Discrete least-norm approximation by nonnegative (trigonometric) polynomials and rational functions

Abstract

Polynomials, trigonometric polynomials, and rational functions are widely used for the discrete approximation of functions or simulation models. Often, it is known beforehand that the underlying unknown function has certain properties, e.g., nonnegative or increasing on a certain region. However, the approximation may not inherit these properties automatically. We present some methodology (using semidefinite programming and results from real algebraic geometry) for least-norm approximation by polynomials, trigonometric polynomials, and rational functions that preserve nonnegativity.

Hierarchical alternating nonlinear least squares for nonnegative matrix factorization using rational functions

“…Existing approaches to approximate (nonnegative) rational functions Solving problem (12) is not trivial, even when neglecting the nonnegativity constraints. If many works exist in the unconstrained case, most of them consider the infinity norm in (12) [35], and there are very few works imposing nonnegativity: to the best of our knowledge this problem is only addressed in [31], [33], for the infinity norm.…”

“…If we fix u, then the problem is a feasibility problem, and therefore it is possible to perform a bisection search on u to find the solution. This is the method used in [31], [33] to solve the problem on nonnegative rational functions. The numerator and the denominator of the rational functions are modeled using Sum Of Squares (SOS), which makes problem ( 16) a SDP feasibility problem for u fixed.…”

“…In this case we consider the infinity norm instead of the squared norm, to express the problem as a bisection search over feasibility problems on polynomials as in (16). These feasibility problems can even have linear constraints if we impose h and g to be nonnegative on points τ i ∈ τ instead of being nonnegative on interval T (this is different from what is done in [31], [33]). The feasibility problem is then:…”