{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,6]],"date-time":"2026-10-06T13:48:26Z","timestamp":1791294506395,"version":"4.1.0"},"reference-count":17,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"5","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Optim."],"published-print":{"date-parts":[[2010,1]]},"abstract":"<jats:p>We propose an algorithm for solving optimization problems defined on a subset of the cone of symmetric positive semidefinite matrices. This algorithm relies on the factorization $X=YY^T$, where the number of columns of Y fixes an upper bound on the rank of the positive semidefinite matrix X. It is thus very effective for solving problems that have a low-rank solution. The factorization $X=YY^T$ leads to a reformulation of the original problem as an optimization on a particular quotient manifold. The present paper discusses the geometry of that manifold and derives a second-order optimization method with guaranteed quadratic convergence. It furthermore provides some conditions on the rank of the factorization to ensure equivalence with the original problem. In contrast to existing methods, the proposed algorithm converges monotonically to the sought solution. Its numerical efficiency is evaluated on two applications: the maximal cut of a graph and the problem of sparse principal component analysis.<\/jats:p>","DOI":"10.1137\/080731359","type":"journal-article","created":{"date-parts":[[2010,5,28]],"date-time":"2010-05-28T18:33:35Z","timestamp":1275071615000},"page":"2327-2351","source":"Crossref","is-referenced-by-count":157,"title":["Low-Rank Optimization on the Cone of Positive Semidefinite Matrices"],"prefix":"10.1137","volume":"20","author":[{"given":"M.","family":"Journ\u00e9e","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"F.","family":"Bach","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"P.-A.","family":"Absil","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"R.","family":"Sepulchre","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2010,5,28]]},"reference":[{"key":"R1","doi-asserted-by":"publisher","DOI":"10.1007\/s10208-005-0179-9"},{"key":"R2","unstructured":"P.A. Absil and J\u00e9r\u00f4me Malick,\n                      Constructing retractions on matrix manifolds\n                      , in preparation, 2009."},{"key":"R3","doi-asserted-by":"publisher","DOI":"10.1162\/neco.2008.04-08-749"},{"key":"R4","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":"R5","doi-asserted-by":"publisher","DOI":"10.1007\/s10107-002-0352-8"},{"key":"R6","doi-asserted-by":"crossref","unstructured":"S. Boyd and L. Vandenberghe,\n                      Convex Optimization\n                      , Cambridge University Press, Cambridge, UK, 2004.","DOI":"10.1017\/CBO9780511804441"},{"key":"R7","doi-asserted-by":"crossref","unstructured":"A. d'Aspremont, F. R. Bach, and L. El Ghaoui,\n                      Full regularization path for sparse principal component analysis\n                      , in Proceedings of the 24th International Conference on Machine Learning ICML 2007, Corvallis, OR, International Machine Learning Society, 2007, pp. 177\u2013184.","DOI":"10.1145\/1273496.1273519"},{"key":"R8","doi-asserted-by":"publisher","DOI":"10.1137\/050645506"},{"key":"R9","doi-asserted-by":"publisher","DOI":"10.1016\/j.laa.2006.11.024"},{"key":"R10","doi-asserted-by":"publisher","DOI":"10.1145\/227683.227684"},{"key":"R11","unstructured":"M. Journ\u00e9e,\n                      Geometric Algorithms for Component Analysis with a View to Gene Expression Data Analysis\n                      , Ph.D. thesis, Universit\u00e9 de Li\u00e8ge, Li\u00e8ge, Belgium, 2009."},{"key":"R12","doi-asserted-by":"publisher","DOI":"10.1287\/moor.21.3.576"},{"key":"R13","doi-asserted-by":"publisher","DOI":"10.1137\/S089547980036838X"},{"key":"R14","unstructured":"J. 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