{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,14]],"date-time":"2026-05-14T14:26:00Z","timestamp":1778768760175,"version":"3.51.4"},"reference-count":20,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2018,12,19]],"date-time":"2018-12-19T00:00:00Z","timestamp":1545177600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["No. 11301296"],"award-info":[{"award-number":["No. 11301296"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Noticing that    E   -convexity, m-convexity and b-invexity have similar structures in their definitions, there are some possibilities to treat these three class of mappings uniformly. For this purpose, the definitions of the     ( E , m )    -convex sets and the b-    ( E , m )    -convex mappings are introduced. The properties concerning operations that preserve the     ( E , m )    -convexity of the proposed mappings are derived. The unconstrained and inequality constrained b-    ( E , m )    -convex programming are considered, where the sufficient conditions of optimality are developed and the uniqueness of the solution to the b-    ( E , m )    -convex programming are investigated. Furthermore, the sufficient optimality conditions and the Fritz\u2013John necessary optimality criteria for nonlinear multi-objective b-    ( E , m )    -convex programming are established. The Wolfe-type symmetric duality theorems under the b-    ( E , m )    -convexity, including weak and strong symmetric duality theorems, are also presented. Finally, we construct two examples in detail to show how the obtained results can be used in b-    ( E , m )    -convex programming.<\/jats:p>","DOI":"10.3390\/sym10120774","type":"journal-article","created":{"date-parts":[[2018,12,19]],"date-time":"2018-12-19T12:12:44Z","timestamp":1545221564000},"page":"774","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Optimality and Duality with Respect to b-(\u2130,m)-Convex Programming"],"prefix":"10.3390","volume":"10","author":[{"given":"Bo","family":"Yu","sequence":"first","affiliation":[{"name":"Department of Mathematics, College of Science, China Three Gorges University, Yichang 443002, China"},{"name":"Three Gorges Mathematical Research Center, China Three Gorges University, Yichang 443002, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jiagen","family":"Liao","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Science, China Three Gorges University, Yichang 443002, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6072-1788","authenticated-orcid":false,"given":"Tingsong","family":"Du","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Science, China Three Gorges University, Yichang 443002, China"},{"name":"Three Gorges Mathematical Research Center, China Three Gorges University, Yichang 443002, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2018,12,19]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"439","DOI":"10.1023\/A:1021792726715","article-title":"E-convex sets, E-convex functions, and E-convex programming","volume":"102","author":"Youness","year":"1999","journal-title":"J. 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