Complexity, Exactness, and Rationality in Polynomial Optimization

TL;DR

This paper investigates the existence and detection of rational solutions in polynomial optimization problems, revealing complexity results and providing certificates for feasibility, with implications for computational hardness and solution verification.

Contribution

It introduces new results on rational solutions in polynomial optimization, including NP-hardness proofs and feasibility certificates in fixed dimensions.

Findings

Rational solutions exist under certain separability conditions.

Detecting rational solutions is NP-hard in general.

Feasibility in fixed dimension is in NP with simple certificates.

Abstract

We focus on rational solutions or nearly-feasible rational solutions that serve as certificates of feasibility for polynomial optimization problems. We show that, under some separability conditions, certain cubic polynomially constrained sets admit rational solutions. However, we show in other cases that it is NP Hard to detect if rational solutions exist or if they exist of any reasonable size. We extend this idea to various settings including near feasible, but super optimal solutions and detecting rational rays on which a cubic function is unbounded. Lastly, we show that in fixed dimension, the feasibility problem over a set defined by polynomial inequalities is in NP by providing a simple certificate to verify feasibility. We conclude with several related examples of irrationality and encoding size issues in QCQPs and SOCPs.