Generating Random Vectors satisfying Linear and Nonlinear Constraints

TL;DR

This paper identifies limitations in the existing Dirichlet-Rescale algorithm for generating uniformly distributed vectors with fixed sums and constraints, and proposes an improved algorithm that handles more complex constraints while maintaining uniformity.

Contribution

The paper introduces the Dirichlet-Rescale-Constraints (DRSC) algorithm, which ensures uniform sampling of vectors under linear and nonlinear constraints, improving upon previous methods.

Findings

DRS algorithm does not produce a true uniform distribution.

DRSC algorithm successfully generates vectors with desired constraints.

Experimental results confirm the effectiveness of DRSC.

Abstract

We consider the problem of generating n-dimensional vectors with a fixed sum, with the goal of generating a uniform distribution of vectors over a valid region. This means that each possible vector has an equal probability of being generated. The Dirichlet-Rescale (DRS) algorithm, introduced by Griffin et al. (2020), aims to generate a uniform distribution of vectors with fixed sum that satisfies lower and upper bounds on the individual entries. However, we demonstrate that the uniform distribution property of the DRS algorithm does not hold in general. Using an analytical procedure and a statistical test, we show that the vectors generated by the DRS algorithm do not appear to be drawn from a uniform distribution. To resolve this issue, we propose the Dirichlet-Rescale-Constraints (DRSC) algorithm, which handles more general constraints, including both linear and nonlinear constraints, while ensuring that the vectors are drawn from a uniform distribution. In our computational experiments we demonstrate the effectiveness of the DRSC algorithm.