[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-82344-en":3,"doc-seo-82344-105":29,"detail-sidebar-cat-0-en-105":91},{"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":20,"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},82344,1374391974585,"Genevieve","https://ap-avatar.wpscdn.com/davatar_276721f389ce27ea32af1340a28f341c",8,"Research & Report","Convex Relaxations for the Optimization of Markov Processes","This paper studies the optimization of Markov processes that interpolate between two prescribed probability distributions while minimizing a given cost across time steps. The key obstacle is dimensionality: representing full intermediate distributions becomes intractable in high-dimensional state spaces. The work reformulates the problem using sequential couplings and introduces convex relaxations based on local marginals and cluster moments, yielding computable lower bounds. Dynamic optimal transport is treated as a special case, and a procedure is proposed to recover Benamou–Brenier dynamics from relaxed solutions, with an application to constrained Ising-model processes.","arXiv :2607 .09423v1 [math .OC] 10 Jul 2026  \nConvex Relaxations for the Optimization of Markov Processes  \nHongyi Zhang∗, Yuehaw Khoo †, Tianyun Tang ‡  \nAbstract  \nIn this paper, we study the problem of optimizing Markov processes that interpolate between two prescribed probability distributions while minimizing a given cost. The main computational challenge is the curse of dimensionality: in high-dimensional state spaces, representing the full distribution is intractable. To address this, we reformulate the problem in terms of sequential couplings and develop convex relaxations based on local marginals and cluster moments. These relaxations exploit locality and sparse interaction structure, provide computable lower bounds, and recover low-order statistics of the intermediate laws. We identify dynamic optimal transport as a special case of our Markov process optimization problem and develop a procedure for recovering the underlying Benamou–Brenier dynamics from the relaxed solution. We also show that the procedure extends to more general Markov processes and illustrate it with a constrained process between Ising models.  \n1 Introduction  \n1.1 Optimizing over Markov processes  \nIn this paper, we study the problem of finding a Markov process that connects two prescribed distributions while minimizing a given cost. Let (Ω , B(Ω)) be a Borel state space. We denote by M(Ω) the set of finite signed Borel measures on Ω, and by  \nP(Ω) := {ρ ∈ M(Ω) : ρ ≥ 0, ρ(Ω) = 1}  \nthe set of probability measures on Ω . Given initial and terminal distributions µ,ν ∈ P(Ω), and one-step cost functions cs : Ω × Ω → R for s = 0 ,..., T − 1, we seek intermediate laws ρs ∈ P(Ω) and Markov transition kernels Ks(x, dy) solving  \nT−1  \n{Ks }Tsn10,f{ρs }Ts=0 Xs=0 ZΩ ZΩ cs(x, y) Ks(x, dy)dρs(x) (1.1)  \ns.t. ρ0 = µ, ρT = ν, (1 . 1-a)  \nρs+1 = (Ks)∗ ρs , s = 0 , ... , T − 1 , (1 . 1-b)  \nKs ∈ Ks, s = 0 ,..., T − 1 , (1.1-c)  \nρs ≥ 0, ρs(Ω) = 1, s = 0 ,..., T. (1.1-d)  \n∗ Department of Statistics, University of Chicago,([hongyi518@uchicago.edu](hongyi518@uchicago.edu)).  \n†Department of Statistics, University of Chicago,([ykhoo@uchicago.edu](ykhoo@uchicago.edu)) . The research of this author is partially funded by NSF DMS-2339439, DOE DE-SC0022232, DARPA The Right Space HR0011-25-9-0031, and a Sloan research fellowship.  \n‡Department of Statistics, University of Chicago,([ttang@u.nus.edu](ttang@u.nus.edu)) .  \nHere Ks denotes the admissible class of Markov kernels at step s. The kernels are allowed to depend on s, so the process is generally time-inhomogeneous. In the Markov evolution constraint (1.1-b) ,(Ks )∗ denotes the adjoint action of the kernel on probability measures: if Ks acts on test functions by  \nKsf(x) = ZΩ f (y) Ks(x, dy),  \nthen, for every Borel set B ∈ B(Ω) and probability measure ρ ∈ P(Ω),  \n(Ks)∗ ρ(B) = ZΩ Ks(x, B)dρ(x) .  \nA special case of problem (1.1) is dynamic optimal transport. Classical optimal transport chooses a coupling between two endpoint distributions and minimizes a transportation cost [11, 27] . Dynamic optimal transport instead seeks a curve of probability measures {q(·, t) ∈ P(Ω) : t ∈ [0 , 1]} with prescribed initial and terminal laws q ( · , 0) and q (· , 1) . When Ω ⊂ Rd , the Benamou–Brenier formula realizes this path through a velocity field [3]:  \nWpp(µ,ν) = inq,fv 􀀚Z01 ZΩ ∥v(x, t)∥p dq(x, t)dt : ∂tq + ∇ · (v q) = 0, q(· , 0) = µ, q(· , 1) = ν􀀛 . (BB)  \nHere Wp denotes the p-Wasserstein distance and 􀀈v(·, t) : Ω → Rd | t ∈ [0 , 1] 􀀉 is the velocity field transporting the mass. In Section 3, we show that, after time discretization, the unconstrained kinetic-cost case of problem (1.1) exactly recovers the optimal value and the grid-time measures of the Benamou–Brenier problem (BB) .  \nBeyond the unconstrained kinetic-cost case, Problem (1.1) also describes more general timediscrete distributional dynamics through the admissible kernel classes Ks. Depending on the application, these classes may encode support, transition","cbCaicXvYBlKRjOH","https://ap.wps.com/l/cbCaicXvYBlKRjOH","pdf",1812979,1,38,"English","en",105,"# Introduction\n## Optimizing over Markov processes\n## Sequential-coupling formulation","[{\"question\":\"What optimization problem does the paper address for Markov processes?\",\"answer\":\"It seeks a Markov process that connects two prescribed probability distributions while minimizing a specified one-step cost over intermediate time steps.\"},{\"question\":\"Why is the problem computationally difficult, and how does the paper mitigate it?\",\"answer\":\"High-dimensional state spaces make full distribution representation intractable. The paper uses sequential couplings and convex relaxations based on local marginals and cluster moments to obtain computable lower bounds.\"},{\"question\":\"How is dynamic optimal transport related to the Markov process optimization framework?\",\"answer\":\"Dynamic optimal transport is identified as a special case of the paper’s Markov process optimization problem. The paper further shows how to recover Benamou–Brenier dynamics from the relaxed solution after time discretization.\"}]",1784179780,96,{"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":86,"head_meta":88,"extra_data":90,"updated_unix":27},"convex-relaxations-for-the-optimization-of-markov-processes","",{"@graph":35,"@context":85},[36,53,68],{"@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/convex-relaxations-for-the-optimization-of-markov-processes/82344/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-17","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What optimization problem does the paper address for Markov processes?","Question",{"text":75,"@type":76},"It seeks a Markov process that connects two prescribed probability distributions while minimizing a specified one-step cost over intermediate time steps.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"Why is the problem computationally difficult, and how does the paper mitigate it?",{"text":80,"@type":76},"High-dimensional state spaces make full distribution representation intractable. The paper uses sequential couplings and convex relaxations based on local marginals and cluster moments to obtain computable lower bounds.",{"name":82,"@type":73,"acceptedAnswer":83},"How is dynamic optimal transport related to the Markov process optimization framework?",{"text":84,"@type":76},"Dynamic optimal transport is identified as a special case of the paper’s Markov process optimization problem. The paper further shows how to recover Benamou–Brenier dynamics from the relaxed solution after time discretization.","https://schema.org",{"og:url":51,"og:type":87,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":89,"canonical":51},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,135],{"id":20,"doc_module":4,"doc_module_name":45,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":46,"doc_module":4,"doc_module_name":45,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":45,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":45,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":45,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":45,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":45,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":45,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":45,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":45,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":45,"category_name":137,"show_sort_weight":106,"slug":138},19,"General","general"]