A FAST ALGORITHM FOR FUJITA-OGAWA MODEL IN DISCRETE TWO-DIMENSIONAL SPACE

Abstract

 This study proposes an efficient algorithm for the Fujita-Ogawa (FO) model in large-scale discrete space. We first generalize the choice behaviors of households and firms using the random utility theory (logit model) and introduce a stochastic FO model, which includes the original FO model as a special case. Subsequently, we reveal that this stochastic FO model has an equivalent optimization problem. We then transform the equivalent problem into a hierarchical optimization problem consisting of a master problem for firms and a subproblem for households. The subproblem, which determines the commuting pattern of households, has the structure of an optimal transport problem with an entropy regularization term. The master problem, which determines the location pattern of firms, is a nonconvex programming problem with constraints. By exploiting these mathematical structures, we propose a hierarchical optimization algorithm composed of the balancing method for the subproblem and the accelerated gradient method for the master problem. Finally, numerical experiments verify the efficiency and accuracy of the proposed method.