{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,4]],"date-time":"2026-02-04T18:04:05Z","timestamp":1770228245419,"version":"3.49.0"},"reference-count":37,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2024,11,13]],"date-time":"2024-11-13T00:00:00Z","timestamp":1731456000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"MICIN","award":["MCIN\/AEI\/PID2021-123051NB-100"],"award-info":[{"award-number":["MCIN\/AEI\/PID2021-123051NB-100"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>In this paper, we model uncertainty in both the objective function and the constraints for the robust semi-infinite interval equilibrium problem involving data uncertainty. We particularize these conditions for the robust semi-infinite mathematical programming problem with interval-valued functions by extending the results from the literature. We introduce the dual robust version of the above problem, prove the Mond\u2013Weir-type weak and strong duality theorems, and illustrate our results with an example.<\/jats:p>","DOI":"10.3390\/axioms13110781","type":"journal-article","created":{"date-parts":[[2024,11,13]],"date-time":"2024-11-13T06:23:16Z","timestamp":1731478996000},"page":"781","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Robust Semi-Infinite Interval Equilibrium Problem Involving Data Uncertainty: Optimality Conditions and Duality"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-0639-3776","authenticated-orcid":false,"given":"Gabriel","family":"Ruiz-Garz\u00f3n","sequence":"first","affiliation":[{"name":"Instituto de Desarrollo Social y Sostenible (INDESS), Universidad de C\u00e1diz, Campus de Jerez de la Frontera, Avda. de la Universidad s\/n, 11405 Jerez de la Frontera, Spain"}]},{"given":"Rafaela","family":"Osuna-G\u00f3mez","sequence":"additional","affiliation":[{"name":"Departamento de Estad\u00edstica e I.O., Universidad de Sevilla, 41012 Sevilla, Spain"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1286-2823","authenticated-orcid":false,"given":"Antonio","family":"Rufi\u00e1n-Lizana","sequence":"additional","affiliation":[{"name":"Departamento de Estad\u00edstica e I.O., Universidad de Sevilla, 41012 Sevilla, Spain"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9325-949X","authenticated-orcid":false,"given":"Antonio","family":"Beato-Moreno","sequence":"additional","affiliation":[{"name":"Departamento de Estad\u00edstica e I.O., Universidad de Sevilla, 41012 Sevilla, Spain"}]}],"member":"1968","published-online":{"date-parts":[[2024,11,13]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"305","DOI":"10.1007\/BF01353421","article-title":"A generalization of Tychonoff\u2019s fixed point theorem","volume":"142","author":"Fan","year":"1961","journal-title":"Math. Ann."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"391","DOI":"10.1007\/BF01585753","article-title":"Exchange price equilibria and variational inequalities","volume":"46","author":"Dafermos","year":"1990","journal-title":"Math. Program."},{"key":"ref_3","first-page":"807","article-title":"Gap functions and descend methods for equilibrium problems on Hadamard manifolds","volume":"17","author":"Li","year":"2016","journal-title":"J. Nonlinear Convex Anal."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"85","DOI":"10.1016\/j.apnum.2022.05.007","article-title":"An extragradient method for non-monotone equilibrium problems on Hadamard manifolds with applications","volume":"180","author":"Babu","year":"2022","journal-title":"Appl. Numer. Math."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"749","DOI":"10.1080\/02331930601122876","article-title":"Extragradient algorithms extended to equilibrium problems","volume":"57","author":"Tran","year":"2008","journal-title":"Optimization"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1069","DOI":"10.1007\/s10898-024-01377-1","article-title":"Simple proximal-type algorithms for equilibrium problems","volume":"89","author":"Yao","year":"2024","journal-title":"J. Glob. Optim."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"809","DOI":"10.1007\/s10957-013-0400-y","article-title":"Hybrid methods for solving simultaneously an equilibrium problem and countably many fixed point problems in a Hilbert space","volume":"160","author":"Nguyen","year":"2014","journal-title":"J. Optim. Theory Appl."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1402","DOI":"10.1137\/130939596","article-title":"Robust solutions of multiobjective linear semi-infinite programs under constraint data uncertainty","volume":"24","author":"Goberna","year":"2014","journal-title":"SIAM J. Optim."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"471","DOI":"10.1016\/j.ejor.2013.09.036","article-title":"Recent advances in robust optimization: An overview","volume":"235","author":"Gabrel","year":"2014","journal-title":"Eur. J. Oper. Res."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"346","DOI":"10.1016\/j.apenergy.2018.02.121","article-title":"Robust optimization for energy transactions in multi-microgrids under uncertainty","volume":"217","author":"Zhang","year":"2018","journal-title":"Appl. Energy"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"487","DOI":"10.1007\/s10479-017-2751-5","article-title":"Robust multiobjective optimization with application to Internet routing","volume":"271","author":"Doolittle","year":"2018","journal-title":"Ann. Oper. Res."},{"key":"ref_12","first-page":"1","article-title":"Uber lineare Ungleichungen","volume":"2","author":"Haar","year":"1924","journal-title":"Acta Math. Szeged"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"783","DOI":"10.1073\/pnas.48.5.783","article-title":"Duality, Haar programs and finite sequence spaces","volume":"48","author":"Charnes","year":"1962","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_14","unstructured":"Jeyakumar, V., and Rubinov, A.M. (2005). Linear semi-infinite optimization: Recent advances. Continuous Optimization, Current Trends and Modern Applications, Springer Science & Business Media. Applied Optimization Series."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"390","DOI":"10.1016\/S0377-2217(02)00327-2","article-title":"Linear semi-infinite programming theory: An updated survey","volume":"143","author":"Goberna","year":"2002","journal-title":"Eur. J. Oper. Res."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"491","DOI":"10.1016\/j.ejor.2006.08.045","article-title":"Semi-infinite programming","volume":"180","author":"Still","year":"2007","journal-title":"Eur. J. Oper. Res."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"607","DOI":"10.1016\/S0377-2217(03)00266-2","article-title":"Robot trajectory planning with semi-infinite programming","volume":"153","author":"Vaz","year":"2004","journal-title":"Eur. J. Oper. Res."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"723","DOI":"10.1007\/s10898-024-01367-3","article-title":"Efficiency conditions and duality for multiobjective semi-infinite programming problems on Hadamard manifolds","volume":"89","author":"Upadhyay","year":"2024","journal-title":"J. Glob. Optim."},{"key":"ref_19","unstructured":"Moore, R.E. (1966). Interval Analysis, Prentice Hall."},{"key":"ref_20","unstructured":"Lodwick, W.A. (1999). Constrained Interval Arithmetic, University of Colorado at Denver, Center for Computational Mathematics."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"137","DOI":"10.1007\/s12351-015-0172-2","article-title":"Sufficiency and duality for optimization problems involving interval-valued invex function in parametric form","volume":"15","author":"Jayswal","year":"2015","journal-title":"Oper. Res.\u2014Int. J. (ORIJ)"},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Lodwick, W., Newman, F., and Neumaier, A. (2001, January 2\u20135). Optimization under uncertainty: Methods and applications in radiation therapy. Proceedings of the 10th IEEE International Conference on Fuzzy Systems (Cat. No.01CH37297), Melbourne, VIC, Australia.","DOI":"10.1109\/FUZZ.2001.1008877"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"871","DOI":"10.1016\/j.ins.2021.08.045","article-title":"Interval nonlinear initial-valued problem using constraint intervals: Theory and an application to the SARS-CoV-2 outbreak","volume":"577","author":"Cecconello","year":"2021","journal-title":"Inf. Sci."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"262","DOI":"10.1016\/j.jmatprotec.2006.08.002","article-title":"The optimization of the variable binder force in U-shaped forming with uncertain friction coefficien","volume":"182","author":"Jiang","year":"2007","journal-title":"J. Mater. Process. Technol."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"41","DOI":"10.1016\/j.ins.2017.06.015","article-title":"Calculating the possible conformations arising from uncertainty in the molecular distance geometry problem using constraint interval analysis","volume":"425\u2013416","author":"Costa","year":"2017","journal-title":"Inform. Sci."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"235","DOI":"10.1016\/j.ins.2017.08.022","article-title":"New efficiency conditions for multiobjective interval-valued programming problems","volume":"420","year":"2017","journal-title":"Inf. Sci."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"246","DOI":"10.1016\/S0022-247X(02)00535-8","article-title":"Generalized vector quasi-equilibrium problems with applications","volume":"277","author":"Ansari","year":"2003","journal-title":"J. Math. Anal. Appl."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"842715","DOI":"10.1155\/2010\/842715","article-title":"Kuhn-Tucker optimality conditions for vector equilibrium problems","volume":"2010","author":"Wei","year":"2010","journal-title":"J. Inequal. Appl."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"597","DOI":"10.4134\/CKMS.2013.28.3.597","article-title":"Duality theorem and vector saddle point theorem for robust multiobjective optimization problems","volume":"28","author":"Kim","year":"2013","journal-title":"Commun. Korean Math. Soc."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"67","DOI":"10.1007\/s12190-019-01274-x","article-title":"Karush-Kuhn-Tucker optimality conditions and duality for convex semi-infinite programming with multiple interval-valued objective functions","volume":"62","author":"Tung","year":"2020","journal-title":"J. Appl. Math. Comput."},{"key":"ref_31","first-page":"87","article-title":"Robust optimality conditions and duality in semi-infinite multiobjective programming","volume":"91","author":"Ahmad","year":"2022","journal-title":"Acta Math. Univ. Comen."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"1391","DOI":"10.1007\/s40840-015-0237-7","article-title":"Nonsmooth interval-valued optimization and saddle-point optimality criteria","volume":"39","author":"Jayswal","year":"2016","journal-title":"Bull. Malays. Math. Sci. Soc."},{"key":"ref_33","first-page":"475","article-title":"Robust semi-infinite interval-valued optimization problem with uncertain inequality constraints","volume":"30","author":"Jaichander","year":"2022","journal-title":"Korean J. Math."},{"key":"ref_34","first-page":"8","article-title":"On nondifferentiable semi-infinite multiobjective programming with interval-valued functions","volume":"19","author":"Antczak","year":"2022","journal-title":"J. Ind. Manag. Optim."},{"key":"ref_35","doi-asserted-by":"crossref","unstructured":"Ruiz-Garz\u00f3n, G., Osuna-G\u00f3mez, R., and Ruiz-Zapatero, J. (2019). Necessary and sufficient optimality conditions for vector equilibrium problem on Hadamard manifolds. Symmetry, 11.","DOI":"10.20944\/preprints201907.0177.v1"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"248","DOI":"10.1007\/s40314-023-02378-8","article-title":"Semi-infinite interval equilibrium problems: Optimality conditions and existence results","volume":"42","year":"2023","journal-title":"Comp. Appl. Math."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"2641","DOI":"10.1007\/s12190-024-02067-7","article-title":"Robust optimality conditions for semi-infinite equilibrium problems involving data uncertainty","volume":"70","author":"Tripathi","year":"2024","journal-title":"J. Appl. Math. Comput."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/11\/781\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T16:31:29Z","timestamp":1760113889000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/11\/781"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,11,13]]},"references-count":37,"journal-issue":{"issue":"11","published-online":{"date-parts":[[2024,11]]}},"alternative-id":["axioms13110781"],"URL":"https:\/\/doi.org\/10.3390\/axioms13110781","relation":{},"ISSN":["2075-1680"],"issn-type":[{"value":"2075-1680","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,11,13]]}}}