Fractional Calculus Approach for Variational Problems: Characterization of Sufficient Optimality Conditions and Duality
In this paper, we present a Wolfe-type dual model containing the Caputo-Fabrizio fractional derivative, weak and strong duality results, number of Kuhn-Tucker type sufficient optimality conditions and duality results for variational problems (VPs) with Caputo-Fabrizio (CF) fractional derivative oper...
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Elsevier
2024-12-01
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| Series: | Partial Differential Equations in Applied Mathematics |
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| Online Access: | http://www.sciencedirect.com/science/article/pii/S2666818124003851 |
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| author | Ved Prakash Dubey Devendra Kumar Jagdev Singh Dumitru Baleanu |
| author_facet | Ved Prakash Dubey Devendra Kumar Jagdev Singh Dumitru Baleanu |
| author_sort | Ved Prakash Dubey |
| collection | DOAJ |
| description | In this paper, we present a Wolfe-type dual model containing the Caputo-Fabrizio fractional derivative, weak and strong duality results, number of Kuhn-Tucker type sufficient optimality conditions and duality results for variational problems (VPs) with Caputo-Fabrizio (CF) fractional derivative operator under weaker invexity assumptions. This newly developed fractional derivative operator delivers an exponential type kernel of nonsingular nature which characterizes the dynamics of physical systems and engineering processes with memory characteristics in a better way. This derivative operator is a convolution of first-order derivative and an exponential function. The proposed work also derives the global optimality criterion of the primal problem, Mond-Weir type duality results, and Mangasarian type strict converse duality theorem in view of this fractional differential operator possessing an exponential type kernel. The derived theorems investigate the global optimal solution of the primal problem. The main results of the present work are duality theorems and sufficient optimality conditions for VPs possessing the CF derivative. In view of applications of the derived optimality theorems, Mond-Weir type duality results have been established subjected to invexity assumptions. These applications and results generalize other important duality results of VPs and also provide results connected to duality with generalized invexity in mathematical programming. Several conventional results can also be seen as a special case of the obtained results in this work. 2010 Mathematics Subject Classification: 90C29; 90C46; 26A33 |
| format | Article |
| id | doaj-art-1ebc1539ed74463ca280dfe45746d8ee |
| institution | OA Journals |
| issn | 2666-8181 |
| language | English |
| publishDate | 2024-12-01 |
| publisher | Elsevier |
| record_format | Article |
| series | Partial Differential Equations in Applied Mathematics |
| spelling | doaj-art-1ebc1539ed74463ca280dfe45746d8ee2025-08-20T02:38:10ZengElsevierPartial Differential Equations in Applied Mathematics2666-81812024-12-011210099910.1016/j.padiff.2024.100999Fractional Calculus Approach for Variational Problems: Characterization of Sufficient Optimality Conditions and DualityVed Prakash Dubey0Devendra Kumar1Jagdev Singh2Dumitru Baleanu3Department of Bachelor of Computer Application, L.N.D. College (B.R. Ambedkar Bihar University, Muzaffarpur), Motihari 845401, Bihar, IndiaDepartment of Mathematics, University of Rajasthan, Jaipur 302004, Rajasthan, India; Department of Mathematics, Kyung Hee University, 26 Kyungheedae-ro, Dongdaemun-gu, Seoul, 02447, Korea; Corresponding author: Tel. +91-9460905223Department of Mathematics, JECRC University, Jaipur 303905, Rajasthan, India; Department of Computer Science and Mathematics, Lebanese American University, Beirut, LebanonDepartment of Computer Science and Mathematics, Lebanese American University, Beirut, Lebanon; Institute of Space Sciences, Magurele-Bucharest, RomaniaIn this paper, we present a Wolfe-type dual model containing the Caputo-Fabrizio fractional derivative, weak and strong duality results, number of Kuhn-Tucker type sufficient optimality conditions and duality results for variational problems (VPs) with Caputo-Fabrizio (CF) fractional derivative operator under weaker invexity assumptions. This newly developed fractional derivative operator delivers an exponential type kernel of nonsingular nature which characterizes the dynamics of physical systems and engineering processes with memory characteristics in a better way. This derivative operator is a convolution of first-order derivative and an exponential function. The proposed work also derives the global optimality criterion of the primal problem, Mond-Weir type duality results, and Mangasarian type strict converse duality theorem in view of this fractional differential operator possessing an exponential type kernel. The derived theorems investigate the global optimal solution of the primal problem. The main results of the present work are duality theorems and sufficient optimality conditions for VPs possessing the CF derivative. In view of applications of the derived optimality theorems, Mond-Weir type duality results have been established subjected to invexity assumptions. These applications and results generalize other important duality results of VPs and also provide results connected to duality with generalized invexity in mathematical programming. Several conventional results can also be seen as a special case of the obtained results in this work. 2010 Mathematics Subject Classification: 90C29; 90C46; 26A33http://www.sciencedirect.com/science/article/pii/S2666818124003851Variational problemsGeneralized invexityKuhn-Tucker type sufficient optimality conditionsMond-Weir type duality, Caputo-Fabrizio fractional derivative |
| spellingShingle | Ved Prakash Dubey Devendra Kumar Jagdev Singh Dumitru Baleanu Fractional Calculus Approach for Variational Problems: Characterization of Sufficient Optimality Conditions and Duality Partial Differential Equations in Applied Mathematics Variational problems Generalized invexity Kuhn-Tucker type sufficient optimality conditions Mond-Weir type duality, Caputo-Fabrizio fractional derivative |
| title | Fractional Calculus Approach for Variational Problems: Characterization of Sufficient Optimality Conditions and Duality |
| title_full | Fractional Calculus Approach for Variational Problems: Characterization of Sufficient Optimality Conditions and Duality |
| title_fullStr | Fractional Calculus Approach for Variational Problems: Characterization of Sufficient Optimality Conditions and Duality |
| title_full_unstemmed | Fractional Calculus Approach for Variational Problems: Characterization of Sufficient Optimality Conditions and Duality |
| title_short | Fractional Calculus Approach for Variational Problems: Characterization of Sufficient Optimality Conditions and Duality |
| title_sort | fractional calculus approach for variational problems characterization of sufficient optimality conditions and duality |
| topic | Variational problems Generalized invexity Kuhn-Tucker type sufficient optimality conditions Mond-Weir type duality, Caputo-Fabrizio fractional derivative |
| url | http://www.sciencedirect.com/science/article/pii/S2666818124003851 |
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