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We give an algorithm for computing saddle points. It is based on solving Lasserre\u2019s hierarchy of semidefinite relaxations. Under some genericity assumptions on defining polynomials, we show that: (i) if there exists a saddle point, our algorithm can get one by solving a finite hierarchy of Lasserre-type semidefinite relaxations; (ii) if there is no saddle point, our algorithm can detect its nonexistence.<\/jats:p>","DOI":"10.1007\/s10208-021-09526-8","type":"journal-article","created":{"date-parts":[[2021,7,9]],"date-time":"2021-07-09T19:03:46Z","timestamp":1625857426000},"page":"1133-1169","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":12,"title":["The Saddle Point Problem of Polynomials"],"prefix":"10.1007","volume":"22","author":[{"given":"Jiawang","family":"Nie","sequence":"first","affiliation":[]},{"given":"Zi","family":"Yang","sequence":"additional","affiliation":[]},{"given":"Guangming","family":"Zhou","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2021,7,9]]},"reference":[{"key":"9526_CR1","doi-asserted-by":"crossref","unstructured":"T.\u00a0Ba\u015far and G.\u00a0Olsder, Dynamic noncooperative game theory, SIAM, 1998","DOI":"10.1137\/1.9781611971132"},{"key":"9526_CR2","doi-asserted-by":"crossref","unstructured":"S.\u00a0Basu, R.\u00a0Pollack and M.-F.\u00a0Roy, Algorithms in Real Algebraic Geometry, Algorithms and Computation in Mathematics, 10, Springer-Verlag, 2003.","DOI":"10.1007\/978-3-662-05355-3"},{"key":"9526_CR3","doi-asserted-by":"publisher","unstructured":"D.\u00a0Bates, J.\u00a0Hauenstein, A.\u00a0Sommese and C.\u00a0Wampler, Bertini: Software for Numerical Algebraic Geometry, Available at bertini.nd.edu with permanent https:\/\/doi.org\/10.7274\/R0H41PB5.","DOI":"10.7274\/R0H41PB5"},{"key":"9526_CR4","unstructured":"D.\u00a0Bertsekas, Nonlinear programming, second edition, Athena Scientific, 1995."},{"key":"9526_CR5","volume-title":"Convex Analysis and Optimization","author":"D Bertsekas","year":"2003","unstructured":"D.\u00a0Bertsekas, A.\u00a0Nedi\u0107 and A.\u00a0Ozdaglar, Convex Analysis and Optimization, Athena Scientific, Belmont, 2003."},{"key":"9526_CR6","doi-asserted-by":"crossref","unstructured":"D.\u00a0Bertsimas, O.\u00a0Nohadani and M.\u00a0Teo, Robust optimization for unconstrained simulation-based problems, Operations Research \u00a0 58(2010), no.\u00a01, 161\u2013178.","DOI":"10.1287\/opre.1090.0715"},{"key":"9526_CR7","doi-asserted-by":"crossref","unstructured":"J. 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