{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,8]],"date-time":"2026-03-08T22:16:38Z","timestamp":1773008198924,"version":"3.50.1"},"reference-count":19,"publisher":"Wiley","issue":"3","license":[{"start":{"date-parts":[[2026,1,10]],"date-time":"2026-01-10T00:00:00Z","timestamp":1768003200000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/onlinelibrary.wiley.com\/termsAndConditions#vor"},{"start":{"date-parts":[[2026,1,10]],"date-time":"2026-01-10T00:00:00Z","timestamp":1768003200000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/doi.wiley.com\/10.1002\/tdm_license_1.1"}],"content-domain":{"domain":["onlinelibrary.wiley.com"],"crossmark-restriction":true},"short-container-title":["Networks"],"published-print":{"date-parts":[[2026,4]]},"abstract":"<jats:title>ABSTRACT<\/jats:title>\n                  <jats:p>The purpose of this work is to develop an algorithmic optimization approach for a capacitated multi\u2010commodity flow problem, where the objective is to minimize the total link costs, where the cost of each arc increases convexly with its utilization. This objective is particularly relevant in telecommunication networks, where device performance can deteriorate significantly as the available bandwidth on a link becomes limited. By optimizing this convex function, traffic is efficiently distributed across the network, ensuring optimal use of available resources and preserving capacity for future demands. This paper describes the convex multi\u2010commodity flow problem and presents methodologies to solve both its Splittable and Unsplittable variants. In the Splittable version, flows can be fractionally distributed across multiple paths, while in the Unsplittable version, each commodity must be routed through a single path. Our approach employs Column\u2010Generation techniques to address the convexly increasing cost functions associated with arc utilization, effectively accommodating various forms of convex increasing cost functions, including non\u2010differentiable or black\u2010box convex increasing functions. The proposed methods demonstrate strong computational efficiency, offering a robust framework for managing network flows in complex telecommunication environments.<\/jats:p>","DOI":"10.1002\/net.70025","type":"journal-article","created":{"date-parts":[[2026,1,10]],"date-time":"2026-01-10T09:33:58Z","timestamp":1768037638000},"page":"322-338","update-policy":"https:\/\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["On the Multi\u2010Commodity Flow With Convex Objective Function: Column\u2010Generation Approaches"],"prefix":"10.1002","volume":"87","author":[{"ORCID":"https:\/\/orcid.org\/0009-0000-6686-8476","authenticated-orcid":false,"given":"Beraud\u2010Sudreau","family":"Guillaume","sequence":"first","affiliation":[{"name":"Huawei Technologies Ltd., Paris Research Center  Boulogne\u2010Billancourt France"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4919-9036","authenticated-orcid":false,"given":"L\u00e9tocart","family":"Lucas","sequence":"additional","affiliation":[{"name":"Universit\u00e9 Sorbonne Paris Nord, CNRS Laboratoire d'Informatique de Paris Nord, LIPN  Villetaneuse France"}]},{"given":"Magnouche","family":"Youcef","sequence":"additional","affiliation":[{"name":"Huawei Technologies Ltd., Paris Research Center  Boulogne\u2010Billancourt France"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8528-6698","authenticated-orcid":false,"given":"Martin","family":"S\u00e9bastien","sequence":"additional","affiliation":[{"name":"Huawei Technologies Ltd., Paris Research Center  Boulogne\u2010Billancourt France"}]}],"member":"311","published-online":{"date-parts":[[2026,1,10]]},"reference":[{"key":"e_1_2_7_2_1","doi-asserted-by":"publisher","DOI":"10.1109\/COMST.2008.4483669"},{"key":"e_1_2_7_3_1","doi-asserted-by":"publisher","DOI":"10.1287\/mnsc.5.1.97"},{"key":"e_1_2_7_4_1","volume-title":"Mathematical Programming Algorithms for Large Scale Network Equilibrium and Network Design Problems","author":"Leblanc L. 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