{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,18]],"date-time":"2026-03-18T06:22:47Z","timestamp":1773814967713,"version":"3.50.1"},"reference-count":37,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2013,3,12]],"date-time":"2013-03-12T00:00:00Z","timestamp":1363046400000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/2.0"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J Glob Optim"],"published-print":{"date-parts":[[2013,11]]},"DOI":"10.1007\/s10898-013-0051-4","type":"journal-article","created":{"date-parts":[[2013,3,11]],"date-time":"2013-03-11T07:23:45Z","timestamp":1362986625000},"page":"695-705","source":"Crossref","is-referenced-by-count":7,"title":["Strongly convex set-valued maps"],"prefix":"10.1007","volume":"57","author":[{"given":"Hugo","family":"Leiva","sequence":"first","affiliation":[]},{"given":"Nelson","family":"Merentes","sequence":"additional","affiliation":[]},{"given":"Kazimierz","family":"Nikodem","sequence":"additional","affiliation":[]},{"given":"Jos\u00e9 Luis","family":"S\u00e1nchez","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2013,3,12]]},"reference":[{"key":"51_CR1","first-page":"1","volume":"14","author":"A Aleman","year":"1985","unstructured":"Aleman, A.: On some generalization of convex sets and convex functions. Anal. Num\u00e9r. Th\u00e9or. Approx. 14, 1\u20136 (1985)","journal-title":"Anal. Num\u00e9r. Th\u00e9or. Approx."},{"key":"51_CR2","doi-asserted-by":"crossref","first-page":"15","DOI":"10.7494\/OpMath.2011.31.1.15","volume":"31","author":"A Az\u00f3car","year":"2011","unstructured":"Az\u00f3car, A., Gim\u00e9nez, J., Nikodem, K., S\u00e1nchez, J.L.: On strongly midconvex functions. Opuscula Math. 31, 15\u201326 (2011)","journal-title":"Opuscula Math."},{"key":"51_CR3","doi-asserted-by":"crossref","first-page":"161","DOI":"10.1016\/S0022-247X(03)00588-2","volume":"288","author":"J Benoist","year":"2003","unstructured":"Benoist, J., Popovici, N.: Generalized convex set-valued maps. J. Math. Anal. Appl. 288, 161\u2013166 (2003)","journal-title":"J. Math. Anal. Appl."},{"key":"51_CR4","doi-asserted-by":"crossref","first-page":"183","DOI":"10.1007\/BF01584336","volume":"13","author":"JM Borwein","year":"1977","unstructured":"Borwein, J.M.: Multivalued convexity and optimization: a unified approach to inequality and equality constrains. Math. Program. 13, 183\u2013199 (1977)","journal-title":"Math. Program."},{"key":"51_CR5","first-page":"335","volume-title":"Generalized Convexity in Optimization and Economics","author":"JM Borwein","year":"1981","unstructured":"Borwein, J.M.: Convex relations in analysis and optimization. In: Schaible, S., Ziemba, W. (eds.) Generalized Convexity in Optimization and Economics, pp. 335\u2013377. Academic Press, New York (1981)"},{"key":"51_CR6","first-page":"39","volume":"22","author":"WW Breckner","year":"1993","unstructured":"Breckner, W.W.: Continuity of generalized convex and generalized concave set-valued functions. Rev. Anal. Num\u00e9r. Th\u00e9or. Approx. 22, 39\u201351 (1993)","journal-title":"Rev. Anal. Num\u00e9r. Th\u00e9or. Approx."},{"key":"51_CR7","doi-asserted-by":"crossref","first-page":"185","DOI":"10.4064\/ap-58-2-185-192","volume":"58","author":"T Cardinali","year":"1993","unstructured":"Cardinali, T., Nikodem, K., Papalini, F.: Some results on stability and on characterization of $$K$$ -convexity of set-valued functions. Ann. Polon. Math. 58, 185\u2013192 (1993)","journal-title":"Ann. Polon. Math."},{"key":"51_CR8","first-page":"5","volume":"11","author":"Z Dar\u00f3czy","year":"1987","unstructured":"Dar\u00f3czy, Z., Zs, P\u00e1les: Convexity with given infinite weight sequences. Stochastica 11, 5\u201312 (1987)","journal-title":"Stochastica"},{"key":"51_CR9","doi-asserted-by":"crossref","first-page":"85","DOI":"10.1007\/s10898-005-3271-4","volume":"35","author":"R Enkhbat","year":"2006","unstructured":"Enkhbat, R., Barsbold, B., Kamada, M.: A numerical approach for solving some convex maximization problems. J. Glob. Optim. 35, 85\u2013101 (2006)","journal-title":"J. Glob. Optim."},{"key":"51_CR10","unstructured":"Frankowska, H., Olech, C.: $$R$$ -convexity of the integral of the set-valued functions. In: Clark, D.N., Pecelli, G., Sacksteder, R. (eds.) Contributions to Analysis and Geometry, pp. 117\u2013129. Johns Hopkins University Press, Baltimore (1981)"},{"key":"51_CR11","doi-asserted-by":"crossref","first-page":"417","DOI":"10.1142\/S0219199710003865","volume":"12","author":"H Huang","year":"2010","unstructured":"Huang, H.: Global error bounds with exponents for multifunctions with set constraints. Commun. Contemp. Math. 12, 417\u2013435 (2010)","journal-title":"Commun. Contemp. Math."},{"key":"51_CR12","first-page":"841","volume":"13","author":"H Huang","year":"2010","unstructured":"Huang, H.: Inversion theorem for nonconvex multifunctions. Math. Inequal. Appl. 13, 841\u2013849 (2010)","journal-title":"Math. Inequal. Appl."},{"key":"51_CR13","doi-asserted-by":"crossref","first-page":"143","DOI":"10.1017\/S1446788700027130","volume":"40","author":"V Jeyakumar","year":"1986","unstructured":"Jeyakumar, V.: On subgradient duality with strong and weak convex functions. J. Aust. Math. Soc. 40, 143\u2013152 (1986)","journal-title":"J. Aust. Math. Soc."},{"key":"51_CR14","doi-asserted-by":"crossref","first-page":"167","DOI":"10.1007\/BF00930577","volume":"4","author":"S Karamardian","year":"1969","unstructured":"Karamardian, S.: The nonlinear complementarity problem with applications, Part 2. J. Optim. Theory Appl. 4, 167\u2013181 (1969)","journal-title":"J. Optim. Theory Appl."},{"key":"51_CR15","doi-asserted-by":"crossref","unstructured":"Kuczma, M.: An Introduction to the Theory of Functional Equations and Inequalities. Cauchy\u2019s Equation and Jensen\u2019s Inequality, PWN\u2014Uniwersytet \u015ala\u0327ski, Warszawa-Krak\u00f3w-Katowice, 1985, 2nd edn. Birkh\u00e4user, Basel (2009)","DOI":"10.1007\/978-3-7643-8749-5"},{"key":"51_CR16","doi-asserted-by":"crossref","unstructured":"Kuhn, N.: A note on t-convex functions. In: Walter, W. (ed.) General Inequalities, 4 (Oberwolfach, 1983), International Series of Numerical Mathematics, vol. 71, pp. 269\u2013276. Birkh\u00e4user, Basel (1984)","DOI":"10.1007\/978-3-0348-6259-2_25"},{"key":"51_CR17","doi-asserted-by":"crossref","first-page":"641","DOI":"10.1007\/s10898-011-9752-8","volume":"52","author":"N Langenberg","year":"2012","unstructured":"Langenberg, N., Tichatschke, R.: Interior proximal methods for quasiconvex optimization. J. Glob. Optim. 52, 641\u2013661 (2012)","journal-title":"J. Glob. Optim."},{"key":"51_CR18","doi-asserted-by":"crossref","first-page":"123","DOI":"10.4064\/ap-36-2-123-137","volume":"37","author":"S \u0141ojasiewicz Jr","year":"1979","unstructured":"\u0141ojasiewicz Jr, S.: Some properties of accessible sets in non linear control systems. Ann. Polon. Math 37, 123\u2013127 (1979)","journal-title":"Ann. Polon. Math"},{"key":"51_CR19","doi-asserted-by":"crossref","first-page":"67","DOI":"10.1023\/A:1024787424532","volume":"118","author":"GH Lin","year":"2003","unstructured":"Lin, G.H., Fukushima, M.: Some exact penalty results for nonlinear programs and mathematical programs with equilibrium constraints. J. Optim. Theory Appl. 118, 67\u201380 (2003)","journal-title":"J. Optim. Theory Appl."},{"key":"51_CR20","doi-asserted-by":"crossref","first-page":"193","DOI":"10.1007\/s00010-010-0043-0","volume":"80","author":"N Merentes","year":"2010","unstructured":"Merentes, N., Nikodem, K.: Remarks on strongly convex functions. Aequationes Math. 80, 193\u2013199 (2010)","journal-title":"Aequationes Math."},{"key":"51_CR21","doi-asserted-by":"crossref","first-page":"46","DOI":"10.1007\/BF01836150","volume":"3","author":"K Nikodem","year":"1987","unstructured":"Nikodem, K.: On midpoint convex set-valued functions. Aequationes Math. 3, 46\u201356 (1987)","journal-title":"Aequationes Math."},{"key":"51_CR22","unstructured":"Nikodem, K.: $$K$$ -Convex and $$K$$ -Concave Set-Valued Functions, Zeszyty Nauk. Politech. \u0141\u00f3dz. Mat. 559 (Rozprawy Nauk 114), \u0141\u00f3d\u017a, pp. 1\u201375 (1989)"},{"key":"51_CR23","unstructured":"Nikodem, K.: Continuity properties of convex-type set-valued maps. J. Inequal. Pure Appl. Math. 4(3), Art. 52 (2003)"},{"issue":"1","key":"51_CR24","doi-asserted-by":"crossref","first-page":"83","DOI":"10.15352\/bjma\/1313362982","volume":"5","author":"K Nikodem","year":"2011","unstructured":"Nikodem, K., Zs, P\u00e1les: Characterizations of inner product spaces by strongly convex functions. Banach J. Math. Anal. 5(1), 83\u201387 (2011)","journal-title":"Banach J. Math. Anal."},{"key":"51_CR25","doi-asserted-by":"crossref","unstructured":"Pli\u015b, A.: Accessible sets in control theory. In: International Conference on Differential Equations (Univ. of Southern California), pp. 646-650. Academic Press, New York (1975)","DOI":"10.1016\/B978-0-12-059650-8.50052-1"},{"key":"51_CR26","doi-asserted-by":"crossref","first-page":"259","DOI":"10.1070\/SM1996v187n02ABEH000111","volume":"187","author":"E Polovinkin","year":"1996","unstructured":"Polovinkin, E.: Strongly convex analysis. Sbornik Math. 187, 259\u2013286 (1996)","journal-title":"Sbornik Math."},{"key":"51_CR27","first-page":"72","volume":"7","author":"BT Polyak","year":"1966","unstructured":"Polyak, B.T.: Existence theorems and convergence of minimizing sequences in extremum problems with restrictions. Sov. Math. Dokl. 7, 72\u201375 (1966)","journal-title":"Sov. Math. Dokl."},{"key":"51_CR28","doi-asserted-by":"crossref","first-page":"165","DOI":"10.2307\/2032477","volume":"3","author":"H R\u00e5dstr\u00f6m","year":"1952","unstructured":"R\u00e5dstr\u00f6m, H.: An embedding theorem for space of convex sets. Proc. Am. Math. Soc. 3, 165\u2013169 (1952)","journal-title":"Proc. Am. Math. Soc."},{"key":"51_CR29","doi-asserted-by":"crossref","unstructured":"Rajba, T., Wa\u0327sowicz, Sz.: Probabilistic characterization of strong convexity. Opuscula Math. 31, 97\u2013103 (2011)","DOI":"10.7494\/OpMath.2011.31.1.97"},{"key":"51_CR30","doi-asserted-by":"crossref","first-page":"519","DOI":"10.1007\/BF01198361","volume":"52","author":"B Ricceri","year":"1989","unstructured":"Ricceri, B.: Remarks on multifunctions with convex graph. Arch. Math. 52, 519\u2013520 (1989)","journal-title":"Arch. Math."},{"key":"51_CR31","volume-title":"Convex Functions","author":"AW Roberts","year":"1973","unstructured":"Roberts, A.W., Varberg, D.E.: Convex Functions. Academic Press, New York (1973)"},{"key":"51_CR32","doi-asserted-by":"crossref","first-page":"130","DOI":"10.1287\/moor.1.2.130","volume":"1","author":"SM Robinson","year":"1976","unstructured":"Robinson, S.M.: Regularity and stability for convex multivalued functions. Math. Oper. Res. 1, 130\u2013143 (1976)","journal-title":"Math. Oper. Res."},{"key":"51_CR33","doi-asserted-by":"crossref","first-page":"799","DOI":"10.1080\/02331938908843500","volume":"20","author":"A Sterna-Karwat","year":"1989","unstructured":"Sterna-Karwat, A.: Convexity of the optimal multifunctions and its consequences in vector optimization. Optimization 20, 799\u2013808 (1989)","journal-title":"Optimization"},{"key":"51_CR34","doi-asserted-by":"crossref","unstructured":"Thibault, L.: Continuity of measurable convex multifunctions. In: Multifunctions and Integrands. Lecture Notes in Math, pp. 216\u2013224. Springer, Berlin (1091)","DOI":"10.1007\/BFb0098814"},{"key":"51_CR35","doi-asserted-by":"crossref","first-page":"187","DOI":"10.1016\/0304-4068(82)90026-X","volume":"9","author":"JP Vial","year":"1982","unstructured":"Vial, J.P.: Strong convexity of sets and functions. J. Math. Econ. 9, 187\u2013205 (1982)","journal-title":"J. Math. Econ."},{"key":"51_CR36","doi-asserted-by":"crossref","first-page":"231","DOI":"10.1287\/moor.8.2.231","volume":"8","author":"JP Vial","year":"1983","unstructured":"Vial, J.P.: Strong and weak convexity of sets and functions. Math. Oper. Res. 8, 231\u2013259 (1983)","journal-title":"Math. Oper. Res."},{"key":"51_CR37","doi-asserted-by":"crossref","first-page":"67","DOI":"10.1007\/BF01100240","volume":"3","author":"SK Zavriev","year":"1993","unstructured":"Zavriev, S.K.: On the global optimization properties of finite difference local descent algorithms. J. Glob. Optim. 3, 67\u201378 (1993)","journal-title":"J. Glob. Optim."}],"container-title":["Journal of Global Optimization"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10898-013-0051-4.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s10898-013-0051-4\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10898-013-0051-4","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,7,10]],"date-time":"2019-07-10T19:51:48Z","timestamp":1562788308000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s10898-013-0051-4"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2013,3,12]]},"references-count":37,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2013,11]]}},"alternative-id":["51"],"URL":"https:\/\/doi.org\/10.1007\/s10898-013-0051-4","relation":{},"ISSN":["0925-5001","1573-2916"],"issn-type":[{"value":"0925-5001","type":"print"},{"value":"1573-2916","type":"electronic"}],"subject":[],"published":{"date-parts":[[2013,3,12]]}}}