{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,7]],"date-time":"2026-03-07T08:36:20Z","timestamp":1772872580236,"version":"3.50.1"},"reference-count":12,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2013,10,28]],"date-time":"2013-10-28T00:00:00Z","timestamp":1382918400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>In the literature, Markowitz\u2019s mean-variance model and its variants have been shown to yield portfolios that put excessive weights on only a few assets. Many diversity constraints were proposed and added to these models to avoid such overly concentrated portfolios. However, since these diversity constraints are formulated differently, it becomes difficult to compare them and study their relationships. This paper proposes a canonical form for the commonly used diversity constraints in the literature, and shows how to transform these diversity constraints into this canonical form. Furthermore, this paper compares these diversity constraints (in the canonical form with the same upper bound) on their ability to shrink the feasible region of the portfolio optimization problem. The results show a subset relation among their feasible regions.<\/jats:p>","DOI":"10.3390\/e15114607","type":"journal-article","created":{"date-parts":[[2013,10,28]],"date-time":"2013-10-28T12:00:27Z","timestamp":1382961627000},"page":"4607-4621","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":15,"title":["On the Diversity Constraints for Portfolio Optimization"],"prefix":"10.3390","volume":"15","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-6844-1182","authenticated-orcid":false,"given":"Jun-Lin","family":"Lin","sequence":"first","affiliation":[{"name":"Department of Information Management, Yuan Ze University, 135 Yuan-Tung Road, Chungli, Taoyuan 32003, Taiwan"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2013,10,28]]},"reference":[{"key":"ref_1","first-page":"77","article-title":"Portfolio selection","volume":"7","author":"Markowitz","year":"1952","journal-title":"J. Finan."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"117","DOI":"10.3390\/e13010117","article-title":"Mean-variance-skewness-entropy measures: A multi-objective approach for portfolio selection","volume":"13","author":"Usta","year":"2011","journal-title":"Entropy"},{"key":"ref_3","unstructured":"Huang, X.X. (2010). Portfolio Analysis from Probabilistic to Credibilistic and Uncertain Approaches, Springer."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1915","DOI":"10.1093\/rfs\/hhm075","article-title":"Optimal versus naive diversification: How inefficient is the 1\/N portfolio strategy?","volume":"22","author":"DeMiguel","year":"2009","journal-title":"Rev. Financ. 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Portfolio Manag."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"798","DOI":"10.1287\/mnsc.1080.0986","article-title":"A generalized approach to portfolio optimization: Improving performance by constraining portfolio norms","volume":"55","author":"DeMiguel","year":"2009","journal-title":"Manag. Sci."},{"key":"ref_10","first-page":"160","article-title":"An entropy method for diversified fuzzy portfolio selection","volume":"14","author":"Huang","year":"2012","journal-title":"Int. J. Fuzzy Syst."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1651","DOI":"10.1111\/1540-6261.00580","article-title":"Risk reduction in large portfolios: Why imposing the wrong constraints helps","volume":"58","author":"Jagannathan","year":"2003","journal-title":"J. Financ."},{"key":"ref_12","unstructured":"Hlouskova, J., and Lee, G.S. Legal restrictions on portfolio holdings: Some empirical results; Reihe \u00d6konomie \/ Economics Series, Institut f\u00fcr H\u00f6here Studien (IHS) 93. Available online: http:\/\/hdl.handle.net\/10419\/71204."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/15\/11\/4607\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T21:50:12Z","timestamp":1760219412000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/15\/11\/4607"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2013,10,28]]},"references-count":12,"journal-issue":{"issue":"11","published-online":{"date-parts":[[2013,11]]}},"alternative-id":["e15114607"],"URL":"https:\/\/doi.org\/10.3390\/e15114607","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2013,10,28]]}}}