[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-81559-en":3,"doc-seo-81559-105":29,"detail-sidebar-cat-0-en-105":90},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":4,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":13,"seo_description":14,"update_tm":27,"read_time":28},81559,549758146520,"Patrick","https://ap-avatar.wpscdn.com/avatar/80002397d8c0411e94?_k=1775819394049821470",8,"Research & Report","Computing Optimal Trajectories for Optimal Transport in Nonuniform Environments","This work addresses a discrete optimal transport problem in a nonuniform environment by computing a cost matrix via optimal trajectories between source and target points. The cost is defined through a calculus of variations formulation, and the associated Euler–Lagrange equations are formulated and solved to obtain path costs. Key contributions include verifiable sufficient conditions for Euler–Lagrange optimality and new a-posteriori algorithms to check optimality, validating exact cost-matrix computation. Numerical experiments in 2D and 3D demonstrate algorithm performance.","Version of Monday 13th July, 2026  \nCOMPUTING OPTIMAL TRAJECTORIES FOR OPTIMAL TRANSPORT IN NONUNIFORM ENVIRONMENTS  \nLUCA DIECI AND DANIYAR OMAROV  \narXiv :2510 . 17170v3 [math .OC] 10 Jul 2026  \nAbstract. In this work, we solve a discrete optimal transport problem in a nonuniform environment. To solve the optimal transport problem, we build the cost matrix and then use classical solvers for discrete optimal transport. The challenge is to form the cost matrix, which requires finding the optimal path between two points, and for this task we formulate and solve the associated Euler-Lagrange equations. A main contribution of ours is to provide verifiable sufficient conditions of optimality of the solution of the EulerLagrange equation and to propose new algorithms to to check optimality a-posteriori, thus validating the (exact) computation of the cost matrix. We illustrate our results and performance of the algorithms on several numerical examples in 2 and 3 dimensions.  \nNotation. Vectors are indicated with boldface and matrices with capital letters. Fora symmetric matrix A, we will write A ≻ 0 to signify that A is positive definite. The standard basis of Rn is indicated with {ei} . A continuous function x : [0 , 1] → Rn will be called a trajectory, or a path. The set P (a, b) will be the set of differentiable trajectories from a ∈ Rn to b ∈ Rn. In particular, if x ∈ P (a, b), then x(0) = a and x(1) = b. A smooth and strictly positive real valued scalar function K (x), from Rn to R+ , is called a weight (or a kernel) . The norm of a vector x will always be the Euclidean norm, unless otherwise stated.  \n1. Introduction  \nMotivated by solving optimal transport problems in the presence of obstacles, in this work we consider solving a discrete optimal transport of masses in a nonuniform environment. More specifically, we will consider the case where the cost function c(a, b) to move one unit of mass from location a ∈ Rn to location b ∈ Rn is given as the solution of a problem of calculus of variations of the form  \n(1) c (a, b) = x (t)(a,b Z01 L (x, x˙ )dt ,  \nand we are interested in the special forms of L given in (5) and (6) . From the theoretical point of view, the model (1) is a classical problem in Calculus of Variations (see Section 1.1 below), and it is considered also by Villani in his book on Optimal Transport, [21, Chapter 7] . Our goal in this work is numerical, that is we want to compute the optimal  \n2020 Mathematics Subject Classification. 49K15, 49Q22, 65K99, 90B80 .  \nKey words and phrases. Optimal transport, numerical computation, optimal trajectory, calculus of variations, necessary and sufficient optimality condition, assignment problem, Sinkhorn method.  \n1  \n2 L. Dieci, D. Omarov  \ntransport plan of moving k0 point masses xi’s into k 1 point masses yj’s, where xi’s andyj’s are points in some subset of Rn, when the cost of moving one into the other is given by (1) .  \nIn a uniform environment, the case that is usually considered in the literature, the transportation cost c (x, y) is typically given by a p-norm, c (x, y) = ∥x − y∥ p, 1 \u003C p \u003C ∞ , the 2-norm ∥x − y∥2 , or simply ∥x − y∥, being the most obvious choice, and also by the 2-norm square, c (x, y) = ~~1~~2 ∥x−y∥22, a case that is known as the 2-Wasserstein distance, or simply Wasserstein distance. So, in a uniform environment in Rn , computation of the cost is straightforward. But, in a nonuniform environment, the cost will generally depend on the path followed to go from x to y, and this makes the computation of c(x, y) a challenge in its own rights.  \nOptimal transport, in one of its several variants, has been receiving a lot of attention in recent decades, in no small part because of its flexibility to adapt to many problems of seemingly different nature, such as the Schr¨odinger bridge problem, unbalanced optimal transport, transport in the presence of physical constraints, and the use of optimal transport in Machine Learning (e.g., see [16]) . Our w","cbCaig3dB5hmJG6e","https://ap.wps.com/l/cbCaig3dB5hmJG6e","pdf",28603317,1,22,"English","en",105,"# Introduction\n## Setup of the discrete optimal transport problem\n## Motivation: optimal transport with obstacles and constraints\n# Notation and problem formulation\n# Cost construction via Euler–Lagrange equations\n# Theoretical optimality conditions\n# Algorithms and simulation results","[{\"question\":\"How is the transport cost c(a,b) defined in this work?\",\"answer\":\"The cost c(a,b) is obtained as the minimum value of a calculus-of-variations problem over trajectories connecting a to b, using a Lagrangian L(x, ẋ) and then integrating over time.\"},{\"question\":\"What role do Euler–Lagrange equations play?\",\"answer\":\"Euler–Lagrange equations characterize optimal trajectories for the variational problem defining the cost. They are formulated and solved so the trajectory information can be used to build the cost matrix.\"},{\"question\":\"How does the paper verify that a computed trajectory is optimal?\",\"answer\":\"The paper provides verifiable sufficient conditions for optimality of solutions to the Euler–Lagrange equations, and it proposes new algorithms that check optimality a-posteriori to validate the exact cost-matrix computation.\"}]",1784174318,55,{"code":4,"msg":30,"data":31},"ok",{"site_id":24,"language":23,"slug":32,"title":13,"keywords":33,"description":14,"schema_data":34,"social_meta":85,"head_meta":87,"extra_data":89,"updated_unix":27},"computing-optimal-trajectories-for-optimal-transport-in-nonuniform-environments","",{"@graph":35,"@context":84},[36,53,67],{"@type":37,"itemListElement":38},"BreadcrumbList",[39,43,47,50],{"item":40,"name":41,"@type":42,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":44,"name":45,"@type":42,"position":46},"https://docshare.wps.com/document/","Document",2,{"item":48,"name":12,"@type":42,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":42,"position":52},"https://docshare.wps.com/document/computing-optimal-trajectories-for-optimal-transport-in-nonuniform-environments/81559/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":61,"encodingFormat":60,"isAccessibleForFree":62,"interactionStatistic":63},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-16",true,{"@type":64,"interactionType":65,"userInteractionCount":4},"InteractionCounter",{"@type":66},"ViewAction",{"@type":68,"mainEntity":69},"FAQPage",[70,76,80],{"name":71,"@type":72,"acceptedAnswer":73},"How is the transport cost c(a,b) defined in this work?","Question",{"text":74,"@type":75},"The cost c(a,b) is obtained as the minimum value of a calculus-of-variations problem over trajectories connecting a to b, using a Lagrangian L(x, ẋ) and then integrating over time.","Answer",{"name":77,"@type":72,"acceptedAnswer":78},"What role do Euler–Lagrange equations play?",{"text":79,"@type":75},"Euler–Lagrange equations characterize optimal trajectories for the variational problem defining the cost. They are formulated and solved so the trajectory information can be used to build the cost matrix.",{"name":81,"@type":72,"acceptedAnswer":82},"How does the paper verify that a computed trajectory is optimal?",{"text":83,"@type":75},"The paper provides verifiable sufficient conditions for optimality of solutions to the Euler–Lagrange equations, and it proposes new algorithms that check optimality a-posteriori to validate the exact cost-matrix computation.","https://schema.org",{"og:url":51,"og:type":86,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":88,"canonical":51},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":91},[92,96,100,104,109,114,119,122,127,130,134],{"id":20,"doc_module":4,"doc_module_name":45,"category_name":93,"show_sort_weight":94,"slug":95},"Story & Novel",90,"story-novel",{"id":46,"doc_module":4,"doc_module_name":45,"category_name":97,"show_sort_weight":98,"slug":99},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":45,"category_name":101,"show_sort_weight":102,"slug":103},"Exam",70,"exam",{"id":105,"doc_module":4,"doc_module_name":45,"category_name":106,"show_sort_weight":107,"slug":108},5,"Comic",60,"comic",{"id":110,"doc_module":4,"doc_module_name":45,"category_name":111,"show_sort_weight":112,"slug":113},6,"Technology",50,"technology",{"id":115,"doc_module":4,"doc_module_name":45,"category_name":116,"show_sort_weight":117,"slug":118},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":45,"category_name":12,"show_sort_weight":120,"slug":121},30,"research-report",{"id":123,"doc_module":4,"doc_module_name":45,"category_name":124,"show_sort_weight":125,"slug":126},9,"Religion & Spirituality",20,"religion-spirituality",{"id":125,"doc_module":4,"doc_module_name":45,"category_name":128,"show_sort_weight":125,"slug":129},"World Cup","world-cup",{"id":131,"doc_module":4,"doc_module_name":45,"category_name":132,"show_sort_weight":131,"slug":133},10,"Lifestyle","lifestyle",{"id":135,"doc_module":4,"doc_module_name":45,"category_name":136,"show_sort_weight":105,"slug":137},19,"General","general"]