{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,29]],"date-time":"2026-09-29T17:21:21Z","timestamp":1790702481546,"version":"4.1.0"},"reference-count":35,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Optim."],"published-print":{"date-parts":[[2018,1]]},"abstract":"<jats:p>In this paper, a Riemannian BFGS method for minimizing a smooth function on a Riemannian manifold is defined, based on a Riemannian generalization of a cautious update and a weak line search condition. It is proven that the Riemannian BFGS method converges (i) globally to stationary points without assuming the objective function to be convex and (ii) superlinearly to a nondegenerate minimizer. Using the weak line search condition removes the need for information from differentiated retraction. The joint matrix diagonalization problem is chosen to demonstrate the performance of the algorithms with various parameters, line search conditions, and pairs of retraction and vector transport. A preliminary version can be found in [ Numerical Mathematics and Advanced Applications: ENUMATH 2015, Lect. Notes Comput. Sci. Eng. 112, Springer, New York, 2016, pp. 627--634].<\/jats:p>","DOI":"10.1137\/17m1127582","type":"journal-article","created":{"date-parts":[[2018,2,15]],"date-time":"2018-02-15T13:50:06Z","timestamp":1518702606000},"page":"470-495","source":"Crossref","is-referenced-by-count":63,"title":["A Riemannian BFGS Method Without Differentiated Retraction for Nonconvex Optimization Problems"],"prefix":"10.1137","volume":"28","author":[{"given":"Wen","family":"Huang","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"P.-A.","family":"Absil","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"K. A.","family":"Gallivan","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2018,2,15]]},"reference":[{"key":"atypb1","doi-asserted-by":"publisher","DOI":"10.1093\/imanum\/22.3.359"},{"key":"atypb2","doi-asserted-by":"crossref","unstructured":"P.A. Absil, R. Mahony, and R. Sepulchre,\n                      Optimization Algorithms on Matrix Manifolds\n                      , Princeton University Press, Princeton, NJ, 2008.","DOI":"10.1515\/9781400830244"},{"key":"atypb3","doi-asserted-by":"publisher","DOI":"10.1016\/j.laa.2015.02.027"},{"key":"atypb4","unstructured":"I. Brace and J. H. Manton,\n                      An improved BFGS-on-manifold algorithm for computing weighted low rank approximations\n                      , in Proceedings of 17th International Symposium on Mathematical Theory of Networks and Systems, 2006, pp. 1735-1738."},{"key":"atypb5","doi-asserted-by":"publisher","DOI":"10.1137\/0726042"},{"key":"atypb6","unstructured":"W. M. Boothby,\n                      An Introduction to Differentiable Manifolds and Riemannian Geometry\n                      , 2nd ed., Academic Press, New York, 1986."},{"key":"atypb7","unstructured":"A. Cherian and S. Sra,\n                      Riemannian Dictionary Learning and Sparse Coding for Positive Definite Matrices\n                      , CoRR,abs\/1507.02772, 2015."},{"key":"atypb8","doi-asserted-by":"publisher","DOI":"10.1007\/s10107-012-0522-2"},{"key":"atypb9","unstructured":"J. E. Dennis and R. B. Schnabel,\n                      Numerical Methods for Unconstrained Optimization and Nonlinear Equations\n                      , Springer, New York, 1983."},{"key":"atypb10","doi-asserted-by":"publisher","DOI":"10.1007\/BF00934767"},{"key":"atypb11","first-page":"281","author":"Gallivan K. A.","year":"2012","journal-title":"New York"},{"key":"atypb12","doi-asserted-by":"publisher","DOI":"10.1007\/s10107-014-0765-1"},{"key":"atypb13","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-319-39929-4_60"},{"key":"atypb14","doi-asserted-by":"publisher","DOI":"10.1007\/s00211-016-0848-4"},{"key":"atypb15","unstructured":"W. Huang, P.A. Absil, K. A. Gallivan, and P. Hand,\n                      ROPTLIB: An Object-Oriented C$++$ Library for Optimization on Riemannian Manifolds\n                      , Technical Report FSU16-14, Florida State University, 2016."},{"key":"atypb16","doi-asserted-by":"publisher","DOI":"10.1137\/140955483"},{"key":"atypb17","doi-asserted-by":"publisher","DOI":"10.1007\/s10851-015-0606-8"},{"key":"atypb18","doi-asserted-by":"publisher","DOI":"10.1137\/16M1072838"},{"key":"atypb19","unstructured":"W. Huang,\n                      Optimization Algorithms on Riemannian Manifolds with Applications\n                      , Ph.D. thesis, Department of Mathematics, Florida State University, 2013."},{"key":"atypb20","doi-asserted-by":"publisher","DOI":"10.1016\/S0377-0427(00)00540-9"},{"key":"atypb21","doi-asserted-by":"publisher","DOI":"10.1137\/S1052623499354242"},{"key":"atypb22","unstructured":"B. Mishra,\n                      A Riemannian Approach to Large-Scale Constrained Least-Squares with Symmetries\n                      , Ph.D. thesis, University of Liege, 2014."},{"key":"atypb23","unstructured":"J. Nocedal and S. J. Wright,\n                      Numerical Optimization\n                      , 2nd ed., Springer, New York, 2006."},{"key":"atypb24","doi-asserted-by":"crossref","unstructured":"C. Qi, K. A. Gallivan, and P.A. Absil,\n                      R\n                      iemannian BFGS algorithm with applications, in Recent Advances in Optimization and Its Applications in Engineering, 2010, Springer, New York, pp. 183-192.","DOI":"10.1007\/978-3-642-12598-0_16"},{"key":"atypb25","doi-asserted-by":"publisher","DOI":"10.1137\/11082885X"},{"key":"atypb26","doi-asserted-by":"publisher","DOI":"10.1137\/120872887"},{"key":"atypb27","unstructured":"M. Seibert, M. Kleinsteuber, and K. Hu\u0308per,\n                      Properties of the BFGS method on Riemannian manifolds\n                      , in Mathematical System Theory: Festschrift in Honor of Uwe Helmke on the Occasion of His Sixtieth Birthday, K. Hu\u0308per and J. Trumpf, eds., 2013, pp. 395-412."},{"key":"atypb28","doi-asserted-by":"publisher","DOI":"10.1137\/090763172"},{"key":"atypb29","first-page":"354","author":"Theis F. J.","year":"2009","journal-title":"New York"},{"key":"atypb30","doi-asserted-by":"publisher","DOI":"10.1137\/110845768"},{"key":"atypb31","doi-asserted-by":"publisher","DOI":"10.1016\/j.amc.2005.08.027"},{"key":"atypb32","doi-asserted-by":"publisher","DOI":"10.1023\/B:COAP.0000044184.25410.39"},{"key":"atypb33","unstructured":"X. Yuan, W. Huang, P.A. Absil, and K. A. Gallivan,\n                      A Riemannian Quasi-Newton Method for Computing the Karcher Mean of Symmetric Positive Definite Matrices\n                      , Technical Report FSU17-02, Florida State University, 2017,http:\/\/www.math.fsu.edu\/~whuang2\/papers\/RMKMSPDM.htm."},{"key":"atypb34","first-page":"727","author":"You Y.","year":"2015","journal-title":"Proceedings of the IEEE Global Conference on Signal and Information Processing"},{"key":"atypb35","doi-asserted-by":"publisher","DOI":"10.1007\/s10589-016-9883-4"}],"container-title":["SIAM Journal on Optimization"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/17M1127582","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T18:15:24Z","timestamp":1787336124000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/17M1127582"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,1]]},"references-count":35,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2018,1]]}},"alternative-id":["10.1137\/17M1127582"],"URL":"https:\/\/doi.org\/10.1137\/17m1127582","relation":{},"ISSN":["1052-6234","1095-7189"],"issn-type":[{"value":"1052-6234","type":"print"},{"value":"1095-7189","type":"electronic"}],"subject":[],"published":{"date-parts":[[2018,1]]}}}