Abstract
We discuss constraint qualifications in Karush–Kuhn–Tucker multiplier rules in nonsmooth semi-infinite multiobjective programming. A version of the Manganarian–Fromovitz constraint qualification is proposed, in terms of the Michel–Penot directional derivative and the Studniarski derivative of order p which is just the order of the directional Hölder metric subregularity which is included also in this proposed qualification version. Using this qualification together with the Pshenichnyi–Levitin–Valadire property, we establish Karush–Kuhn–Tucker optimality conditions for Borwein-proper and firm solutions. We also compare in detail our qualification version with other usually-employed constraint qualifications. Applications to semi-infinite multiobjective fractional problems and minimax problems are provided.
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This work was supported by Vietnam National University-Hochiminh City under the Grant B2018-28-02. A part of the work was completed during a stay of the authors at the Vietnam Institute for Advanced Study in Mathematics (VIASM). The authors would like to thank VIASM for its hospitality and support. They are grateful to the anonymous referees for their very helpful remarks and suggestions.
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Khanh, P.Q., Tung, N.M. On the Mangasarian–Fromovitz constraint qualification and Karush–Kuhn–Tucker conditions in nonsmooth semi-infinite multiobjective programming. Optim Lett 14, 2055–2072 (2020). https://doi.org/10.1007/s11590-019-01529-3
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DOI: https://doi.org/10.1007/s11590-019-01529-3