{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T19:41:17Z","timestamp":1787341277788,"version":"build-2736575974"},"reference-count":34,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Optim."],"published-print":{"date-parts":[[2026,6,30]]},"abstract":"<jats:p>Abstract.<\/jats:p>\n                  <jats:p>We introduce a novel moment-sum-of-squares (SOS) hierarchy of lower bounds for a polynomial optimization problem whose feasible set is defined by polynomial matrix inequalities. Our hierarchy avoids the Kronecker product structure in Hol and Scherer\u2019s hierarchy, thus resulting in smaller-sized semidefinite programs. Our approach involves utilizing a penalty function framework to directly address the matrix-based constraint, which is applicable to both discrete and continuous polynomial optimization problems. We investigate the convergence rates of these bounds for both types of problems. The proposed method yields a variant of Putinar\u2019s theorem, tailored for positive polynomials on a compact set [Formula: see text] defined by a polynomial matrix inequality. More specifically, we derive novel insights into the bounds on the degree of the SOS polynomials required to certify positivity over [Formula: see text], based on Jackson\u2019s theorem and a variant of the \u0141ojasiewicz inequality in the matrix setting.<\/jats:p>","DOI":"10.1137\/24m1670184","type":"journal-article","created":{"date-parts":[[2026,4,29]],"date-time":"2026-04-29T07:06:05Z","timestamp":1777446365000},"page":"760-790","source":"Crossref","is-referenced-by-count":0,"title":["Convergence Rates of Sum-of-Squares Hierarchies for Polynomial Semidefinite Programs"],"prefix":"10.1137","volume":"36","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7418-2527","authenticated-orcid":true,"given":"Hoang Anh","family":"Tran","sequence":"first","affiliation":[{"name":"Department of Mathematics, National University of Singapore, 119076, Singapore."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7204-8933","authenticated-orcid":true,"given":"Kim-Chuan","family":"Toh","sequence":"additional","affiliation":[{"name":"Department of Mathematics, National University of Singapore, 119076, Singapore."}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2026,4,29]]},"reference":[{"key":"ref1","doi-asserted-by":"publisher","DOI":"10.1007\/s10107-022-01877-6"},{"key":"ref2","doi-asserted-by":"publisher","DOI":"10.1016\/j.jalgebra.2024.08.022"},{"key":"ref3","first-page":"49","volume":"2","author":"Bernstein S. 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Scherer, Sum of squares relaxations for polynomial semidefinite programming, in Proceedings of the Symposium on Mathematical Theory of Networks and Systems (MTNS), 2004."},{"key":"ref14","doi-asserted-by":"publisher","DOI":"10.1007\/s11590-022-01940-3"},{"key":"ref15","doi-asserted-by":"publisher","DOI":"10.1137\/24M1675461"},{"key":"ref16","doi-asserted-by":"publisher","DOI":"10.1137\/23M1605430"},{"key":"ref17","doi-asserted-by":"publisher","DOI":"10.1109\/TAC.2009.2017159"},{"key":"ref18","doi-asserted-by":"publisher","DOI":"10.1007\/s11590-015-0868-5"},{"key":"ref19","unstructured":"M. Kojima, Sums of Squares Relaxations of Polynomial Semidefinite Programs, Technical Report, Tokyo Institute of Technology, 2003."},{"key":"ref20","unstructured":"J. Lasserre, A Unified Criterion for Positive Definiteness and Semidefiniteness, Technical report 05283, LAAS-CNRS, Toulouse, France, 2005."},{"key":"ref21","doi-asserted-by":"crossref","unstructured":"J. B. 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