{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,23]],"date-time":"2026-03-23T12:56:23Z","timestamp":1774270583640,"version":"3.50.1"},"reference-count":19,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2026,3,2]],"date-time":"2026-03-02T00:00:00Z","timestamp":1772409600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>We study Maximum Entropy density estimation on continuous domains under finitely many moment constraints, formulated as the minimization of the Kullback\u2013Leibler divergence with respect to a reference measure. To model uncertainty in empirical moments, constraints are relaxed through convex penalty functions, leading to an infinite-dimensional convex optimization problem over probability densities. The main contribution of this work is a rigorous convex-analytic treatment of such relaxed Maximum Entropy problems in a functional setting, without discretization or smoothness assumptions on the density. Using convex integral functionals and an extension of Fenchel duality, we show that, under mild and explicit qualification conditions, the infinite-dimensional primal problem admits a dual formulation involving only finitely many variables. This reduction can be interpreted as a continuous-domain instance of partially finite convex programming. The resulting dual problem yields explicit primal\u2013dual optimality conditions and characterizes Maximum Entropy solutions in exponential form. The proposed framework unifies exact and relaxed moment constraints, including box and quadratic relaxations, within a single variational formulation, and provides a mathematically sound foundation for relaxed Maximum Entropy methods previously studied mainly in finite or discrete settings. A brief numerical illustration demonstrates the practical tractability of the approach.<\/jats:p>","DOI":"10.3390\/e28030282","type":"journal-article","created":{"date-parts":[[2026,3,2]],"date-time":"2026-03-02T14:06:56Z","timestamp":1772460416000},"page":"282","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["On Maximum Entropy Density Estimation with Relaxed Moment Constraints"],"prefix":"10.3390","volume":"28","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-0378-4977","authenticated-orcid":false,"given":"Thi Lich","family":"Nghiem","sequence":"first","affiliation":[{"name":"Informatics Department, Thuongmai University, 79 Ho Tung Mau Street, Cau Giay District, Hanoi 100000, Vietnam"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7121-5699","authenticated-orcid":false,"given":"Pierre","family":"Mar\u00e9chal","sequence":"additional","affiliation":[{"name":"Institut de Math\u00e9matiques de Toulouse, Universit\u00e9 de Toulouse, 118 Route de Narbonne, 31062 Toulouse, Cedex 9, France"},{"name":"\u00c9quipe OPTIM-Optimisation, \u00c9cole Nationale de l\u2019Aviation Civile, 7 Avenue \u00c9douard Belin, 31055 Toulouse, Cedex 4, France"}]}],"member":"1968","published-online":{"date-parts":[[2026,3,2]]},"reference":[{"key":"ref_1","unstructured":"Fisher, N.I. 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Theory, Algorithms and Applications, Springer.","DOI":"10.1007\/978-981-97-8907-8_14"},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Mar\u00e9chal, P. (2001). A note on entropy optimization. Approximation, Optimization and Mathematical Economics, Springer.","DOI":"10.1007\/978-3-642-57592-1_18"},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Beck, A. (2017). First-Order Methods in Optimization, SIAM-Society for Industrial and Applied Mathematics.","DOI":"10.1137\/1.9781611974997"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"67","DOI":"10.1007\/s10107-018-1284-2","article-title":"A simplified view of first order methods for optimization","volume":"170","author":"Teboulle","year":"2018","journal-title":"Math. Program."},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Sun, J., V\u00fb, H., Ellerbroek, J., and Hoekstra, J.M. (2018). Supplemental dataset for \u201cWeather field reconstruction using aircraft surveillance data and a novel meteo-particle model\u201d. PLoS ONE, 13.","DOI":"10.1371\/journal.pone.0205029"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/28\/3\/282\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,3,23]],"date-time":"2026-03-23T12:04:12Z","timestamp":1774267452000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/28\/3\/282"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,3,2]]},"references-count":19,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2026,3]]}},"alternative-id":["e28030282"],"URL":"https:\/\/doi.org\/10.3390\/e28030282","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,3,2]]}}}