisons with the well-known Bernstein constant lower bounding function are given. Finally, with these affine functions, the positivity of polynomials and rational functions can be certified by computing the Bernstein coefficients of their linear lower bounds.

Dollmaker Optimization Algorithm: A Novel Human-Inspired Optimizer for Solving Optimization Problems

“…This concept extends beyond mathematics and finds International Journal of Intelligent Engineering and Systems, Vol.17, No. 3,2024 DOI: 10.22266/ijies2024.0630.63 application in computer science, engineering, finance, and many other fields [4,5]. Metaheuristic algorithms, inspired by natural processes and phenomena, represent a powerful class of optimization techniques that have gained prominence across diverse domains.…”

Approximation Hierarchies for the Copositive Tensor Cone and Their Application to the Polynomial Optimization over the Simplex

“…They occur in applications where one needs to represent multidimensional data such as in signal processing [1][2][3], machine learning [2,4,5], material science [6], and speech recognition [7]. For example, any homogeneous polynomial of degree d in n-variables is associated with a symmetric tensor of order d and dimension n. In polynomial optimization and, likewise, in control theory, checking the non-negativity of a polynomial is a fundamental problem [8,9]. Nonnegativity of a polynomial over the real space or its non-negative orthant results in positive semidefinite and copositive tensors, respectively.…”