A simple parallelizable method for the approximate solution of a quadratic transportation problem of large dimension with additional constraints

TL;DR

The paper introduces a simple, parallelizable iterative algorithm for approximating solutions to large-dimensional quadratic transportation problems with additional constraints, emphasizing efficiency and convergence properties.

Contribution

It presents a novel, easy-to-implement method that converges rapidly and is suitable for parallel computing, addressing large-scale transportation problems with constraints.

Findings

Method converges with geometric rate.

Algorithm is effective for large-scale problems.

Parallel implementation enhances computational efficiency.

Abstract

Complexity of the Operations Research Theory tasks can be often diminished in cases that do not require finding the exact solution. For example, forecasting two-dimensional hierarchical time series leads us to the transportation problem with a quadratic objective function and with additional constraints. While solving this task there is no need to minimize objective function with high accuracy, but it is very important to meet all the constraints. In this article we propose a simple iterative algorithm, which can find a valid transportation flow matrix in a limited number of steps while allowing massively parallel computing. Method's convergence was studied: a convergence criterion was indicated, as well as the solution's accuracy estimation technique. It was proved that the method converges with the speed of geometric progression, whose ratio weakly depends on the problem's dimension. Numerical experiments were performed to demonstrate the method's efficiency for solving specific large scale transportation problems.