{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,8]],"date-time":"2026-05-08T09:10:46Z","timestamp":1778231446177,"version":"3.51.4"},"reference-count":65,"publisher":"Institute for Operations Research and the Management Sciences (INFORMS)","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Mathematics of OR"],"published-print":{"date-parts":[[2026,5]]},"abstract":"<jats:p>This paper studies Positivstellens\u00e4tze and moment problems for sets K that are given by universal quantifiers. Let Q be the closed set of universal quantifiers. Fix a finite nonnegative Borel measure whose support is Q and assume it satisfies the multivariate Carleman condition. First, we prove a Positivstellensatz with universal quantifiers: if a polynomial f is positive on K, then f belongs to the associated quadratic module, under the archimedeanness assumption. Second, we prove some necessary and sufficient conditions for a full (or truncated) multisequence to admit a representing measure supported in K. In particular, the classical flat extension theorem of Curto and Fialkow is generalized to truncated moment problems on such a set K. Third, we present applications of the above Positivstellensatz and moment problems in semi-infinite optimization, where feasible sets are given by infinitely many constraints with universal quantifiers. This results in a new hierarchy of Moment-SOS relaxations. Its convergence is shown under some usual assumptions. The quantifier set Q is allowed to be non-semialgebraic, which makes it possible to solve some optimization problems with non-semialgebraic constraints.<\/jats:p>\n                  <jats:p>Funding: X. Hu and J. Nie are partially supported by the NSF [Grant DMS-2110780]. I. Klep is supported by the Slovenian Research Agency program P1-0222 [also Grants J1-50002, J1-60011, J1-50001, J1-2453, N1-0217, and J1-3004] and was partially supported by the Marsden Fund Council of the Royal Society of New Zealand. I. Klep\u2019s work was partly performed within the project COMPUTE, funded within the QuantERA II program that has received funding from the EU\u2019s H2020 research and innovation program under the GA No 101017733.<\/jats:p>","DOI":"10.1287\/moor.2024.0402","type":"journal-article","created":{"date-parts":[[2025,5,7]],"date-time":"2025-05-07T12:53:02Z","timestamp":1746622382000},"page":"1037-1060","source":"Crossref","is-referenced-by-count":1,"title":["Positivstellens\u00e4tze and Moment Problems with Universal Quantifiers"],"prefix":"10.1287","volume":"51","author":[{"given":"Xiaomeng","family":"Hu","sequence":"first","affiliation":[{"name":"Department of Mathematics, University of California San Diego, La Jolla, California 92093"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5245-8484","authenticated-orcid":false,"given":"Igor","family":"Klep","sequence":"additional","affiliation":[{"name":"Faculty of Mathematics and Physics, University of Ljubljana, 1000 Ljubljana, Slovenia; and Faculty of Mathematics, Science and Information Technology, University of Primorska, 6000 Koper, Slovenia; and Institute of Mathematics, Physics and Mechanics, 1000 Ljubljana, Slovenia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2637-8274","authenticated-orcid":false,"given":"Jiawang","family":"Nie","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of California San Diego, La Jolla, California 92093"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"109","reference":[{"key":"B1","volume-title":"The Classical Moment Problem and Some Related Questions in Analysis","author":"Akhiezer NI","year":"1965"},{"key":"B2","doi-asserted-by":"publisher","DOI":"10.1090\/gsm\/054"},{"key":"B3","doi-asserted-by":"publisher","DOI":"10.1007\/3-540-33099-2"},{"key":"B4","doi-asserted-by":"publisher","DOI":"10.1090\/psapm\/037\/921086"},{"key":"B5","doi-asserted-by":"publisher","DOI":"10.1016\/j.aim.2013.09.020"},{"key":"B6","doi-asserted-by":"publisher","DOI":"10.7900\/jot.2019mar15.2239"},{"key":"B7","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-03718-8"},{"key":"B8","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-08-04588-1"},{"key":"B9","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-2011-05225-6"},{"key":"B10","doi-asserted-by":"publisher","DOI":"10.1007\/BF01582053"},{"key":"B11","doi-asserted-by":"publisher","DOI":"10.1090\/memo\/0568"},{"issue":"1","key":"B12","first-page":"189","volume":"54","author":"Curto RE","year":"2005","journal-title":"J. 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