{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,2,21]],"date-time":"2025-02-21T10:42:25Z","timestamp":1740134545452,"version":"3.37.3"},"reference-count":34,"publisher":"Wiley","license":[{"start":{"date-parts":[[2014,1,1]],"date-time":"2014-01-01T00:00:00Z","timestamp":1388534400000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["70831005","11171237"],"award-info":[{"award-number":["70831005","11171237"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["70831005","11171237"],"award-info":[{"award-number":["70831005","11171237"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Journal of Applied Mathematics"],"published-print":{"date-parts":[[2014]]},"abstract":"<jats:p>We introduce a class of functions called geodesic<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M1\"><mml:mrow><mml:mi>B<\/mml:mi><\/mml:mrow><\/mml:math>-preinvex and geodesic<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M2\"><mml:mrow><mml:mi>B<\/mml:mi><\/mml:mrow><\/mml:math>-invex functions on Riemannian manifolds and generalize the notions to the so-called geodesic quasi\/pseudo<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M3\"><mml:mrow><mml:mi>B<\/mml:mi><\/mml:mrow><\/mml:math>-preinvex and geodesic quasi\/pseudo<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M4\"><mml:mrow><mml:mi>B<\/mml:mi><\/mml:mrow><\/mml:math>-invex functions. We discuss the links among these functions under appropriate conditions and obtain results concerning extremum points of a nonsmooth geodesic<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M5\"><mml:mrow><mml:mi>B<\/mml:mi><\/mml:mrow><\/mml:math>-preinvex function by using the proximal subdifferential. Moreover, we study a differentiable multiobjective optimization problem involving new classes of generalized geodesic<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M6\"><mml:mrow><mml:mi>B<\/mml:mi><\/mml:mrow><\/mml:math>-invex functions and derive Kuhn-Tucker-type sufficient conditions for a feasible point to be an efficient or properly efficient solution. Finally, a Mond-Weir type duality is formulated and some duality results are given for the pair of primal and dual programming.<\/jats:p>","DOI":"10.1155\/2014\/524698","type":"journal-article","created":{"date-parts":[[2014,3,26]],"date-time":"2014-03-26T17:06:30Z","timestamp":1395853590000},"page":"1-12","source":"Crossref","is-referenced-by-count":4,"title":["Geodesic<i>B<\/i>-Preinvex Functions and Multiobjective Optimization Problems on Riemannian Manifolds"],"prefix":"10.1155","volume":"2014","author":[{"given":"Sheng-lan","family":"Chen","sequence":"first","affiliation":[{"name":"Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China"},{"name":"School of Mathematics and Physics, Chongqing University of Posts and Telecommunications, Chongqing 400065, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Nan-Jing","family":"Huang","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Donal","family":"O'Regan","sequence":"additional","affiliation":[{"name":"School of Mathematics, Statistics and Applied Mathematics, National University of Ireland, Galway, Ireland"},{"name":"Department of Mathematics, King Abdulaziz University, Jeddah, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","reference":[{"key":"9","doi-asserted-by":"publisher","DOI":"10.1007\/BF00939919"},{"key":"17","doi-asserted-by":"publisher","DOI":"10.1016\/0022-247X(81)90123-2"},{"key":"11","doi-asserted-by":"publisher","DOI":"10.1017\/S0334270000005142"},{"key":"34","doi-asserted-by":"publisher","DOI":"10.1016\/0022-247X(88)90113-8"},{"key":"10","doi-asserted-by":"publisher","DOI":"10.1007\/BF00939383"},{"key":"31","doi-asserted-by":"publisher","DOI":"10.1007\/BF00939384"},{"key":"20","doi-asserted-by":"publisher","DOI":"10.1007\/BF02207775"},{"key":"29","doi-asserted-by":"publisher","DOI":"10.1080\/02331939708844289"},{"key":"16","doi-asserted-by":"publisher","DOI":"10.1201\/9781420018868"},{"key":"19","doi-asserted-by":"publisher","DOI":"10.1007\/s10957-012-0123-5"},{"key":"26","doi-asserted-by":"publisher","DOI":"10.1006\/jmaa.1995.1057"},{"key":"24","doi-asserted-by":"publisher","DOI":"10.1137\/09075367X"},{"key":"8","doi-asserted-by":"publisher","DOI":"10.1023\/A:1014231603588"},{"issue":"2","key":"14","first-page":"80","volume":"11","year":"2006","journal-title":"Balkan Journal of Geometry and its Applications"},{"key":"33","series-title":"Colloquia Mathematica Societatis J\u00e1nos Bolyai","first-page":"1247","volume-title":"Kuhn-Tucker theorem on Riemannian manifolds","year":"1984"},{"key":"7","doi-asserted-by":"publisher","DOI":"10.1016\/j.jmaa.2006.05.081"},{"key":"6","doi-asserted-by":"publisher","DOI":"10.1016\/j.na.2008.02.085"},{"key":"18","doi-asserted-by":"publisher","DOI":"10.1080\/01630563.2010.520215"},{"issue":"5","key":"2","doi-asserted-by":"crossref","first-page":"1719","DOI":"10.11650\/twjm\/1500406792","volume":"16","year":"2012","journal-title":"Taiwanese Journal of Mathematics"},{"key":"1","doi-asserted-by":"publisher","DOI":"10.1080\/02331934.2010.544314"},{"key":"28","doi-asserted-by":"publisher","DOI":"10.1080\/02331939408843959"},{"key":"5","first-page":"34","volume":"13","year":"2011","journal-title":"Differential Geometry"},{"issue":"1","key":"25","first-page":"21","volume":"3","year":"2001","journal-title":"Differential Geometry"},{"key":"4","doi-asserted-by":"publisher","DOI":"10.1016\/j.jfa.2004.10.008"},{"issue":"3","key":"35","doi-asserted-by":"crossref","first-page":"833","DOI":"10.11650\/tjm.17.2013.1937","volume":"17","year":"2013","journal-title":"Taiwanese Journal of Mathematics"},{"year":"1997","key":"30"},{"volume":"297","volume-title":"Convex functions and optimization methods on riemannian manifolds","year":"1994","key":"32"},{"year":"1993","key":"13"},{"volume":"1","year":"1982","series-title":"Walter de Gruyter Studies in Mathematics","key":"21"},{"key":"22","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-0541-8"},{"key":"23","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-07-04075-5"},{"key":"3","doi-asserted-by":"publisher","DOI":"10.1007\/s00009-005-0056-4"},{"key":"15","doi-asserted-by":"publisher","DOI":"10.1016\/0022-247X(68)90201-1"},{"year":"1983","key":"12"}],"container-title":["Journal of Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/downloads.hindawi.com\/journals\/jam\/2014\/524698.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/jam\/2014\/524698.xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/jam\/2014\/524698.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,8,8]],"date-time":"2019-08-08T19:30:51Z","timestamp":1565292651000},"score":1,"resource":{"primary":{"URL":"http:\/\/www.hindawi.com\/journals\/jam\/2014\/524698\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2014]]},"references-count":34,"alternative-id":["524698","524698"],"URL":"https:\/\/doi.org\/10.1155\/2014\/524698","relation":{},"ISSN":["1110-757X","1687-0042"],"issn-type":[{"type":"print","value":"1110-757X"},{"type":"electronic","value":"1687-0042"}],"subject":[],"published":{"date-parts":[[2014]]}}}