{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,17]],"date-time":"2026-09-17T03:37:33Z","timestamp":1789616253628,"version":"build-2803163510"},"reference-count":31,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"2","funder":[{"DOI":"10.13039\/501100001659","name":"Deutsche Forschungsgemeinschaft","doi-asserted-by":"publisher","award":["SCHW1723\/1-1"],"award-info":[{"award-number":["SCHW1723\/1-1"]}],"id":[{"id":"10.13039\/501100001659","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Optim."],"published-print":{"date-parts":[[2018,1]]},"abstract":"<jats:p>Consider a finite system of nonstrict real polynomial inequalities and suppose its solution set $S\\subseteq\\mathbb R^n$ is convex, has nonempty interior, and is compact. Suppose that the system satisfies the Archimedean condition, which is slightly stronger than the compactness of $S$. Suppose that each defining polynomial satisfies a second order strict quasiconcavity condition where it vanishes on $S$ (which is very natural because of the convexity of $S$) or its Hessian has a certain matrix sums of squares certificate for negative-semidefiniteness on $S$ (fulfilled trivially by linear polynomials). Then we show that the system possesses an exact Lasserre relaxation. In their seminal work of 2009, Helton and Nie showed under the same conditions that $S$ is the projection of a spectrahedron, i.e., it has a semidefinite representation. The semidefinite representation used by Helton and Nie arises from glueing together Lasserre relaxations of many small pieces obtained in a nonconstructive way. By refining and varying their approach, we show that we can simply take a Lasserre relaxation of the original system itself. Such a result was provided by Helton and Nie with much more machinery only under very technical conditions and after changing the description of $S$.<\/jats:p>","DOI":"10.1137\/17m1128290","type":"journal-article","created":{"date-parts":[[2018,6,14]],"date-time":"2018-06-14T12:29:58Z","timestamp":1528979398000},"page":"1796-1816","source":"Crossref","is-referenced-by-count":5,"title":["On the Exactness of Lasserre Relaxations for Compact Convex Basic Closed Semialgebraic Sets"],"prefix":"10.1137","volume":"28","author":[{"given":"Tom-Lukas","family":"Kriel","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Markus","family":"Schweighofer","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2018,6,14]]},"reference":[{"key":"atypb1","doi-asserted-by":"crossref","unstructured":"S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan,\n                      Linear Matrix Inequalities in System and Control Theory\n                      , Stud. Appl. Math. 15, SIAM, Philadelphia, 1994.","DOI":"10.1137\/1.9781611970777"},{"key":"atypb2","doi-asserted-by":"crossref","unstructured":"W. Forst and D. Hoffmann,\n                      Optimization: Theory and Practice\n                      , Springer Undergrad. Texts Math. Technol., Springer, New York, 2010.","DOI":"10.1007\/978-0-387-78977-4"},{"key":"atypb3","doi-asserted-by":"publisher","DOI":"10.1112\/S0024609303003059"},{"key":"atypb4","doi-asserted-by":"publisher","DOI":"10.1137\/100801913"},{"key":"atypb5","doi-asserted-by":"publisher","DOI":"10.1007\/s10107-008-0240-y"},{"key":"atypb6","doi-asserted-by":"publisher","DOI":"10.1137\/07070526X"},{"key":"atypb7","doi-asserted-by":"publisher","DOI":"10.1137\/100814147"},{"key":"atypb8","unstructured":"T.L. Kriel,\n                      A new proof for the existence of degree bounds for Putinar's Positivstellensatz\n                      , in Ordered Algebraic Structures and Related Topics, Contemp. Math 697, F. Broglia, F. Delon, M. Dickmann, D. Gondard-Cozette, and V. A. Powers, eds., Amer. Math. Soc., Providence, RI, 2017, pp. 203-209."},{"key":"atypb9","doi-asserted-by":"crossref","unstructured":"T.L. Kriel and M. Schweighofer,\n                      On the Exactness of Lasserre Relaxations and Pure States over Real Closed Fields\n                      , preprint, arXiv:1710.07521, 2017.","DOI":"10.1007\/s10208-018-9406-z"},{"key":"atypb10","doi-asserted-by":"publisher","DOI":"10.1137\/S1052623400366802"},{"key":"atypb11","doi-asserted-by":"publisher","DOI":"10.1007\/s10107-008-0222-0"},{"key":"atypb12","first-page":"157","author":"Laurent M.","year":"2009","journal-title":"New York"},{"key":"atypb13","doi-asserted-by":"crossref","unstructured":"M. Marshall,\n                      Positive Polynomials and Sums of Squares\n                      , Math. Surveys Monogr. 146, Amer. Math. Soc., Providence, RI, 2008.","DOI":"10.1090\/surv\/146"},{"key":"atypb14","first-page":"413","author":"Nemirovski A.","year":"2006","journal-title":"Zu\u0308rich"},{"key":"atypb15","doi-asserted-by":"publisher","DOI":"10.1016\/j.laa.2010.02.005"},{"key":"atypb16","doi-asserted-by":"crossref","unstructured":"Y. Nesterov and A. Nemirovskii,\n                      Interior-Point Polynomial Algorithms in Convex Programming\n                      , Stud. Appl. Numer. Math. 13, SIAM, Philadelphia, PA, 1994.","DOI":"10.1137\/1.9781611970791"},{"key":"atypb17","doi-asserted-by":"publisher","DOI":"10.1137\/090750196"},{"key":"atypb18","unstructured":"T. Netzer and R. Sinn,\n                      A Note on the Convex Hull of Finitely Many Projections of Spectrahedra\n                      , preprint, arXiv:0908.3386, 2009."},{"key":"atypb19","doi-asserted-by":"publisher","DOI":"10.1016\/j.jco.2006.07.002"},{"key":"atypb20","doi-asserted-by":"crossref","unstructured":"A. Prestel and C. N. Delzell,\n                      Positive Polynomials: From Hilbert's 17th Problem to Real Algebra\n                      , Springer Monogr. Math., Springer, Berlin, 2001.","DOI":"10.1007\/978-3-662-04648-7"},{"key":"atypb21","unstructured":"D. Plaumann,\n                      Geometry of Linear Matrix Inequalities\n                      , Lecture notes, Universita\u0308t Konstanz, 2013; available athttp:\/\/www.math.uni-konstanz.de\/~plaumann\/LMI13\/lmi.pdf."},{"key":"atypb22","doi-asserted-by":"publisher","DOI":"10.1512\/iumj.1993.42.42045"},{"key":"atypb23","first-page":"522","volume":"9","author":"Rubin H.","year":"1958","journal-title":"Proc. Amer. Math. Soc."},{"key":"atypb24","doi-asserted-by":"publisher","DOI":"10.1137\/17M1115113"},{"key":"atypb25","doi-asserted-by":"publisher","DOI":"10.1137\/17M1118981"},{"key":"atypb26","unstructured":"A. Schrijver,\n                      Theory of Linear and Integer Programming\n                      , Wiley-Intersci. Ser. Discrete Math. Optim., Wiley, Hoboken, NJ, 1986."},{"key":"atypb27","doi-asserted-by":"publisher","DOI":"10.1007\/BF01446568"},{"key":"atypb28","doi-asserted-by":"publisher","DOI":"10.1007\/s10107-005-0684-2"},{"key":"atypb29","unstructured":"R. Sinn,\n                      Spectrahedra and a relaxation of convex, semialgebraic sets\n                      , Diplomarbeit, Universita\u0308t Konstanz, Konstanz, 2010; available athttp:\/\/www.math.uni-konstanz.de\/~sinn\/thesis.pdf."},{"key":"atypb30","unstructured":"D. E. Speyer,\n                      User Answer\n                      , MathOverflow, 2009; available athttp:\/\/mathoverflow.net\/a\/4410."},{"key":"atypb31","doi-asserted-by":"publisher","DOI":"10.1090\/noti1116"}],"container-title":["SIAM Journal on Optimization"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/17M1128290","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T19:03:59Z","timestamp":1787339039000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/17M1128290"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,1]]},"references-count":31,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2018,1]]}},"alternative-id":["10.1137\/17M1128290"],"URL":"https:\/\/doi.org\/10.1137\/17m1128290","relation":{},"ISSN":["1052-6234","1095-7189"],"issn-type":[{"value":"1052-6234","type":"print"},{"value":"1095-7189","type":"electronic"}],"subject":[],"published":{"date-parts":[[2018,1]]}}}