UC Irvine

In an increasingly interconnected world, the efficient and economical transportation of individuals and commodities has emerged as a cornerstone of modern society. Optimizing transportation plans has a huge potential for journey planning, congestion reduction, supply chain management, and data exchanges. These strategies hold immense relevance not onlyin the realm of engineering and transportation, but also in other fields, such as physics, computer science, economics, and several subject areas in mathematics. The present thesis aims to elucidate the optimization of transportation strategies, with a particular focus on two classical problems, finding short paths in large networks and solving optimal transport problems with flow-rate constraints.

The so-called shortest path problem seeks an optimal path of transporting one unit of mass between pairs of vertices on graphs. We present a novel formulation of the problem as an l1-regularized regression, often referred to as lasso (Least Absolute Shrinkage and Selection Operator). Based on this formulation, we draw a connection specifically between trees that grow as active edge-sets in the least angle regression (LARS) algorithm of the lasso problem, and respective shortest-path trees that emerge using the bi-directional Dijkstra algorithm. Then, to overcome the dimensionality challenge in large graphs, we explore the alternating direction method of multipliers (ADMM) in the lasso formulation. The resulting derivativeproximal algorithm speeds up the search for the short paths, trading off optimality (i.e., finding shortest paths) that may not be absolutely essential in a variety applications.

The basic transport problem is motivated by the need to transport resources/mass between end-point distributions (supply and demand). We consider the classical Monge-Kantorovich optimal transport problem with a quadratic cost functional to penalize distance of transport, with an added constraint that transported mass is required to pass through constriction points while abiding by specified allowable flow-rate; constriction points may be conceptualized as toll stations with limited throughput. Our contributions in this topic are as follows: (1) we provide a precise Monge formulation for the optimal transport problem with flux constraint at constriction sites along the path that is amenable to generalization in higher dimensions. We work out in detail the case of transport in one dimension by proving existence and uniqueness of solutions. Under suitable regularity assumptions we give an explicit construction of the transport plan; (2) we provide a Kantorovich-type reformulation of the problem by introducing a marginal probability density for the time that mass-elements cross toll stations –a probability density that is to be determined so as to meet given flow-rate constraints. Interestingly, the Kantorovich-type formalism leads to multi-marginal optimal transport problem that is readily solvable by using linear programming. Moreover, existence and uniqueness of solutions are also established in this setting. Then, (3) we propose an entropic penalty term to regularize and reduce the computational cost of resulting multi-marginal problems. Entropic regularization of standard optimal transport leads to an efficient algorithm, the Sinkhorn algorithm, which applies in the present case as well. Leveraging the splittable nature of the cost in our formulation, we proposed a Gluing Sinkhorn algorithm for the multi-marginal optimal transport problem, which reduces the computational cost to a level comparable to that in standard two-marginal problems.