Convergence of the non-uniform directed Physarum model

The directed Physarum dynamics is known to solve positive linear programs: minimize c(T) x subject to Ax = b and x >= 0 for a positive cost vector c. The directed Physarum dynamics evolves a positive vector x according to the dynamics (x)over dot = q(x) - x. Here q(x) is the solution to Af =b that minimizes the "energy" Sigma(i)c(i)f(i)2/x(i). In this paper, we study the non-uniform directed dynamics (x)over dot = D(q(x) - x), where D is a positive diagonal matrix. The non-uniform dynamics is more complex than the uniform dynamics (with D being the identity matrix), as it allows each component of x to react with different speed to the differences between q(x) and x. Our contribution is to show that the non-uniform directed dynamics solves positive linear programs. (C) 2020 Elsevier B.V. All rights reserved.