Lagrange Duality and Saddle-Point Optimality Conditions for Nonsmooth Interval-Valued Multiobjective Semi-Infinite Programming Problems with Vanishing Constraints
This article deals with a class of nonsmooth interval-valued multiobjective semi-infinite programming problems with vanishing constraints (NIMSIPVC). We introduce the VC-Abadie constraint qualification (VC-ACQ) for NIMSIPVC and employ it to establish Karush–Kuhn–Tucker (KKT)-type necessary optimalit...
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2024-08-01
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| author | Balendu Bhooshan Upadhyay Shivani Sain Ioan Stancu-Minasian |
| author_facet | Balendu Bhooshan Upadhyay Shivani Sain Ioan Stancu-Minasian |
| author_sort | Balendu Bhooshan Upadhyay |
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| description | This article deals with a class of nonsmooth interval-valued multiobjective semi-infinite programming problems with vanishing constraints (NIMSIPVC). We introduce the VC-Abadie constraint qualification (VC-ACQ) for NIMSIPVC and employ it to establish Karush–Kuhn–Tucker (KKT)-type necessary optimality conditions for the considered problem. Regarding NIMSIPVC, we formulate interval-valued weak vector, as well as interval-valued vector Lagrange-type dual and scalarized Lagrange-type dual problems. Subsequently, we establish the weak, strong, and converse duality results relating the primal problem NIMSIPVC and the corresponding dual problems. Moreover, we introduce the notion of saddle points for the interval-valued vector Lagrangian and scalarized Lagrangian of NIMSIPVC. Furthermore, we derive the saddle-point optimality criteria for NIMSIPVC by establishing the relationships between the solutions of NIMSIPVC and the saddle points of the corresponding Lagrangians of NIMSIPVC, under convexity assumptions. Non-trivial illustrative examples are provided to demonstrate the validity of the established results. The results presented in this paper extend the corresponding results derived in the existing literature from smooth to nonsmooth optimization problems, and we further generalize them for interval-valued multiobjective semi-infinite programming problems with vanishing constraints. |
| format | Article |
| id | doaj-art-71b476fcf73e47e3b0b7f6db34bbd3d8 |
| institution | OA Journals |
| issn | 2075-1680 |
| language | English |
| publishDate | 2024-08-01 |
| publisher | MDPI AG |
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| series | Axioms |
| spelling | doaj-art-71b476fcf73e47e3b0b7f6db34bbd3d82025-08-20T01:56:10ZengMDPI AGAxioms2075-16802024-08-0113957310.3390/axioms13090573Lagrange Duality and Saddle-Point Optimality Conditions for Nonsmooth Interval-Valued Multiobjective Semi-Infinite Programming Problems with Vanishing ConstraintsBalendu Bhooshan Upadhyay0Shivani Sain1Ioan Stancu-Minasian2Department of Mathematics, Indian Institute of Technology Patna, Patna 801103, Bihar, IndiaDepartment of Mathematics, Indian Institute of Technology Patna, Patna 801103, Bihar, India“Gheorghe Mihoc-Caius Iacob” Institute of Mathematical Statistics and Applied Mathematics of the Romanian Academy, 050711 Bucharest, RomaniaThis article deals with a class of nonsmooth interval-valued multiobjective semi-infinite programming problems with vanishing constraints (NIMSIPVC). We introduce the VC-Abadie constraint qualification (VC-ACQ) for NIMSIPVC and employ it to establish Karush–Kuhn–Tucker (KKT)-type necessary optimality conditions for the considered problem. Regarding NIMSIPVC, we formulate interval-valued weak vector, as well as interval-valued vector Lagrange-type dual and scalarized Lagrange-type dual problems. Subsequently, we establish the weak, strong, and converse duality results relating the primal problem NIMSIPVC and the corresponding dual problems. Moreover, we introduce the notion of saddle points for the interval-valued vector Lagrangian and scalarized Lagrangian of NIMSIPVC. Furthermore, we derive the saddle-point optimality criteria for NIMSIPVC by establishing the relationships between the solutions of NIMSIPVC and the saddle points of the corresponding Lagrangians of NIMSIPVC, under convexity assumptions. Non-trivial illustrative examples are provided to demonstrate the validity of the established results. The results presented in this paper extend the corresponding results derived in the existing literature from smooth to nonsmooth optimization problems, and we further generalize them for interval-valued multiobjective semi-infinite programming problems with vanishing constraints.https://www.mdpi.com/2075-1680/13/9/573interval-valued multiobjective programmingvanishing constraintsClarke subdifferentialLagrange dualitysaddle point |
| spellingShingle | Balendu Bhooshan Upadhyay Shivani Sain Ioan Stancu-Minasian Lagrange Duality and Saddle-Point Optimality Conditions for Nonsmooth Interval-Valued Multiobjective Semi-Infinite Programming Problems with Vanishing Constraints Axioms interval-valued multiobjective programming vanishing constraints Clarke subdifferential Lagrange duality saddle point |
| title | Lagrange Duality and Saddle-Point Optimality Conditions for Nonsmooth Interval-Valued Multiobjective Semi-Infinite Programming Problems with Vanishing Constraints |
| title_full | Lagrange Duality and Saddle-Point Optimality Conditions for Nonsmooth Interval-Valued Multiobjective Semi-Infinite Programming Problems with Vanishing Constraints |
| title_fullStr | Lagrange Duality and Saddle-Point Optimality Conditions for Nonsmooth Interval-Valued Multiobjective Semi-Infinite Programming Problems with Vanishing Constraints |
| title_full_unstemmed | Lagrange Duality and Saddle-Point Optimality Conditions for Nonsmooth Interval-Valued Multiobjective Semi-Infinite Programming Problems with Vanishing Constraints |
| title_short | Lagrange Duality and Saddle-Point Optimality Conditions for Nonsmooth Interval-Valued Multiobjective Semi-Infinite Programming Problems with Vanishing Constraints |
| title_sort | lagrange duality and saddle point optimality conditions for nonsmooth interval valued multiobjective semi infinite programming problems with vanishing constraints |
| topic | interval-valued multiobjective programming vanishing constraints Clarke subdifferential Lagrange duality saddle point |
| url | https://www.mdpi.com/2075-1680/13/9/573 |
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