Positivity and Optimization: Beyond Polynomials

Abstract

The recent progress optimization theory, due to a novel synthesis of real algebraic geometry and moment problems techniques, can naturally be reduced to positivity certificates for polynomial functions defined on basic semi-algebraic sets. There are however classical problems of applied mathematics which require exact positivity criteria for non-polynomial functions, such as splines, wavelets, periodic or almost periodic functions. While we do not lack fine analysis results referring to the positivity of such functions, traditionally stated in terms of Fourier-Laplace transforms type, the algebraic machinery of modern optimization theory based on polynomial algebra fails when applied to this more general context. A notorious example being the stability problem of differential equations with delays in the argument. In all these cases, the exact algebraic certificates must be complemented by approximation theory results. Without aiming at completeness, the present chapter offers a glimpse at a series of specific non-polynomial optimization problems, by identifying in every instance the specific results needed to run a robust algebraic relaxation scheme.