Bounds on the infimum of polynomials over a generic semi-algebraic set using asymptotic critical values

TL;DR

This paper establishes precise exponential bounds on the optimal value of polynomial optimization problems over semi-algebraic sets, with applications to algorithms for non-attained optima and bifurcation sets.

Contribution

It generalizes existing bounds to non-compact semi-algebraic sets and introduces specialized algorithms for two large classes of polynomial optimization problems.

Findings

Single exponential bounds depending on degree and bitsize.

Effective algorithms for non-attained optima.

Improved bounds for bifurcation sets.

Abstract

We present precise bit and degree estimates for the optimal value of the polynomial optimization problem $f^*:=\text{inf}_{x\in \mathscr{X}}~f(x)$, where $\mathscr{X}$ is a semi-algebraic set satisfying some non-degeneracy conditions. Our bounds depend on the degree, the bitsize of $f$, and the polynomials defining $\mathscr{X}$, and are single exponential with respect to the number of variables. They generalize the single exponential bounds from Jeronimo, Perrucci, and Tsigaridas (SIAM Journal on Optimization, 23(1):241--255, 2013) for the minimum of a polynomial function on a compact connected component of a basic closed semi-algebraic set. The tools that we use allow us to obtain specialized bounds and dedicated algorithms for two large families of polynomial optimization problems in which the optimum value might not be attained. The first family forms a dense set of real polynomial functions with a fixed collection of Newton polytopes; we provide the best approximation yet for the bifurcation set, which contains the optimal value, and we deduce an effective method for computations. As for the second family, we consider any unconstrained polynomial optimization problem; we present more precise bounds, together with a better bit complexity estimate of an algorithm to compute the optimal value.