{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,5,4]],"date-time":"2022-05-04T08:06:32Z","timestamp":1651651592090},"reference-count":45,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2019,5,17]],"date-time":"2019-05-17T00:00:00Z","timestamp":1558051200000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2019,5,17]],"date-time":"2019-05-17T00:00:00Z","timestamp":1558051200000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Optim Theory Appl"],"published-print":{"date-parts":[[2019,12]]},"DOI":"10.1007\/s10957-019-01534-7","type":"journal-article","created":{"date-parts":[[2019,5,17]],"date-time":"2019-05-17T06:28:15Z","timestamp":1558074495000},"page":"963-976","update-policy":"http:\/\/dx.doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Optimality Conditions and Constraint Qualifications for Quasiconvex Programming"],"prefix":"10.1007","volume":"183","author":[{"given":"Satoshi","family":"Suzuki","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2019,5,17]]},"reference":[{"key":"1534_CR1","doi-asserted-by":"crossref","unstructured":"Bo\u0163, R.I.: Conjugate Duality in Convex Optimization. Lecture Notes in Economics and Mathematical Systems, vol. 637. Springer, Berlin, (2010)","DOI":"10.1007\/978-3-642-04900-2"},{"key":"1534_CR2","doi-asserted-by":"crossref","first-page":"57","DOI":"10.1016\/0167-6377(91)90087-6","volume":"10","author":"JV Burke","year":"1991","unstructured":"Burke, J.V., Ferris, M.C.: Characterization of solution sets of convex programs. Oper. Res. Lett. 10, 57\u201360 (1991)","journal-title":"Oper. Res. Lett."},{"key":"1534_CR3","doi-asserted-by":"crossref","first-page":"163","DOI":"10.1137\/060676982","volume":"19","author":"C Li","year":"2008","unstructured":"Li, C., Ng, K.F., Pong, T.K.: Constraint qualifications for convex inequality systems with applications in constrained optimization. SIAM J. Optim. 19, 163\u2013187 (2008)","journal-title":"SIAM J. Optim."},{"key":"1534_CR4","doi-asserted-by":"crossref","first-page":"21","DOI":"10.1016\/0167-6377(88)90047-8","volume":"7","author":"OL Mangasarian","year":"1988","unstructured":"Mangasarian, O.L.: A simple characterization of solution sets of convex programs. Oper. Res. Lett. 7, 21\u201326 (1988)","journal-title":"Oper. Res. Lett."},{"key":"1534_CR5","doi-asserted-by":"crossref","DOI":"10.1515\/9781400873173","volume-title":"Convex Analysis","author":"RT Rockafellar","year":"1970","unstructured":"Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970)"},{"key":"1534_CR6","first-page":"339","volume":"130","author":"ZL Wu","year":"2006","unstructured":"Wu, Z.L., Wu, S.Y.: Characterizations of the solution sets of convex programs and variational inequality problems. J. Optim. Theory Appl. 130, 339\u2013358 (2006)","journal-title":"J. Optim. Theory Appl."},{"key":"1534_CR7","first-page":"1","volume":"29","author":"VI Ivanov","year":"2003","unstructured":"Ivanov, V.I.: Characterizations of the solution sets of generalized convex minimization problems. Serdica Math. J. 29, 1\u201310 (2003)","journal-title":"Serdica Math. J."},{"key":"1534_CR8","doi-asserted-by":"crossref","first-page":"677","DOI":"10.1007\/s10898-012-9956-6","volume":"57","author":"VI Ivanov","year":"2013","unstructured":"Ivanov, V.I.: Characterizations of pseudoconvex functions and semistrictly quasiconvex ones. J. Global Optim. 57, 677\u2013693 (2013)","journal-title":"J. Global Optim."},{"key":"1534_CR9","doi-asserted-by":"crossref","first-page":"65","DOI":"10.1007\/s10957-012-0243-y","volume":"158","author":"VI Ivanov","year":"2013","unstructured":"Ivanov, V.I.: Optimality conditions and characterizations of the solution sets in generalized convex problems and variational inequalities. J. Optim. Theory Appl. 158, 65\u201384 (2013)","journal-title":"J. Optim. Theory Appl."},{"key":"1534_CR10","doi-asserted-by":"crossref","first-page":"333","DOI":"10.1016\/j.cam.2013.11.004","volume":"261","author":"TQ Son","year":"2014","unstructured":"Son, T.Q., Kim, D.S.: A new approach to characterize the solution set of a pseudoconvex programming problem. J. Comput. Appl. Math. 261, 333\u2013340 (2014)","journal-title":"J. Comput. Appl. Math."},{"key":"1534_CR11","doi-asserted-by":"crossref","first-page":"537","DOI":"10.1007\/s10957-008-9470-7","volume":"140","author":"XM Yang","year":"2009","unstructured":"Yang, X.M.: On characterizing the solution sets of pseudoinvex extremum problems. J. Optim. Theory Appl. 140, 537\u2013542 (2009)","journal-title":"J. Optim. Theory Appl."},{"key":"1534_CR12","doi-asserted-by":"crossref","first-page":"685","DOI":"10.1007\/s11590-012-0471-y","volume":"7","author":"KQ Zhao","year":"2013","unstructured":"Zhao, K.Q., Yang, X.M.: Characterizations of the solution set for a class of nonsmooth optimization problems. Optim. Lett. 7, 685\u2013694 (2013)","journal-title":"Optim. Lett."},{"key":"1534_CR13","doi-asserted-by":"crossref","first-page":"55","DOI":"10.1007\/978-1-4684-7600-2_3","volume-title":"Generalized Concavity","author":"Mordecai Avriel","year":"1988","unstructured":"Avriel, M., Diewert, W.E., Schaible, S., Zang, I.: Generalized concavity. Math. Concepts Methods Sci. Engrg. Plenum Press, New York (1988)"},{"issue":"1","key":"1534_CR14","doi-asserted-by":"publisher","first-page":"144","DOI":"10.1007\/s10957-018-1379-1","volume":"181","author":"Vsevolod I. Ivanov","year":"2018","unstructured":"Ivanov, V.I.: Characterizations of solution sets of differentiable quasiconvex programming problems. J. Optim. Theory Appl. \n                    https:\/\/doi.org\/10.1007\/s10957-018-1379-1","journal-title":"Journal of Optimization Theory and Applications"},{"key":"1534_CR15","doi-asserted-by":"crossref","first-page":"500","DOI":"10.1137\/040621843","volume":"17","author":"NTH Linh","year":"2006","unstructured":"Linh, N.T.H., Penot, J.P.: Optimality conditions for quasiconvex programs. SIAM J. Optim. 17, 500\u2013510 (2006)","journal-title":"SIAM J. Optim."},{"key":"1534_CR16","doi-asserted-by":"crossref","first-page":"627","DOI":"10.1023\/A:1023905907248","volume":"117","author":"JP Penot","year":"2003","unstructured":"Penot, J.P.: Characterization of solution sets of quasiconvex programs. J. Optim. Theory Appl. 117, 627\u2013636 (2003)","journal-title":"J. Optim. Theory Appl."},{"key":"1534_CR17","doi-asserted-by":"crossref","first-page":"1279","DOI":"10.1016\/j.na.2010.09.066","volume":"74","author":"S Suzuki","year":"2011","unstructured":"Suzuki, S., Kuroiwa, D.: Optimality conditions and the basic constraint qualification for quasiconvex programming. Nonlinear Anal. 74, 1279\u20131285 (2011)","journal-title":"Nonlinear Anal."},{"key":"1534_CR18","doi-asserted-by":"crossref","first-page":"677","DOI":"10.1016\/j.jmaa.2011.06.015","volume":"384","author":"S Suzuki","year":"2011","unstructured":"Suzuki, S., Kuroiwa, D.: Subdifferential calculus for a quasiconvex function with generator. J. Math. Anal. Appl. 384, 677\u2013682 (2011)","journal-title":"J. Math. Anal. Appl."},{"key":"1534_CR19","doi-asserted-by":"crossref","first-page":"539","DOI":"10.1007\/s10898-011-9807-x","volume":"55","author":"S Suzuki","year":"2013","unstructured":"Suzuki, S., Kuroiwa, D.: Some constraint qualifications for quasiconvex vector-valued systems. J. Global Optim. 55, 539\u2013548 (2013)","journal-title":"J. Global Optim."},{"key":"1534_CR20","doi-asserted-by":"crossref","first-page":"431","DOI":"10.1007\/s10898-014-0255-2","volume":"62","author":"S Suzuki","year":"2015","unstructured":"Suzuki, S., Kuroiwa, D.: Characterizations of the solution set for quasiconvex programming in terms of Greenberg-Pierskalla subdifferential. J. Global Optim. 62, 431\u2013441 (2015)","journal-title":"J. Global Optim."},{"key":"1534_CR21","doi-asserted-by":"crossref","first-page":"1699","DOI":"10.1007\/s11590-016-1084-7","volume":"11","author":"S Suzuki","year":"2017","unstructured":"Suzuki, S., Kuroiwa, D.: Characterizations of the solution set for non-essentially quasiconvex programming. Optim. Lett. 11, 1699\u20131712 (2017)","journal-title":"Optim. Lett."},{"key":"1534_CR22","doi-asserted-by":"crossref","first-page":"489","DOI":"10.11650\/tjm\/7256","volume":"21","author":"S Suzuki","year":"2017","unstructured":"Suzuki, S.: Duality theorems for quasiconvex programming with a reverse quasiconvex constraint. Taiwanese J. Math. 21, 489\u2013503 (2017)","journal-title":"Taiwanese J. Math."},{"key":"1534_CR23","doi-asserted-by":"crossref","first-page":"1184","DOI":"10.1016\/j.na.2006.12.014","volume":"68","author":"MA Goberna","year":"2008","unstructured":"Goberna, M.A., Jeyakumar, V., L\u00f3pez, M.A.: Necessary and sufficient constraint qualifications for solvability of systems of infinite convex inequalities. Nonlinear Anal. 68, 1184\u20131194 (2008)","journal-title":"Nonlinear Anal."},{"key":"1534_CR24","doi-asserted-by":"crossref","first-page":"31","DOI":"10.1007\/s10957-007-9294-x","volume":"136","author":"V Jeyakumar","year":"2008","unstructured":"Jeyakumar, V.: Constraint qualifications characterizing Lagrangian duality in convex optimization. J. Optim. Theory Appl. 136, 31\u201341 (2008)","journal-title":"J. Optim. Theory Appl."},{"key":"1534_CR25","unstructured":"Jeyakumar, V., Dinh, N., Lee, G. M.: A new closed cone constraint qualification for convex optimization. Research Report AMR 04\/8, Department of Applied Mathematics, University of New South Wales, (2004)"},{"key":"1534_CR26","doi-asserted-by":"crossref","first-page":"554","DOI":"10.1007\/s10957-011-9804-8","volume":"149","author":"S Suzuki","year":"2011","unstructured":"Suzuki, S., Kuroiwa, D.: On set containment characterization and constraint qualification for quasiconvex programming. J. Optim. Theory Appl. 149, 554\u2013563 (2011)","journal-title":"J. Optim. Theory Appl."},{"key":"1534_CR27","doi-asserted-by":"crossref","first-page":"2851","DOI":"10.1016\/j.na.2011.11.025","volume":"75","author":"S Suzuki","year":"2012","unstructured":"Suzuki, S., Kuroiwa, D.: Necessary and sufficient conditions for some constraint qualifications in quasiconvex programming. Nonlinear Anal. 75, 2851\u20132858 (2012)","journal-title":"Nonlinear Anal."},{"key":"1534_CR28","doi-asserted-by":"crossref","first-page":"366","DOI":"10.1007\/s10957-011-9893-4","volume":"152","author":"S Suzuki","year":"2012","unstructured":"Suzuki, S., Kuroiwa, D.: Necessary and sufficient constraint qualification for surrogate duality. J. Optim. Theory Appl. 152, 366\u2013367 (2012)","journal-title":"J. Optim. Theory Appl."},{"key":"1534_CR29","first-page":"2101","volume":"18","author":"S Suzuki","year":"2017","unstructured":"Suzuki, S., Kuroiwa, D.: Generators and constraint qualifications for quasiconvex inequality systems. J. Nonlinear Convex Anal. 18, 2101\u20132121 (2017)","journal-title":"J. Nonlinear Convex Anal."},{"key":"1534_CR30","doi-asserted-by":"crossref","first-page":"669","DOI":"10.1007\/s10957-016-1003-1","volume":"172","author":"S Suzuki","year":"2017","unstructured":"Suzuki, S., Kuroiwa, D.: Duality Theorems for Separable Convex Programming without Qualifications. J. Optim. Theory Appl. 172, 669\u2013683 (2017)","journal-title":"J. Optim. Theory Appl."},{"key":"1534_CR31","doi-asserted-by":"crossref","first-page":"603","DOI":"10.1080\/02331938808843379","volume":"19","author":"JE Mart\u00ednez-Legaz","year":"1988","unstructured":"Mart\u00ednez-Legaz, J.E.: Quasiconvex duality theory by generalized conjugation methods. Optimization. 19, 603\u2013652 (1988)","journal-title":"Optimization."},{"key":"1534_CR32","doi-asserted-by":"crossref","first-page":"597","DOI":"10.1287\/moor.15.4.597","volume":"15","author":"JP Penot","year":"1990","unstructured":"Penot, J.P., Volle, M.: On quasi-convex duality. Math. Oper. Res. 15, 597\u2013625 (1990)","journal-title":"Math. Oper. Res."},{"key":"1534_CR33","doi-asserted-by":"crossref","first-page":"193","DOI":"10.1007\/BF01583788","volume":"23","author":"JP Crouzeix","year":"1982","unstructured":"Crouzeix, J.P., Ferland, J.A.: Criteria for quasiconvexity and pseudoconvexity: relationships and comparisons. Math. Programming. 23, 193\u2013205 (1982)","journal-title":"Math. Programming."},{"key":"1534_CR34","first-page":"203","volume":"27","author":"VI Ivanov","year":"2001","unstructured":"Ivanov, V.I.: First order characterizations of pseudoconvex functions. Serdica Math. J. 27, 203\u2013218 (2001)","journal-title":"Serdica Math. J."},{"key":"1534_CR35","doi-asserted-by":"crossref","first-page":"93","DOI":"10.1007\/s10957-018-1223-7","volume":"177","author":"S Al-Homidan","year":"2018","unstructured":"Al-Homidan, S., Hadjisavvas, N., Shaalan, L.: Transformation of quasiconvex functions to eliminate local minima. J. Optim. Theory Appl. 177, 93\u2013105 (2018)","journal-title":"J. Optim. Theory Appl."},{"key":"1534_CR36","first-page":"437","volume":"15","author":"HJ Greenberg","year":"1973","unstructured":"Greenberg, H.J., Pierskalla, W.P.: Quasi-conjugate functions and surrogate duality. Cah. Cent. \u00c9tud. Rech. Op\u00e9r. 15, 437\u2013448 (1973)","journal-title":"Cah. Cent. \u00c9tud. Rech. Op\u00e9r."},{"key":"1534_CR37","doi-asserted-by":"crossref","first-page":"407","DOI":"10.1137\/S1052623400371399","volume":"12","author":"A Daniilidis","year":"2001","unstructured":"Daniilidis, A., Hadjisavvas, N., Mart\u00ednez-Legaz, J.E.: An appropriate subdifferential for quasiconvex functions. SIAM J. Optim. 12, 407\u2013420 (2001)","journal-title":"SIAM J. Optim."},{"key":"1534_CR38","first-page":"523","volume":"78","author":"JM Guti\u00e9rrez D\u00edez","year":"1984","unstructured":"Guti\u00e9rrez D\u00edez, J.M.: Infragradientes $$y$$ Direcciones de Decrecimiento. Rev. Real Acad. Cienc. Exact. F\u00eds. Natur. Madrid. 78, 523\u2013532 (1984)","journal-title":"Rev. Real Acad. Cienc. Exact. F\u00eds. Natur. Madrid."},{"key":"1534_CR39","doi-asserted-by":"crossref","first-page":"315","DOI":"10.1016\/j.ejor.2014.05.017","volume":"240","author":"Y Hu","year":"2015","unstructured":"Hu, Y., Yang, X., Sim, C.K.: Inexact subgradient methods for quasi-convex optimization problems. European J. Oper. Res. 240, 315\u2013327 (2015)","journal-title":"European J. Oper. Res."},{"key":"1534_CR40","unstructured":"Mart\u00ednez-Legaz, J.E.: A generalized concept of conjugation. Lecture Notes in Pure and Appl. Math. 86, 45\u201359 (1983)"},{"key":"1534_CR41","unstructured":"Mart\u00ednez-Legaz, J.E.: A new approach to symmetric quasiconvex conjugacy. Lecture Notes in Econom. and Math. Systems. 226, 42\u201348 (1984)"},{"key":"1534_CR42","first-page":"1","volume":"6","author":"JE Mart\u00ednez-Legaz","year":"1999","unstructured":"Mart\u00ednez-Legaz, J.E., Sach, P.H.: A new subdifferential in quasiconvex analysis. J. Convex Anal. 6, 1\u201311 (1999)","journal-title":"J. Convex Anal."},{"key":"1534_CR43","first-page":"109","volume":"49","author":"JJ Moreau","year":"1970","unstructured":"Moreau, J.J.: Inf-convolution, sous-additivit\u00e9, convexit\u00e9 des fonctions num\u00e9riques. J. Math. Pures Appl. 49, 109\u2013154 (1970)","journal-title":"J. Math. Pures Appl."},{"key":"1534_CR44","doi-asserted-by":"crossref","first-page":"35","DOI":"10.1080\/02331930008844469","volume":"47","author":"JP Penot","year":"2000","unstructured":"Penot, J.P.: What is quasiconvex analysis? Optimization. 47, 35\u2013110 (2000)","journal-title":"Optimization."},{"key":"1534_CR45","doi-asserted-by":"crossref","first-page":"37","DOI":"10.1007\/BF00938758","volume":"46","author":"F Plastria","year":"1985","unstructured":"Plastria, F.: Lower subdifferentiable functions and their minimization by cutting planes. J. Optim. Theory Appl. 46, 37\u201353 (1985)","journal-title":"J. Optim. Theory Appl."}],"container-title":["Journal of Optimization Theory and Applications"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10957-019-01534-7.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s10957-019-01534-7\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10957-019-01534-7.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,5,15]],"date-time":"2020-05-15T23:27:12Z","timestamp":1589585232000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s10957-019-01534-7"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,5,17]]},"references-count":45,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2019,12]]}},"alternative-id":["1534"],"URL":"https:\/\/doi.org\/10.1007\/s10957-019-01534-7","relation":{},"ISSN":["0022-3239","1573-2878"],"issn-type":[{"value":"0022-3239","type":"print"},{"value":"1573-2878","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,5,17]]},"assertion":[{"value":"17 October 2018","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"8 May 2019","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"17 May 2019","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}