Generator Matrices by Solving Integer Linear Programs

TL;DR

This paper introduces a greedy algorithm that formulates the construction of generator matrices for low discrepancy sequences as an integer linear programming problem, enabling the creation of high-quality sequences in complex settings.

Contribution

The authors present a novel integer linear programming approach to design generator matrices, improving the construction of low discrepancy sequences for high-dimensional quasi-Monte Carlo methods.

Findings

Successfully constructs generator matrices in challenging scenarios

Produces low discrepancy sequences surpassing traditional methods

Demonstrates effectiveness through empirical results

Abstract

In quasi-Monte Carlo methods, generating high-dimensional low discrepancy sequences by generator matrices is a popular and efficient approach. Historically, constructing or finding such generator matrices has been a hard problem. In particular, it is challenging to take advantage of the intrinsic structure of a given numerical problem to design samplers of low discrepancy in certain subsets of dimensions. To address this issue, we devise a greedy algorithm allowing us to translate desired net properties into linear constraints on the generator matrix entries. Solving the resulting integer linear program yields generator matrices that satisfy the desired net properties. We demonstrate that our method finds generator matrices in challenging settings, offering low discrepancy sequences beyond the limitations of classic constructions.