{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T00:58:10Z","timestamp":1760144290349,"version":"build-2065373602"},"reference-count":51,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2024,4,1]],"date-time":"2024-04-01T00:00:00Z","timestamp":1711929600000},"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>It is well known that in information theory\u2014as well as in the adjacent fields of statistics, machine learning and artificial intelligence\u2014it is essential to quantify the dissimilarity between objects of uncertain\/imprecise\/inexact\/vague information; correspondingly, constrained optimization is of great importance, too. In view of this, we define the dissimilarity-measure-natured generalized \u03c6\u2013divergences between fuzzy sets, \u03bd\u2013rung orthopair fuzzy sets, extended representation type \u03bd\u2013rung orthopair fuzzy sets as well as between those fuzzy set types and vectors. For those, we present how to tackle corresponding constrained minimization problems by appropriately applying our recently developed dimension-free bare (pure) simulation method. An analogous program is carried out by defining and optimizing generalized \u03c6\u2013divergences between (rescaled) basic belief assignments as well as between (rescaled) basic belief assignments and vectors.<\/jats:p>","DOI":"10.3390\/e26040312","type":"journal-article","created":{"date-parts":[[2024,4,2]],"date-time":"2024-04-02T01:26:10Z","timestamp":1712021170000},"page":"312","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Some Theoretical Foundations of Bare-Simulation Optimization of Some Directed Distances between Fuzzy Sets Respectively Basic Belief Assignments"],"prefix":"10.3390","volume":"26","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-6301-5531","authenticated-orcid":false,"given":"Michel","family":"Broniatowski","sequence":"first","affiliation":[{"name":"Laboratoire de Probabilit\u00e9s, Statistique et Mod\u00e9lisation, Sorbonne Universit\u00e9, 4 Place Jussieu, 75252 Paris, France"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7831-4558","authenticated-orcid":false,"given":"Wolfgang","family":"Stummer","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Friedrich-Alexander-Universit\u00e4t Erlangen\u2013N\u00fcrnberg, Cauerstrasse 11, 91058 Erlangen, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,4,1]]},"reference":[{"key":"ref_1","first-page":"85","article-title":"Eine informationstheoretische Ungleichung und ihre Anwendung auf den Beweis der Ergodizit\u00e4t von Markoffschen Ketten","volume":"A-8","year":"1963","journal-title":"Publ. 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