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Optim."],"published-print":{"date-parts":[[2025,12,31]]},"abstract":"<jats:p>Abstract.<\/jats:p>\n                  <jats:p>We determine the Euclidean distance degrees of the three most common manifolds arising in manifold optimization: flag, Grassmann, and Stiefel manifolds. For the Grassmannian, we will also determine the Euclidean distance degree of an important class of Schubert varieties that often appear in applications. Our technique goes further than furnishing the value of the Euclidean distance degree; it will also yield closed-form expressions for all stationary points of the Euclidean distance function in each instance. We will discuss the implications of these results on the tractability of manifold optimization problems.<\/jats:p>","DOI":"10.1137\/25m1735032","type":"journal-article","created":{"date-parts":[[2025,10,22]],"date-time":"2025-10-22T07:10:28Z","timestamp":1761117028000},"page":"2402-2422","source":"Crossref","is-referenced-by-count":5,"title":["Euclidean Distance Degree in Manifold Optimization"],"prefix":"10.1137","volume":"35","author":[{"given":"Zehua","family":"Lai","sequence":"first","affiliation":[{"name":"Department of Mathematics, University of Texas, Austin, TX 78712 USA."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Lek-Heng","family":"Lim","sequence":"additional","affiliation":[{"name":"Computational and Applied Mathematics Initiative, Department of Statistics, University of Chicago, Chicago, IL 60637-1514 USA."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3193-7264","authenticated-orcid":true,"given":"Ke","family":"Ye","sequence":"additional","affiliation":[{"name":"State Key Laboratory of Mathematical Sciences, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190 China."}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2025,10,22]]},"reference":[{"key":"ref1","doi-asserted-by":"publisher","DOI":"10.1515\/9781400830244"},{"key":"ref2","doi-asserted-by":"publisher","DOI":"10.2140\/ant.2018.12.2005"},{"key":"ref3","doi-asserted-by":"publisher","DOI":"10.1016\/j.laa.2014.11.012"},{"key":"ref4","doi-asserted-by":"publisher","DOI":"10.1007\/s11856-018-1767-0"},{"key":"ref5","doi-asserted-by":"publisher","DOI":"10.1137\/S106482759630992X"},{"key":"ref6","doi-asserted-by":"publisher","DOI":"10.1007\/s10208-021-09534-8"},{"key":"ref7","doi-asserted-by":"publisher","DOI":"10.1137\/20M1323527"},{"key":"ref8","doi-asserted-by":"publisher","DOI":"10.1007\/s10107-019-01399-8"},{"key":"ref9","doi-asserted-by":"publisher","DOI":"10.1137\/18M1222235"},{"key":"ref10","doi-asserted-by":"crossref","unstructured":"J. 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