{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,30]],"date-time":"2026-03-30T19:15:13Z","timestamp":1774898113417,"version":"3.50.1"},"reference-count":105,"publisher":"Emerald","issue":"1-2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2016,8,4]]},"abstract":"<jats:p>Finding low-rank solutions of semidefinite programs is important in many applications. For example, semidefinite programs that arise as relaxations of polynomial optimization problems are exact relaxations when the semidefinite program has a rank-1 solution. Unfortunately, computing a minimum-rank solution of a semidefinite program is an NP-hard problem. In this paper we review the theory of low-rank semidefinite programming, presenting theorems that guarantee the existence of a low-rank solution, heuristics for computing low-rank solutions, and algorithms for finding low-rank approximate solutions. Then we present applications of the theory to trust-region problems and signal processing.<\/jats:p>","DOI":"10.1561\/2400000009","type":"journal-article","created":{"date-parts":[[2016,8,4]],"date-time":"2016-08-04T11:04:51Z","timestamp":1470308691000},"page":"1-156","source":"Crossref","is-referenced-by-count":28,"title":["Low-Rank Semidefinite Programming: Theory and Applications"],"prefix":"10.1561","volume":"2","author":[{"given":"Alex","family":"Lemon","sequence":"first","affiliation":[{"name":"Stanford University"}]},{"given":"Anthony Man-Cho","family":"So","sequence":"additional","affiliation":[{"name":"The Chinese University of Hong Kong"}]},{"given":"Yinyu","family":"Ye","sequence":"additional","affiliation":[{"name":"Stanford University"}]}],"member":"140","published-online":{"date-parts":[[2016,8,4]]},"reference":[{"issue":"4","key":"2026033014415864500_ref001","doi-asserted-by":"crossref","first-page":"965","DOI":"10.1287\/moor.1080.0331","article-title":"On the low rank solutions for linear matrix inequalities","volume":"33","author":"Ai","year":"2008","journal-title":"Mathematics of Operations Research"},{"issue":"3","key":"2026033014415864500_ref002","doi-asserted-by":"crossref","first-page":"253","DOI":"10.1007\/s00365-007-9003-x","article-title":"A simple proof of the restricted isometry property for random matrices","volume":"28","author":"Baraniuk","year":"2008","journal-title":"Constructive Approximation"},{"issue":"2","key":"2026033014415864500_ref003","doi-asserted-by":"crossref","first-page":"189","DOI":"10.1007\/BF02574037","article-title":"Problems of distance geometry and convex properties of quadratic maps","volume":"13","author":"Barvinok","year":"1995","journal-title":"Discrete & Computational Geometry"},{"issue":"1","key":"2026033014415864500_ref004","doi-asserted-by":"crossref","first-page":"23","DOI":"10.1007\/s004540010074","article-title":"A remark on the rank of positive semidefinite matrices subject to affine constraints","volume":"25","author":"Barvinok","year":"2001","journal-title":"Discrete & Computational Geometry"},{"issue":"3","key":"2026033014415864500_ref005","doi-asserted-by":"crossref","first-page":"844","DOI":"10.1137\/050644471","article-title":"Strong duality in nonconvex quadratic optimization with two quadratic constraints","volume":"17","author":"Beck","year":"2006","journal-title":"SIAM Journal on Optimization"},{"key":"2026033014415864500_ref006","volume-title":"Robust Optimization. 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