Optimal Transport Plan Research Articles

Quantitative stability and error estimates for optimal transport plans

Abstract Optimal transport maps and plans between two absolutely continuous measures $\mu$ and $\nu$ can be approximated by solving semidiscrete or fully discrete optimal transport problems. These two problems ensue from approximating $\mu$ or both $\mu$ and $\nu$ by Dirac measures. Extending an idea from Gigli (2011, On Hölder continuity-in-time of the optimal transport map towards measures along a curve. Proc. Edinb. Math. Soc. (2), 54, 401–409), we characterize how transport plans change under the perturbation of both $\mu$ and $\nu$. We apply this insight to prove error estimates for semidiscrete and fully discrete algorithms in terms of errors solely arising from approximating measures. We obtain weighted $L^2$ error estimates for both types of algorithms with a convergence rate $O(h^{1/2})$. This coincides with the rate in Theorem 5.4 in Berman (2018, Convergence rates for discretized Monge–Ampère equations and quantitative stability of optimal transport. Preprint available at arXiv:1803.00785) for semidiscrete methods, but the error notion is different.

Ollivier Ricci curvature of directed hypergraphs

Many empirical networks incorporate higher order relations between elements and therefore are naturally modelled as, possibly directed and/or weighted, hypergraphs, rather than merely as graphs. In order to develop a systematic tool for the statistical analysis of such hypergraph, we propose a general definition of Ricci curvature on directed hypergraphs and explore the consequences of that definition. The definition generalizes Ollivier’s definition for graphs. It involves a carefully designed optimal transport problem between sets of vertices. While the definition looks somewhat complex, in the end we shall be able to express our curvature in a very simple formula, kappa =mu _0-mu _2-2mu _3. This formula simply counts the fraction of vertices that have to be moved by distances 0, 2 or 3 in an optimal transport plan. We can then characterize various classes of hypergraphs by their curvature.

Optimization model of freight transportation on the routes of international transport corridors

The article deals with the modified Dijkstra’s algorithm of searching the shortest routes between all transport nodes of the road-transport network, which allows presenting the transport problem in the classical matrix form. This makes it possible to apply each of the known methods of optimal transport plans to solve it. The object of study is the transport process of freight transportation on the transport network by routes of international transport corridors. The purpose of the work is to improve the methods of solving the problems of finding the shortest routes on the transport network, including sections of international transport corridors. The research method is the analysis and modeling of freight transportation on road networks. The modified Dijkstra’s method of finding the shortest paths between all nodes of the road-transport network was work out, which allows to represent the transport problem in the classical matrix form, i.e. in the form of a table of connections. This makes it possible to apply each of the known methods of constructing optimal plans of cargo transportation in the table of connections. The software complex based on the developed algorithm was designed in the algorithmic language Delphi, which was tested on the example of a transport problem set in the form of a road network, as well as complex testing and debugging of a computer system to support decision-making on the optimization of freight traffic on Ukrainian and Western Europe transport systems.

A hybrid fix-and-optimize heuristic for integrated inventory-transportation problem in a multi-region multi-facility supply chain

In this work, we study an integrated inventory-transportation problem in a supply chain consisting of region-bound warehouses located in different regions. The supply chain deals with multiple items that compete for storage space and transportation capacity with multi-modal transportation considering regional capacity constraint for each mode of transportation. The objective is to determine an optimal storage and transportation plan to satisfy the demand of all regions without shortages for known procurement plan for all items. The problem is formulated as a mixed integer programming (MIP) model for minimizing the total costs over a finite planning horizon. An MIP-based fix-and-optimize (F&O) heuristic with several decomposition schemes is proposed to solve the problem efficiently. The performance of the decomposition schemes is investigated against the structure of the sub-problems obtained. To enhance the performance, F&O is crossbred with two metaheuristics – genetic algorithm (GA) and iterated local search (ILS) separately, which lead to hybrid heuristic approach. Extensive numerical experiments are carried out to analyze the performance of the proposed solution methodology by randomly generating several problem instances built using data collected from the Indian Public Distribution System. The proposed solution approach is found to be computationally efficient and effective, and outperforming state of the art MIP solver Cplex for practical size problem instances. Also, the hybridization of F&O heuristic with GA and ILS boosts its performance although with a justified increase in the computational time.

Semi-discrete optimal transport: a solution procedure for the unsquared Euclidean distance case

We consider the problem of finding an optimal transport plan between an absolutely continuous measure and a finitely supported measure of the same total mass when the transport cost is the unsquared Euclidean distance. We may think of this problem as closest distance allocation of some resource continuously distributed over Euclidean space to a finite number of processing sites with capacity constraints. This article gives a detailed discussion of the problem, including a comparison with the much better studied case of squared Euclidean cost. We present an algorithm for computing the optimal transport plan, which is similar to the approach for the squared Euclidean cost by Aurenhammer et al. (Algorithmica 20(1):61–76, 1998) and Mérigot (Comput Graph Forum 30(5):1583–1592, 2011). We show the necessary results to make the approach work for the Euclidean cost, evaluate its performance on a set of test cases, and give a number of applications. The later include goodness-of-fit partitions, a novel visual tool for assessing whether a finite sample is consistent with a posited probability density.

An Integrated Multi-Criteria and Multi-Objective Optimization Approach for Establishing the Transport Plan of Intercity Trains

The development of the transport plan must take into account various criteria impacting the transport process. The main objective of the study is to propose an integrated approach to determine the transport plan of passenger trains. The methodology consists of five steps. In the first step, the criteria for optimization of the transport plan were defined. In the second step, variants of the transport plan were formulated. In the third step, the weights of the criteria are determined by applying the step-wise weight assessment ratio analysis method (SWARA) multi-criteria method. The multi-objective optimization was conducted in the fourth step. The following multi-objective optimization approaches were used and compared: weighted sum method (WSM), compromise programming method (CP), and the epsilon–constraint method (EC). The study proposes a modified epsilon–constraint method (MEC) by applying normalization of each objective function according to the maximal value of the solution by individual optimization for each objective function, and hybrid methods: hybrid WSM and EC, hybrid WSM and MEC, hybrid CP and EC, and Hybrid CP and MEC. The impact of the variation of passenger flows on the choice of an optimal transport plan was studied in the fifth step. The Laplace’s criterion, Hurwitz’s criterion, and Savage’s criterion were applied to come to a decision. The approbation of the methodology was demonstrated through the case study of Bulgaria’s railway network. Suitable variant of transport plan is proposed.

Quadratically Regularized Optimal Transport

We investigate the problem of optimal transport in the so-called Kantorovich form, i.e. given two Radon measures on two compact sets, we seek an optimal transport plan which is another Radon measure on the product of the sets that has these two measures as marginals and minimizes a certain cost function. We consider quadratic regularization of the problem, which forces the optimal transport plan to be a square integrable function rather than a Radon measure. We derive the dual problem and show strong duality and existence of primal and dual solutions to the regularized problem. Then we derive two algorithms to solve the dual problem of the regularized problem: A Gauss–Seidel method and a semismooth quasi-Newton method and investigate both methods numerically. Our experiments show that the methods perform well even for small regularization parameters. Quadratic regularization is of interest since the resulting optimal transport plans are sparse, i.e. they have a small support (which is not the case for the often used entropic regularization where the optimal transport plan always has full measure).

Optimal transportation planning for prefabricated products in construction

AbstractPrefabrication‐based construction has several advantages, such as short building time and superior quality. The transportation of prefabricated products from factories to construction sites has two key issues to consider. One is how to efficiently utilize the capacity of trucks in order to use as few trucks as possible, and the other is how to postpone the transportation of prefabricated products so as to pay for them as late as possible. This study proposes a mathematical programming model to optimize the transportation planning of prefabricated products to minimize the sum of trucking transportation costs and inventory holding costs. Extensive numerical experiments show that the optimal transport plan generated by the model outperforms the plan obtained by a greedy approach by 10% of cost savings, implying the effectiveness of the model for the construction industry.

Targeted Adversarial Learning Optimized Sampling.

Boosting transitions of rare events is critical to simulations of chemical and biophysical dynamic systems in order to close the time scale gaps between theoretical modeling and experiments. We present a novel approach, called targeted adversarial learning optimized sampling (TALOS), to modify the potential energy surface in order to drive the system to a user-defined target distribution where the free-energy barrier is lowered. Combining statistical mechanics and generative learning, TALOS formulates a competing game between a sampling engine and a virtual discriminator, enables unsupervised construction of bias potentials, and seeks for an optimal transport plan that transforms the system into a target. Through multiple experiments, we show that on-the-fly training of TALOS benefits from the state-of-art optimization techniques in deep learning and thus is efficient, robust, and interpretable. TALOS is also closely connected to the actor-critic reinforcement learning and hence leads to a new way of flexibly manipulating the many-body Hamiltonian systems.

Statistical Physics Approach to the Optimal Transport Problem.

Originally defined for the optimal allocation of resources, optimal transport (OT) has found many theoretical and practical applications in multiple domains of science and physics. In this Letter we develop a new method for solving the discrete version of this problem using techniques derived from statistical physics. We derive a strongly concave free energy function that captures the constraints of the OT problem at a finite temperature. Its maximum defines an optimal transport plan, or registration between the two discrete probability measures that are compared, as well as a pseudodistance between those measures that satisfies the triangular inequalities. The computation of this pseudodistance is fast and numerically stable. The temperature dependent OT pseudodistance is shown to decrease monotonically with respect to the inverse of the temperature and to converge to the standard OT distance at zero temperature, providing a robust framework for temperature annealing. We illustrate applications of this framework to the problem of image comparison.

Block-matrix-based approach for the vehicle routing problem with transportation type selection under an uncertain environment

ABSTRACTThe vehicle routing problem has become a fundamental part of supply chains in competitive environments. Many studies have been conducted on the uncertain vehicle routing problem to improve transportation plans. However, few have concentrated on the selection of transportation type under uncertain environments. In this study, a novel vehicle routing model that considers transportation type selection between milk-run and cross-dock strategies under uncertain environments with fuzzy travel time is proposed. Furthermore, a novel block-matrix-based approach for the transportation type selection is presented to explore optimal transportation plans in an intuitive, reasonable, effective and efficient form. An extended biogeography-based optimization algorithm is proposed to derive an optimal transportation plan by extending the migration and mutation operators, and introducing a novel self-adaptive mutation rate and a secondary mutation operator. Finally, simulation experiments are performed to validate the effectiveness and practicality of this approach in solving the proposed model.

Mass-Driven Topology-Aware Curve Skeleton Extraction from Incomplete Point Clouds.

We introduce a mass-driven curve skeleton as a curve skeleton representation for 3D point cloud data. The mass-driven curve skeleton presents geometric properties and mass distribution of a curve skeleton simultaneously. The computation of the mass-driven curve skeleton is formulated as a minimization of Wasserstein distance, with an entropic regularization term, between mass distributions of point clouds and curve skeletons. Assuming that the mass of one sampling point should be transported to a line-like structure, a topology-aware rough curve skeleton is extracted via the optimal transport plan. A Dirichlet energy regularization term is then used to obtain a smooth curve skeleton via geometric optimization. Given that rough curve skeleton extraction does not depend on complete point clouds, our algorithm can be directly applied to curve skeleton extraction from incomplete point clouds. We demonstrate that a mass-driven curve skeleton can be directly applied to an unoriented raw point scan with significant noise, outliers and large areas of missing data. In comparison with state-of-the-art methods on curve skeleton extraction, the performance of the proposed mass-driven curve skeleton is more robust in terms of extracting a correct topology.

A study of the dual problem of the one-dimensional L∞-optimal transport problem with applications

The Monge-Kantorovich problem for the W∞ distance presents several peculiarities. Among them the lack of convexity and then of a direct duality. We study in dimension 1 the dual problem introduced by Barron, Bocea and Jensen in [2]. We construct a couple of Kantorovich potentials which is non trivial in the best possible way. More precisely, we build a potential which is non constant around any point that the restrictable, minimizing plan moves at maximal distance. As an application, we show that the set of points which are displaced at maximal distance by a “locally optimal” transport plan is shared by all the other optimal transport plans, and we describe the general structure of all the one-dimensional optimal transport plans.

Structure of optimal martingale transport plans in general dimensions

Given two probability measures $\mu$ and $\nu$ in “convex order” on $\mathbb{R}^{d}$, we study the profile of one-step martingale plans $\pi$ on $\mathbb{R}^{d}\times\mathbb{R}^{d}$ that optimize the expected value of the modulus of their increment among all martingales having $\mu$ and $\nu$ as marginals. While there is a great deal of results for the real line (i.e., when $d=1$), much less is known in the richer and more delicate higher-dimensional case that we tackle in this paper. We show that many structural results can be obtained, provided the initial measure $\mu$ is absolutely continuous with respect to the Lebesgue measure. One such a property is that $\mu$-almost every $x$ in $\mathbb{R}^{d}$ is transported by the optimal martingale plan into a probability measure $\pi_{x}$ concentrated on the extreme points of the closed convex hull of its support. This will be established for the distance cost $c(x,y)=\vert x-y\vert $ in the two-dimensional case, and also for any $d\geq3$ as long as the marginals are in “subharmonic order.” In some cases, $\pi_{x}$ is supported on the vertices of a $k(x)$-dimensional polytope, such as when the target measure is discrete. Duality plays a crucial role in our approach, even though, in contrast to standard optimal transports, the dual extremal problem may not be attained in general. We show however that “martingale supporting” Borel subsets of $\mathbb{R}^{d}\times\mathbb{R}^{d}$ can be decomposed into a collection of mutually disjoint components by means of a “convex paving” of the source space, in such a way that when the martingale is optimal for a general cost function, each of the components then supports a restricted optimal martingale transport whose dual problem is attained. This decomposition is used to obtain structural results in cases where global duality is not attained. On the other hand, it shows that certain “optimal martingale supporting” Borel sets can be viewed as higher-dimensional versions of Nikodym-type sets. The paper focuses on the distance cost, but much of the results hold for general Lipschitz cost functions.

The Task of Combinatorial Optimization: the Search for an Optimal Production and Transport Plan When Organizing Production in New Territories

The Task of Combinatorial Optimization: the Search for an Optimal Production and Transport Plan When Organizing Production in New Territories

Optimum Cost of Transporting Problems with Hexagonal Fuzzy Numbers

Transportation problems can be used in managing shipments straightforwardly from the supply point to the demand point. More recently, there has been widespread interest in applying transportation problems to optimize the fuzzy number systems. The ability to distinguish and make use of particular structures is a significant factor in the efficacious application of optimization models. In this paper, fuzzy methods of hexagonal are used to solve transportation problems when the demand and destination are uncertain. The most important findings of this paper are reaching the optimum transportation cost minimization through magnitude hexagonal fuzzy numbers. The results verified that the optimal transport plan of the company succeeded in minimizing the total cost to 1222$. The outcomes of this paper can be optimized by particle swarm optimization as a future trend.

On a mixture of Brenier and Strassen Theorems

We give a characterization of optimal transport plans for a variant of the usual quadratic transport cost introduced in [33]. Optimal plans are composition of a deterministic transport given by the gradient of a continuously differentiable convex function followed by a martingale coupling. We also establish some connections with Caffarelli's contraction theorem [14].

A new class of costs for optimal transport planning

We study a class of optimal transport planning problems where the reference cost involves a non-linear functionG(x, p) representing the transport cost between the Dirac measureδxand a target probabilityp. This allows to consider interesting models which favour multi-valued transport maps in contrast with the classical linear case ($G(x,p)=\int c(x,y)dp$) where finding single-valued optimal transport is a key issue. We present an existence result and a general duality principle which apply to many examples. Moreover, under a suitable subadditivity condition, we derive a Kantorovich–Rubinstein version of the dual problem allowing to show existence in some regular cases. We also consider the well studied case of Martingale transport and present some new perspectives for the existence of dual solutions in connection with Γ-convergence theory.

OPTIMIZATION OF PLANT INTERNAL TRANSPORTATION ACCORDING TO THE CONTACT SCHEDULE

For the English full text of the article please see the attached PDF-File (English version follows Russian version).ABSTRACT The presented article is a continuation of research on development of an alternative model for organization of transportation according to the contact schedule i n the conditions of a metallurgical enterprise. The mathematical model is based on a dynamic transport problem with delays (DTPD). It is shown in what way an optimal transportation plan is obtained and minimization of transportation costs is ensured with the agreed production programs of supply shops and consumer shops. The improvement of the system of transportation of empty cargo flow according to the contact schedule with the help of DTPD, as the authors argue, allows to abandon the strict assignment of cars to a specific transportation, supply of empty cars is carried out according to the most favorable option for a consumer (with a minimum of transport and production costs). The cars can be delivered to the consumer for loading onto the cargo front from under any used load. The use of a dynamic transportation problem with delays as the basis of a mathematical model for development of an optimal plan for transportation of cars according to the contact schedule helps to more than halve the transport and production costs of the shops of the metallurgical plant. Keywords: dynamic transport problem with delays, internal transportation, optimization of transportation, contact schedule, metallurgical plant, railway transport.

Coefficient Groups Inducing Nonbranched Optimal Transport

In this work we consider an optimal transport problem with coefficients in a normed Abelian group $G$, and extract a purely intrinsic condition on $G$ that guarantees that the optimal transport (or the corresponding minimum filling) is not branching. The condition turns out to be equivalent to the nonbranching of minimum fillings in geodesic metric spaces. We completely characterize finitely generated normed groups and finite-dimensional normed vector spaces of coefficients that induce nonbranching optimal transport plans. We also provide a complete classification of normed groups for which the optimal transport plans, besides being nonbranching, have acyclic support. This seems to initiate a new geometric classifications of certain normed groups. In the nonbranching case we also provide a global version of calibration, i.e. a generalization of Monge-Kantorovich duality.