On the complexity of nonlinear mixed-integer optimization

TL;DR

This survey reviews the computational complexity landscape of nonlinear mixed-integer optimization, covering incomputability, approximation schemes, and specialized algorithms across various problem classes.

Contribution

It provides a comprehensive overview of the complexity results and recent advances in algorithms for nonlinear mixed-integer optimization.

Findings

Incomputability results linked to number theory and logic.

Existence of fully polynomial-time approximation schemes in fixed dimensions.

Strongly polynomial algorithms for specific problem subclasses.

Abstract

This is a survey on the computational complexity of nonlinear mixed-integer optimization. It highlights a selection of important topics, ranging from incomputability results that arise from number theory and logic, to recently obtained fully polynomial time approximation schemes in fixed dimension, and to strongly polynomial-time algorithms for special cases.