Nonlinear Integer Programming

TL;DR

This paper explores the computational complexity of nonlinear integer programming with linear constraints, highlighting boundary cases, recent approaches, and the interplay between theory and practical solution methods.

Contribution

It provides a comprehensive analysis of the complexity landscape and discusses recent successful methods for solving nonlinear integer problems.

Findings

Complexity varies significantly with nonlinear objective types.

Some boundary cases are solvable in polynomial time.

Recent approaches are promising for practical problem solving.

Abstract

Research efforts of the past fifty years have led to a development of linear integer programming as a mature discipline of mathematical optimization. Such a level of maturity has not been reached when one considers nonlinear systems subject to integrality requirements for the variables. This chapter is dedicated to this topic. The primary goal is a study of a simple version of general nonlinear integer problems, where all constraints are still linear. Our focus is on the computational complexity of the problem, which varies significantly with the type of nonlinear objective function in combination with the underlying combinatorial structure. Numerous boundary cases of complexity emerge, which sometimes surprisingly lead even to polynomial time algorithms. We also cover recent successful approaches for more general classes of problems. Though no positive theoretical efficiency results are available, nor are they likely to ever be available, these seem to be the currently most successful and interesting approaches for solving practical problems. It is our belief that the study of algorithms motivated by theoretical considerations and those motivated by our desire to solve practical instances should and do inform one another. So it is with this viewpoint that we present the subject, and it is in this direction that we hope to spark further research.