{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,2,21]],"date-time":"2025-02-21T13:28:27Z","timestamp":1740144507226,"version":"3.37.3"},"reference-count":45,"publisher":"EDP Sciences","license":[{"start":{"date-parts":[[2021,3,2]],"date-time":"2021-03-02T00:00:00Z","timestamp":1614643200000},"content-version":"vor","delay-in-days":60,"URL":"https:\/\/www.edpsciences.org\/en\/authors\/copyright-and-licensing"}],"funder":[{"DOI":"10.13039\/100007224","name":"National Foundation for Science and Technology Development","doi-asserted-by":"publisher","award":["101.01-2020.11"],"award-info":[{"award-number":["101.01-2020.11"]}],"id":[{"id":"10.13039\/100007224","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["RAIRO-Oper. Res."],"accepted":{"date-parts":[[2019,12,22]]},"published-print":{"date-parts":[[2021]]},"abstract":"<jats:p>In this paper we consider stochastic equilibrium problems involving parameter of probability measures. Employing the fixed point theorem of Knaster, Kuratowski, Mazurkiewicz, and Fan (KKM-Fan), conditions for the existence of solutions to such problems are established. We then propose new metric concepts on the underlying stochastic spaces and study some properties corresponding to these metrics. Afterwards, we study sufficient conditions for the solution mappings of such problems, that are closed, upper (lower) semicontinuous and continuous with respect to the mentioned metrics. Finally, the special cases of stochastic optimization problems are taken into account as the applications.<\/jats:p>","DOI":"10.1051\/ro\/2020001","type":"journal-article","created":{"date-parts":[[2020,1,2]],"date-time":"2020-01-02T19:47:55Z","timestamp":1577994475000},"page":"S705-S718","source":"Crossref","is-referenced-by-count":2,"special_numbering":"Supplement","title":["On the existence and stability of solutions to stochastic equilibrium problems"],"prefix":"10.1051","volume":"55","author":[{"given":"Lam Quoc","family":"Anh","sequence":"first","affiliation":[]},{"given":"Nguyen Xuan","family":"Hai","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7522-4557","authenticated-orcid":false,"given":"Kien Trung","family":"Nguyen","sequence":"additional","affiliation":[]},{"given":"Nguyen Hong","family":"Quan","sequence":"additional","affiliation":[]},{"given":"Dang Thi My","family":"Van","sequence":"additional","affiliation":[]}],"member":"250","published-online":{"date-parts":[[2021,3,2]]},"reference":[{"key":"R1","doi-asserted-by":"crossref","first-page":"7","DOI":"10.1080\/02331934.2013.784765","volume":"63","author":"Alparslan G\u00f6k","year":"2014","journal-title":"Optimization"},{"key":"R2","doi-asserted-by":"crossref","first-page":"1929","DOI":"10.1080\/02331934.2016.1209673","volume":"65","author":"Anh","year":"2016","journal-title":"Optimization"},{"key":"R3","doi-asserted-by":"crossref","first-page":"371","DOI":"10.1287\/opre.1090.0754","volume":"58","author":"Ankenman","year":"2010","journal-title":"Oper. Res"},{"key":"R4","unstructured":"Barbagallo A. and Scilla G., Stochastic weighted variational inequalities in non-pivot Hilbert spaces with applications to a transportation model. In: New trends in optimization and variational analysis and applications; an international conference in honor of Prof. Michel Th\u00e9ra for his 70th birthday, Quy Nhon, Viet Nam, 2016 December 7\u201310 (2016)."},{"key":"R5","first-page":"123","volume":"63","author":"Blum","year":"1994","journal-title":"Math. Student"},{"key":"R6","first-page":"287","volume":"32","author":"Cho","year":"1995","journal-title":"Bull. Korean Math. Soc"},{"key":"R7","doi-asserted-by":"crossref","first-page":"649","DOI":"10.1137\/110825248","volume":"22","author":"Chen","year":"2012","journal-title":"SIAM J. Optim"},{"key":"R8","doi-asserted-by":"crossref","first-page":"65","DOI":"10.1023\/B:JOTA.0000026131.04418.b7","volume":"121","author":"Evgrafov","year":"2004","journal-title":"J. Optim. Theory Appl"},{"key":"R9","unstructured":"Fan K., A minimax inequality and applications, In: Inequality III, edited by Shisha O.. Academic Press, New York, NY (1972) 103\u2013113."},{"key":"R10","doi-asserted-by":"crossref","first-page":"519","DOI":"10.1007\/BF01458545","volume":"266","author":"Fan","year":"1984","journal-title":"Math. Annalen"},{"key":"R11","doi-asserted-by":"crossref","first-page":"1099","DOI":"10.1016\/j.jmaa.2005.09.079","volume":"322","author":"Farajzadeh","year":"2006","journal-title":"J. Math. Anal. Appl"},{"key":"R12","doi-asserted-by":"crossref","first-page":"55","DOI":"10.1007\/s10107980016a","volume":"84","author":"Henrion","year":"1999","journal-title":"Math. Program"},{"key":"R13","doi-asserted-by":"crossref","unstructured":"Henrion R. and R\u00f6misch W., Problem-based optimal scenario generation and reduction in stochastic programming. To appear in: Math. Program. (2018) DOI: 10.1007\/s10107-018-1337-6.","DOI":"10.1007\/s10107-018-1337-6"},{"key":"R14","doi-asserted-by":"crossref","first-page":"455","DOI":"10.1023\/A:1008306431147","volume":"13","author":"Jones","year":"1998","journal-title":"J. Global Optim"},{"key":"R15","first-page":"339","volume":"14","author":"Ka\u0148kov\u00e1","year":"1978","journal-title":"Kybernetika"},{"key":"R16","unstructured":"Ka\u0148kov\u00e1 V., On stability in two-stage stochastic nonlinear programming, edited by Mandl P. and Hu\u0161kov\u00e1 M.. In: Vol. 5 of Asymptotic Statistics. Springer, Berlin (1994) 329\u2013340."},{"key":"R17","first-page":"55","volume":"5","author":"Ka\u0148kov\u00e1","year":"1994","journal-title":"Informatica"},{"key":"R18","doi-asserted-by":"crossref","first-page":"132","DOI":"10.4064\/fm-14-1-132-137","volume":"14","author":"Knaster","year":"1929","journal-title":"Fundam. Math"},{"key":"R19","doi-asserted-by":"crossref","first-page":"347","DOI":"10.1080\/02331938408842947","volume":"15","author":"Muu","year":"1984","journal-title":"Math Operationsforsch Statist. Ser. Optim"},{"key":"R20","doi-asserted-by":"crossref","first-page":"1159","DOI":"10.1016\/0362-546X(92)90159-C","volume":"18","author":"Muu","year":"1992","journal-title":"Nonlinear Anal"},{"key":"R21","doi-asserted-by":"crossref","first-page":"1574","DOI":"10.1137\/070704277","volume":"19","author":"Nemirovski","year":"2009","journal-title":"SIAM J. Optim"},{"key":"R22","doi-asserted-by":"crossref","first-page":"159","DOI":"10.1016\/S0167-6377(99)00052-8","volume":"25","author":"Patriksson","year":"1999","journal-title":"Oper. Res. Lett"},{"key":"R23","unstructured":"Rachev S.T., Probability Metrics and the Stability of Stochastic Models. Wiley, Chichester, UK (1991)."},{"key":"R24","doi-asserted-by":"crossref","first-page":"792","DOI":"10.1287\/moor.27.4.792.304","volume":"27","author":"Rachev","year":"2002","journal-title":"Math. Oper. Res"},{"key":"R25","doi-asserted-by":"crossref","first-page":"1168","DOI":"10.1137\/100792644","volume":"21","author":"Ravat","year":"2011","journal-title":"SIAM J. Optim"},{"key":"R26","doi-asserted-by":"crossref","first-page":"291","DOI":"10.1007\/s10107-017-1179-7","volume":"165","author":"Ravat","year":"2017","journal-title":"Math. Program"},{"key":"R27","doi-asserted-by":"crossref","first-page":"301","DOI":"10.1016\/0022-247X(77)90020-8","volume":"60","author":"Rockafellar","year":"1977","journal-title":"J. Math. Anal. Appl"},{"key":"R28","doi-asserted-by":"crossref","unstructured":"Rockafellar R.T. and Wets R.J.-B., Variational Analysis. Springer, Berlin (1998).","DOI":"10.1007\/978-3-642-02431-3"},{"key":"R29","unstructured":"R\u00f6misch W., Stability of stochastic programming problems, In: Stochastic Programming, edited by Rusczy\u0144ski A. and Shapiro A.. Vol. 10 of Handbooks in Operations Research and Management Science (2003) 483\u2013554."},{"key":"R30","doi-asserted-by":"crossref","first-page":"961","DOI":"10.1137\/060657716","volume":"18","author":"R\u00f6misch","year":"2007","journal-title":"SIAM J. Optim"},{"key":"R31","doi-asserted-by":"crossref","unstructured":"Rusczy\u0144ski A. and Shapiro A., Stochastic programming models, Stochastic Programming, edited by Rusczy\u0144ski A. and Shapiro A.. In Vol. 10 of Handbooks in Operations Research and Management Science (2003) 1\u201364.","DOI":"10.1016\/S0927-0507(03)10001-1"},{"key":"R32","doi-asserted-by":"crossref","unstructured":"Rusczy\u0144ski A. and Shapiro A., Optimality and duality in stochastic programming, Stochastic Programming. edited by Rusczy\u0144ski A. and Shapiro A.. In Vol. 10 of Handbooks in Operations Research and Management Science (2003) 65\u2013139.","DOI":"10.1016\/S0927-0507(03)10002-3"},{"key":"R33","first-page":"71","volume":"2013","author":"Shanbhag","year":"2013","journal-title":"TUTORIALS Oper. Res"},{"key":"R34","doi-asserted-by":"crossref","first-page":"183","DOI":"10.1007\/s10107-006-0090-4","volume":"112","author":"Shapiro","year":"2008","journal-title":"Math. Program Ser. B"},{"key":"R35","doi-asserted-by":"crossref","first-page":"395","DOI":"10.1080\/02331930801954177","volume":"57","author":"Shapiro","year":"2008","journal-title":"Optimization"},{"key":"R36","doi-asserted-by":"crossref","first-page":"168","DOI":"10.2969\/jmsj\/02810168","volume":"28","author":"Takahashi","year":"1976","journal-title":"J. Math. Soc. Jpn"},{"key":"R37","doi-asserted-by":"crossref","first-page":"569","DOI":"10.1016\/j.jmaa.2009.07.062","volume":"363","author":"Ter\u00e1n","year":"2010","journal-title":"J. Math. Anal. Appl"},{"key":"R38","doi-asserted-by":"crossref","first-page":"27","DOI":"10.1007\/s10898-011-9739-5","volume":"54","author":"Wangkeeree","year":"2012","journal-title":"J. Global Optim"},{"key":"R39","unstructured":"Weber G.-W., Kruger A., Mart\u00ednez-Legaz J.E., Mordukhovich B. and Sakalauskas L., Special Issue on recent advances in continuous optimization on the occasion of the 25th European conference on Operational Research (EURO XXV 2012). Optimization 63 (2014) 1\u2013166."},{"key":"R40","doi-asserted-by":"crossref","unstructured":"Wets R.J.-B., Stochastic programming, edited by Nemhauser G.L.. In Vol. 1 of Handbooks in Operations Research and Management Science (1989) 573\u2013629.","DOI":"10.1016\/S0927-0507(89)01009-1"},{"key":"R41","doi-asserted-by":"crossref","first-page":"692","DOI":"10.1016\/j.jmaa.2010.03.021","volume":"368","author":"Xu","year":"2010","journal-title":"J. Math. Anal. Appl"},{"key":"R42","doi-asserted-by":"crossref","first-page":"103","DOI":"10.1142\/S0217595910002569","volume":"27","author":"Xu","year":"2010","journal-title":"Asia Pac. J. Oper. Res"},{"key":"R43","doi-asserted-by":"crossref","first-page":"1685","DOI":"10.1137\/090748974","volume":"20","author":"Xu","year":"2010","journal-title":"SIAM J. Optim"},{"key":"R44","doi-asserted-by":"crossref","first-page":"597","DOI":"10.1007\/s10589-013-9538-7","volume":"55","author":"Xu","year":"2013","journal-title":"Comput. Optim. Appl"},{"key":"R45","doi-asserted-by":"crossref","first-page":"69","DOI":"10.1016\/j.simpat.2017.09.006","volume":"79","author":"Zhang","year":"2017","journal-title":"Simul. Model Pract. Theory"}],"container-title":["RAIRO - Operations Research"],"original-title":[],"link":[{"URL":"https:\/\/www.rairo-ro.org\/10.1051\/ro\/2020001\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,3,2]],"date-time":"2021-03-02T09:25:46Z","timestamp":1614677146000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.rairo-ro.org\/10.1051\/ro\/2020001"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021]]},"references-count":45,"alternative-id":["ro190233"],"URL":"https:\/\/doi.org\/10.1051\/ro\/2020001","relation":{},"ISSN":["0399-0559","1290-3868"],"issn-type":[{"type":"print","value":"0399-0559"},{"type":"electronic","value":"1290-3868"}],"subject":[],"published":{"date-parts":[[2021]]}}}