{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,21]],"date-time":"2025-11-21T12:35:01Z","timestamp":1763728501270,"version":"build-2065373602"},"reference-count":23,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2023,6,27]],"date-time":"2023-06-27T00:00:00Z","timestamp":1687824000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Fundamental Research Funds for the Central Universities","award":["2021KYQD23","2022XYZSX03","2022AAC03260","2022BSB03046","11861002","ZDZX201804","YCX23081"],"award-info":[{"award-number":["2021KYQD23","2022XYZSX03","2022AAC03260","2022BSB03046","11861002","ZDZX201804","YCX23081"]}]},{"name":"Natural Science Foundation of Ningxia Provincial of China","award":["2021KYQD23","2022XYZSX03","2022AAC03260","2022BSB03046","11861002","ZDZX201804","YCX23081"],"award-info":[{"award-number":["2021KYQD23","2022XYZSX03","2022AAC03260","2022BSB03046","11861002","ZDZX201804","YCX23081"]}]},{"name":"Key Research and Development Program of Ningxia","award":["2021KYQD23","2022XYZSX03","2022AAC03260","2022BSB03046","11861002","ZDZX201804","YCX23081"],"award-info":[{"award-number":["2021KYQD23","2022XYZSX03","2022AAC03260","2022BSB03046","11861002","ZDZX201804","YCX23081"]}]},{"name":"Natural Science Foundation of China","award":["2021KYQD23","2022XYZSX03","2022AAC03260","2022BSB03046","11861002","ZDZX201804","YCX23081"],"award-info":[{"award-number":["2021KYQD23","2022XYZSX03","2022AAC03260","2022BSB03046","11861002","ZDZX201804","YCX23081"]}]},{"name":"Key Project of North Minzu University","award":["2021KYQD23","2022XYZSX03","2022AAC03260","2022BSB03046","11861002","ZDZX201804","YCX23081"],"award-info":[{"award-number":["2021KYQD23","2022XYZSX03","2022AAC03260","2022BSB03046","11861002","ZDZX201804","YCX23081"]}]},{"name":"North Minzu University","award":["2021KYQD23","2022XYZSX03","2022AAC03260","2022BSB03046","11861002","ZDZX201804","YCX23081"],"award-info":[{"award-number":["2021KYQD23","2022XYZSX03","2022AAC03260","2022BSB03046","11861002","ZDZX201804","YCX23081"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>This paper is devoted to the investigation of optimality conditions and saddle point theorems for robust approximate quasi-weak efficient solutions for a nonsmooth uncertain multiobjective fractional semi-infinite optimization problem (NUMFP). Firstly, a necessary optimality condition is established by using the properties of the Gerstewitz\u2019s function. Furthermore, a kind of approximate pseudo\/quasi-convex function is defined for the problem (NUMFP), and under its assumption, a sufficient optimality condition is obtained. Finally, we introduce the notion of a robust approximate quasi-weak saddle point to the problem (NUMFP) and prove corresponding saddle point theorems.<\/jats:p>","DOI":"10.3390\/axioms12070635","type":"journal-article","created":{"date-parts":[[2023,6,28]],"date-time":"2023-06-28T00:50:52Z","timestamp":1687913452000},"page":"635","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Optimality Conditions of the Approximate Efficiency for Nonsmooth Robust Multiobjective Fractional Semi-Infinite Optimization Problems"],"prefix":"10.3390","volume":"12","author":[{"given":"Liu","family":"Gao","sequence":"first","affiliation":[{"name":"School of Mathematics and Information Science, North Minzu University, Yinchuan 750021, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Guolin","family":"Yu","sequence":"additional","affiliation":[{"name":"School of Mathematics and Information Science, North Minzu University, Yinchuan 750021, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Wenyan","family":"Han","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, Ningxia University, Yinchuan 750021, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,6,27]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"417","DOI":"10.37193\/CJM.2019.03.16","article-title":"On quasi approximate solutions for nonsmooth robust semi-infinite optimization problems","volume":"35","author":"Khantree","year":"2019","journal-title":"Carpathian J. Math."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"170","DOI":"10.1016\/j.jmaa.2008.10.009","article-title":"Nonsmooth semi-infinite programming problems with mixed constraints","volume":"351","author":"Kanzi","year":"2009","journal-title":"J. Math. Anal. Appl."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"2109","DOI":"10.1080\/02331934.2020.1763990","article-title":"Robust approximate optimal solutions for nonlinear semi-infinite programming with uncertainty","volume":"69","author":"Sun","year":"2020","journal-title":"Optimization"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"73","DOI":"10.1007\/s40840-023-01466-6","article-title":"On Isolated\/Properly Efficient Solutions in Nonsmooth Robust Semi-infinite Multiobjective Optimization","volume":"46","author":"Pham","year":"2023","journal-title":"Bull. Malays. Math. Sci. Soc."},{"key":"ref_5","first-page":"260","article-title":"Nondifferentiable fractional semi-infinite multiobjective optimization problems","volume":"44","author":"Chuong","year":"2016","journal-title":"Oper. Res."},{"key":"ref_6","first-page":"69","article-title":"Optimality and duality for nonsmooth multiobjective fractional semi-infinite programming problem","volume":"16","author":"Mishra","year":"2013","journal-title":"Adv. Nonlinear Var. Inequalities"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"3649","DOI":"10.2298\/FIL1614649A","article-title":"Sufficient optimality conditions for semi-infinite multiobjective fractional programming under (\u03a6,\u03c1)-V-invexity and generalized (\u03a6,\u03c1)-V-invexity","volume":"30","author":"Antczak","year":"2017","journal-title":"Filomat"},{"key":"ref_8","first-page":"1","article-title":"Optimality conditions for generalized convex nonsmooth uncertain multi-objective fractional programming","volume":"1","author":"Pan","year":"2022","journal-title":"J. Oper. Res. Soc."},{"key":"ref_9","unstructured":"Dem\u2019yanov, V.F., and Vasil\u2019ev, L.V. (1985). Optimization Software, Inc. Publications Division."},{"key":"ref_10","first-page":"1","article-title":"Optimality conditions for a nonsmooth uncertain multiobjective programming problem","volume":"2020","author":"Han","year":"2020","journal-title":"Complexity"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"287","DOI":"10.3934\/jimo.2008.4.287","article-title":"Optimality conditions, duality and saddle points for nondifferentiable multiobjective fractional programs","volume":"4","author":"Long","year":"2017","journal-title":"J. Ind. Manag. Optim."},{"key":"ref_12","first-page":"305","article-title":"On robust multiobjective optimization","volume":"40","author":"Kuroiwa","year":"2012","journal-title":"J. Nonlinear Convex. Anal."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Ben-Tal, A., Ghaoui, L.E., and Nemirovski, A. (2009). Robust Optimization, Princeton University Press.","DOI":"10.1515\/9781400831050"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"103","DOI":"10.1007\/s10203-017-0186-8","article-title":"Necessary conditions for nonsmooth multiobjective semi-infinite problems using Michel-Penot subdifferential","volume":"40","author":"Caristi","year":"2017","journal-title":"Decis. Econ. Financ."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"217","DOI":"10.1080\/02331934.2017.1393675","article-title":"Characterization of (weakly\/properly\/robust) efficient solutions in nonsmooth semi-infinite multiobjective optimization using convexificators","volume":"67","author":"Kabgani","year":"2018","journal-title":"Optimization"},{"key":"ref_16","unstructured":"Clarke, F.H. (1983). Optimization and Nonsmooth Analysis, Wiley."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"39","DOI":"10.1016\/j.ejor.2017.08.003","article-title":"On nonsmooth robust multiobjective optimization under generalized convexity with applications to portfolio optimization","volume":"265","author":"Fakhar","year":"2018","journal-title":"Eur. J. Oper. Res."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"275","DOI":"10.1007\/978-3-642-21114-0_8","article-title":"Optimality Conditions for Approximate Solutions of Convex Semi-Infinite Vector Optimization Problems","volume":"1","author":"Lee","year":"2012","journal-title":"Recent Dev. Vector Optim."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"411","DOI":"10.1007\/s10957-018-1437-8","article-title":"Optimality Conditions and Duality for Robust Nonsmooth Multiobjective Optimization Problems with Constraints","volume":"265","author":"Chen","year":"2019","journal-title":"J. Optim. Theory Appl."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"297","DOI":"10.1007\/BF00940478","article-title":"Nonconvex separation theorems and some applications in vector optimization","volume":"67","author":"Gerth","year":"1990","journal-title":"J. Optim. Theory Appl."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"4819","DOI":"10.1080\/02331934.2021.1969393","article-title":"Connectedness of the approximate solution sets for set optimization problems","volume":"71","author":"Han","year":"2022","journal-title":"Optimization"},{"key":"ref_22","unstructured":"Rockafellar, R.T. (1997). Convex Analysis, Princeton University Press."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"151","DOI":"10.1007\/s10288-022-00506-4","article-title":"Constraint qualifications and optimality conditions for robust nonsmooth semi-infinite multiobjective optimization problems","volume":"21","author":"Tung","year":"2022","journal-title":"4OR"}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/12\/7\/635\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T20:02:09Z","timestamp":1760126529000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/12\/7\/635"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,6,27]]},"references-count":23,"journal-issue":{"issue":"7","published-online":{"date-parts":[[2023,7]]}},"alternative-id":["axioms12070635"],"URL":"https:\/\/doi.org\/10.3390\/axioms12070635","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2023,6,27]]}}}