变分问题的分数阶微积分方法：充分最优性条件和对偶性的表征

Q1 Mathematics

Partial Differential Equations in Applied Mathematics Pub Date : 2024-12-01 DOI:10.1016/j.padiff.2024.100999

Ved Prakash Dubey , Devendra Kumar , Jagdev Singh , Dumitru Baleanu

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引用次数: 0

摘要

本文针对Caputo-Fabrizio （CF）分数阶导数算子的变分问题，给出了一个包含Caputo-Fabrizio分数阶导数、弱和强对偶结果、Kuhn-Tucker型充分最优性条件的个数和对偶结果的wolfe型对偶模型。分数阶导数算子提供了一种非奇异性质的指数型核，可以更好地表征具有记忆特性的物理系统和工程过程的动力学特性。这个导数算子是一阶导数与指数函数的卷积。本文还针对具有指数型核的分数阶微分算子，导出了原始问题的全局最优性判据、Mond-Weir型对偶结果和Mangasarian型严格逆对偶定理。导出的定理研究了原始问题的全局最优解。本文的主要结果是具有CF导数的vp的对偶定理和充分最优性条件。鉴于所导出的最优性定理的应用，在指数性假设下建立了Mond-Weir型对偶结果。这些应用和结果推广了VPs的其他重要对偶结果，并提供了数学规划中具有广义不变性的对偶结果。几个常规结果也可以看作是本工作所得结果的特例。2010数学学科分类90c29；90 c46;26日,a33

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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 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.