{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,5]],"date-time":"2026-01-05T15:27:33Z","timestamp":1767626853198,"version":"build-2065373602"},"reference-count":44,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2025,2,20]],"date-time":"2025-02-20T00:00:00Z","timestamp":1740009600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"TAIF University","award":["TU-DSPP-2024-258"],"award-info":[{"award-number":["TU-DSPP-2024-258"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>The present paper investigates new connections and characterization results on interval-valued minimization models. Specifically, we describe the solution set of the considered control problem with mixed constraints by employing the solution set associated with a class of controlled split variational inequalities. These equivalence results are also accompanied by suitable numerical experiments.<\/jats:p>","DOI":"10.3390\/axioms14030151","type":"journal-article","created":{"date-parts":[[2025,2,20]],"date-time":"2025-02-20T11:03:37Z","timestamp":1740049417000},"page":"151","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Characterization Results of Extremization Models with Interval Values"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-8209-3869","authenticated-orcid":false,"given":"Savin","family":"Trean\u0163\u0103","sequence":"first","affiliation":[{"name":"Department Applied Mathematics, National University of Science and Technology Politehnica Bucharest, 060042 Bucharest, Romania"},{"name":"Academy of Romanian Scientists, 54 Splaiul Independentei, 050094 Bucharest, Romania"},{"name":"Fundamental Sciences Applied in Engineering Research Center, National University of Science and Technology Politehnica Bucharest, 060042 Bucharest, Romania"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7530-6264","authenticated-orcid":false,"given":"Omar Mutab","family":"Alsalami","sequence":"additional","affiliation":[{"name":"Department of Electrical Engineering, College of Engineering, Taif University, Taif 21944, Saudi Arabia"}]}],"member":"1968","published-online":{"date-parts":[[2025,2,20]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"271","DOI":"10.1007\/BF02392210","article-title":"On some non-linear elliptic differential-functional equations","volume":"115","author":"Hartman","year":"1966","journal-title":"Acta Math."},{"key":"ref_2","unstructured":"Giannessi, F. (1980). Theorems of alternative, quadratic programs and complementarity problems. Variational Inequalities and Complementarity Problems (Proc. Internat. School, Erice, 1978), Wiley."},{"key":"ref_3","unstructured":"Moore, R.E. (1966). Interval Analysis, Prentice-Hall."},{"key":"ref_4","unstructured":"Moore, R.E. (1979). Methods and Applications of Interval Analysis, SIAM. SIAM Studies in Applied Mathematics 2."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"1311","DOI":"10.1016\/j.na.2008.12.005","article-title":"Generalized Hukuhara differentiability of interval-valued functions and interval differential equations","volume":"71","author":"Stefanini","year":"2009","journal-title":"Nonlinear Anal."},{"key":"ref_6","first-page":"4119","article-title":"On sufficiency and duality for a class of interval-valued programming problems","volume":"218","author":"Jayswal","year":"2011","journal-title":"Appl. Math. Comput."},{"key":"ref_7","unstructured":"Liu, Y. (2015). Variational Inequalities and Optimization Problems. [Ph.D. Thesis, University of Liverpool]."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"14434","DOI":"10.3934\/math.2022795","article-title":"On some vector variational inequalities and optimization problems","volume":"7","year":"2022","journal-title":"AIMS Math."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"739","DOI":"10.1016\/j.ejor.2016.05.006","article-title":"Multitime multiobjective variational problems and vector variational-like inequalities","volume":"254","author":"Jayswal","year":"2016","journal-title":"Eur. J. Oper. Res."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"33","DOI":"10.1007\/s10700-015-9212-x","article-title":"Relationships between interval-valued vector optimization problems and vector variational inequalities","volume":"15","author":"Zhang","year":"2016","journal-title":"Fuzzy Optim. Decis. Mak."},{"key":"ref_11","first-page":"100044","article-title":"Characterization of LU-efficiency and saddle-point criteria for F-approximated multiobjective interval-valued variational problems","volume":"4","author":"Jha","year":"2021","journal-title":"Res. Control Optim."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Trean\u0163\u0103, S. (2021). On a new class of interval-valued variational control problems. Metric Fixed Point Theory, Springer.","DOI":"10.1007\/978-981-16-4896-0_10"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"913","DOI":"10.1007\/s10957-021-01815-0","article-title":"On a class of constrained interval-valued optimization problems governed by mechanical work cost functionals","volume":"188","year":"2021","journal-title":"J. Optim. Theory Appl."},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Guo, Y., Ye, G., Liu, W., Zhao, D., and Trean\u0163\u0103, S. (2021). Optimality conditions and duality for a class of generalized convex interval-valued optimization problems. Mathematics, 9.","DOI":"10.3390\/math9222979"},{"key":"ref_15","first-page":"5","article-title":"Interval and fuzzy analysis: A unified approach","volume":"142","author":"Lodwick","year":"2007","journal-title":"Adv. Imaging Electron Phys."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"84","DOI":"10.1016\/0022-247X(64)90086-1","article-title":"Bounds for functionally convex optimal control problems","volume":"8","author":"Hanson","year":"1964","journal-title":"J. Math. Anal. Appl."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1133","DOI":"10.1016\/S0252-9602(17)30062-0","article-title":"Optimality conditions and duality results for nonsmooth vector optimization problems with the multiple interval-valued objective function","volume":"37","author":"Antczak","year":"2017","journal-title":"Acta Math. Sci."},{"key":"ref_18","first-page":"879","article-title":"An iterative algorithm for testing solvability of max-min interval systems","volume":"48","author":"Myskova","year":"2012","journal-title":"Kybernetika"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"46","DOI":"10.1016\/j.ejor.2005.09.007","article-title":"The Karush-Kuhn-Tucker optimality conditions in an optimization problem with interval-valued objective function","volume":"176","author":"Wu","year":"2007","journal-title":"Eur. J. Oper. Res."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"607","DOI":"10.1007\/s11590-012-0601-6","article-title":"The KKT optimality conditions in a class of generalized convex optimization problems with an interval-valued objective function","volume":"8","author":"Zhang","year":"2014","journal-title":"Optim. Lett."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"113083","DOI":"10.1016\/j.chaos.2022.113083","article-title":"On variational methods for interval-valued functions with some applications","volume":"167","author":"Zhang","year":"2023","journal-title":"Chaos Solitons Fractals"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"421","DOI":"10.1016\/S0377-0427(00)00342-3","article-title":"Interval analysis: Theory and applications","volume":"121","author":"Alefeld","year":"2000","journal-title":"J. Comput. Appl. Math."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"219","DOI":"10.1016\/0377-2217(90)90375-L","article-title":"Multi-objective programming in optimization of the interval objective function","volume":"48","author":"Ishibuchi","year":"1990","journal-title":"Eur. J. Oper. Res."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"49","DOI":"10.1016\/j.ejor.2008.03.012","article-title":"The Karush-Kuhn-Tucker optimality conditions in multiobjective programming problems with interval-valued objective functions","volume":"196","author":"Wu","year":"2009","journal-title":"Eur. J. Oper. Res."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"7783","DOI":"10.1002\/rnc.5153","article-title":"Resilient interval consensus in robust networks","volume":"30","author":"Shang","year":"2020","journal-title":"Int. J. Robust Nonlinear Control"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"351","DOI":"10.1080\/00207721.2021.1954721","article-title":"Interval consensus of switched multiagent systems","volume":"53","author":"Shang","year":"2022","journal-title":"Int. J. Syst. Sci."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"125","DOI":"10.1186\/s13660-020-02393-x","article-title":"New Hermite-Hadamard type inequalities for n-polynomial harmonically convex functions","volume":"2020","author":"Awan","year":"2020","journal-title":"J. Inequalities Appl."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"317","DOI":"10.1186\/s13660-019-2272-7","article-title":"Hermite-Hadamard type inequalities for co-ordinated convex and quasi-convex functions and their applications","volume":"2019","author":"Latif","year":"2019","journal-title":"J. Inequalities Appl."},{"key":"ref_29","first-page":"9487823","article-title":"Integral majorization type inequalities for the functions in the sense of strong convexity","volume":"2019","author":"Ullah","year":"2019","journal-title":"J. Funct. Space"},{"key":"ref_30","first-page":"130","article-title":"p-convex functions and their properties","volume":"23","author":"Zhang","year":"2007","journal-title":"Pure Appl. Math."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"53","DOI":"10.1002\/mana.200810197","article-title":"The Schur multiplicative and harmonic convexities of the complete symmetric function","volume":"284","author":"Chu","year":"2011","journal-title":"Math. Nachrichten"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"20","DOI":"10.1007\/s00041-012-9223-8","article-title":"On the Gagliardo-Nirenberg type inequality in the critical Sobolev-Orrey space","volume":"19","author":"Sawano","year":"2013","journal-title":"J. Fourier Anal. Appl."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"365","DOI":"10.1016\/j.aim.2014.12.040","article-title":"Hardy and uncertainty inequalities on stratified Lie groups","volume":"277","author":"Ciatti","year":"2015","journal-title":"Adv. Math."},{"key":"ref_34","first-page":"33","article-title":"On some Ostrowski type inequalities","volume":"18","author":"Gavrea","year":"2010","journal-title":"Gen. Math."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"31","DOI":"10.5666\/KMJ.2009.49.1.031","article-title":"Fractional integrals and generalized Olsen inequalities","volume":"49","author":"Gunawan","year":"2009","journal-title":"Kyungpook Math. J."},{"key":"ref_36","first-page":"675","article-title":"Some quantum estimates for Hermite\u2013Hadamard inequalities","volume":"251","author":"Noor","year":"2015","journal-title":"Appl. Math. Comput."},{"key":"ref_37","first-page":"242","article-title":"Some quantum integral inequalities via preinvex functions","volume":"269","author":"Noor","year":"2015","journal-title":"Appl. Math. Comput."},{"key":"ref_38","doi-asserted-by":"crossref","unstructured":"Trean\u0163\u0103, S., and Saeed, T. (2023). On Weak Variational Control Inequalities via Interval Analysis. Mathematics, 11.","DOI":"10.3390\/math11092177"},{"key":"ref_39","doi-asserted-by":"crossref","unstructured":"Saunders, D.J. (1989). The Geometry of Jet Bundles, Cambridge University Press.","DOI":"10.1017\/CBO9780511526411"},{"key":"ref_40","unstructured":"Valentin, F.A. (1937). The problem of Lagrange with differentiable inequality as added side conditions. Contributions to the Calculus of Variations, University of Chicago Press."},{"key":"ref_41","unstructured":"Chankong, V., and Haimes, Y.Y. (1983). Multiobjective Decision Making: Theory and Methodology, Courier Dover Publications."},{"key":"ref_42","first-page":"100300","article-title":"On some connections between interval-valued variational control problems and the associated inequalities","volume":"12","author":"Ciontescu","year":"2023","journal-title":"Res. Control Optim."},{"key":"ref_43","first-page":"100324","article-title":"New classes of interval-valued variational problems and inequalities","volume":"13","author":"Saeed","year":"2023","journal-title":"Res. Control Optim."},{"key":"ref_44","doi-asserted-by":"crossref","unstructured":"Jayswal, A., and Trean\u0163\u0103, S. (2022). Multi-Dimensional Control Problems: Robust Approach, Springer.","DOI":"10.1007\/978-981-19-6561-6"}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/3\/151\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T16:38:52Z","timestamp":1760027932000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/3\/151"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,2,20]]},"references-count":44,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2025,3]]}},"alternative-id":["axioms14030151"],"URL":"https:\/\/doi.org\/10.3390\/axioms14030151","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2025,2,20]]}}}