Stationary Condition for Borwein Proper Efficient Solutions of Nonsmooth Multiobjective Problems with Vanishing Constraints
This paper discusses optimality conditions for Borwein proper efficient solutions of nonsmooth multiobjective optimization problems with vanishing constraints. A new notion in terms of contingent cone and upper directional derivative is introduced, and a necessary condition for the Borwein proper ef...
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MDPI AG
2022-12-01
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| Online Access: | https://www.mdpi.com/2227-7390/10/23/4569 |
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| author | Hui Huang Haole Zhu |
| author_facet | Hui Huang Haole Zhu |
| author_sort | Hui Huang |
| collection | DOAJ |
| description | This paper discusses optimality conditions for Borwein proper efficient solutions of nonsmooth multiobjective optimization problems with vanishing constraints. A new notion in terms of contingent cone and upper directional derivative is introduced, and a necessary condition for the Borwein proper efficient solution of the considered problem is derived. The concept of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ε</mi></semantics></math></inline-formula> proper Abadie data qualification is also introduced, and a necessary condition which is called a strictly strong stationary condition for Borwein proper efficient solutions is obtained. In view of the strictly strong stationary condition, convexity of the objective functions, and quasi-convexity of constrained functions, sufficient conditions for the Borwein proper efficient solutions are presented. Some examples are given to illustrate the reasonability of the obtained results. |
| first_indexed | 2024-03-09T17:41:04Z |
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| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-09T17:41:04Z |
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| spelling | doaj.art-da01b01e7c70413a9ecf359d0587894d2023-11-24T11:35:45ZengMDPI AGMathematics2227-73902022-12-011023456910.3390/math10234569Stationary Condition for Borwein Proper Efficient Solutions of Nonsmooth Multiobjective Problems with Vanishing ConstraintsHui Huang0Haole Zhu1Department of Mathematics, Yunnan University, Kunming 650091, ChinaDepartment of Mathematics, Yunnan University, Kunming 650091, ChinaThis paper discusses optimality conditions for Borwein proper efficient solutions of nonsmooth multiobjective optimization problems with vanishing constraints. A new notion in terms of contingent cone and upper directional derivative is introduced, and a necessary condition for the Borwein proper efficient solution of the considered problem is derived. The concept of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ε</mi></semantics></math></inline-formula> proper Abadie data qualification is also introduced, and a necessary condition which is called a strictly strong stationary condition for Borwein proper efficient solutions is obtained. In view of the strictly strong stationary condition, convexity of the objective functions, and quasi-convexity of constrained functions, sufficient conditions for the Borwein proper efficient solutions are presented. Some examples are given to illustrate the reasonability of the obtained results.https://www.mdpi.com/2227-7390/10/23/4569Borwein proper efficient solutionnonsmooth multiobjective optimizationstationary conditionClarke subdifferential |
| spellingShingle | Hui Huang Haole Zhu Stationary Condition for Borwein Proper Efficient Solutions of Nonsmooth Multiobjective Problems with Vanishing Constraints Mathematics Borwein proper efficient solution nonsmooth multiobjective optimization stationary condition Clarke subdifferential |
| title | Stationary Condition for Borwein Proper Efficient Solutions of Nonsmooth Multiobjective Problems with Vanishing Constraints |
| title_full | Stationary Condition for Borwein Proper Efficient Solutions of Nonsmooth Multiobjective Problems with Vanishing Constraints |
| title_fullStr | Stationary Condition for Borwein Proper Efficient Solutions of Nonsmooth Multiobjective Problems with Vanishing Constraints |
| title_full_unstemmed | Stationary Condition for Borwein Proper Efficient Solutions of Nonsmooth Multiobjective Problems with Vanishing Constraints |
| title_short | Stationary Condition for Borwein Proper Efficient Solutions of Nonsmooth Multiobjective Problems with Vanishing Constraints |
| title_sort | stationary condition for borwein proper efficient solutions of nonsmooth multiobjective problems with vanishing constraints |
| topic | Borwein proper efficient solution nonsmooth multiobjective optimization stationary condition Clarke subdifferential |
| url | https://www.mdpi.com/2227-7390/10/23/4569 |
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