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

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.