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A study of interval optimization problems

optimization LETTERS 2021年第3期15卷 859-877页

作者： Aguirre-Cipe, Ivan Lopez, Ruben Mallea-Zepeda, Exequiel Vasquez, Lautaro Univ Tarapaca Dept Matemat Arica Chile

We study optimization problems with interval objective functions. We focus on set-type solution notions defined using the Kulisch-Miranker order between intervals. We obtain bounds for the asymptotic cones of level, colevel and solution sets that allow us to deduce coercivity properties and coercive existence results. Finally, we obtain various noncoercive existence results. Our results are easy to check since they are given in terms of the asymptotic cone of the constraint set and the asymptotic functions of the end point functions. This work extends, unifies and sheds new light on the theory of these problems.

关键词： Asymptotic cones Asymptotic functions Coercivity properties Coercive and noncoercive existence results Set-type solutions interval optimization problems

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An Approach for Solving interval optimization problems 37th

An Approach for Solving Interval Optimization Problems

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37th Conference of the North-American-Fuzzy-Information-Processing-Society (NAFIPS)

作者： Villanueva, Fabiola Roxana de Oliveira, Valeriano Antunes Sao Paulo State Univ UNESP Inst Biosci Letters & Exact Sci Dept Appl Math Sao Jose Do Rio Preto SP Brazil

ISBN: (纸本)9783319953120;9783319953113

This work considers an optimization problem where the objective function possesses interval uncertainty in the coefficients. In this sense, first, an order relation will be defined for the interval space and, from this, it will be defined a solution concept for the interval problem in question. Subsequently, it will be shown that an interval problem is equivalent to a bi-objective problem. Finally, it will be established the necessary conditions of Fritz John and Karush-Kuhn-Tucker types for the interval-valued optimization problem.

关键词： Multi-objective optimization problems Karush-Kuhn-Tucker LU order relation LU solutions Classical necessary condition of Fritz John interval optimization problems Necessary optimality conditions for the LU-solution

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Generalized-Hukuhara penalty method for optimization problem with interval-valued functions and its application in interval-valued portfolio optimization problems

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OPERATIONS RESEARCH LETTERS 2022年第5期50卷 602-609页

作者： Debnath, Amit Kumar Ghosh, Debdas Indian Inst Technol BHU Varanasi Dept Math Sci Varanasi 221005 Uttar Pradesh India

In this study, a gH-penalty method is developed to obtain efficient solutions to constrained optimization problems with interval-valued functions. The algorithmic implementation of the proposed method is illustrated. In order to develop the gH-penalty method, an interval-valued penalty function is defined and the characterization of efficient solutions of a CIOP is done. As an application of the proposed method, a portfolio optimization problem with interval-valued return is solved.(c) 2022 Elsevier B.V. All rights reserved.

关键词： gH-penalty method interval-valued functions interval optimization problems interval-valued portfolio optimization

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Generalized Hukuhara Hadamard derivative of interval-valued functions and its applications to interval optimization

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SOFT COMPUTING 2023年第5期28卷 4107页

作者： Chauhan, Ram Surat Ghosh, Debdas Ansari, Qamrul Hasan Jaypee Inst Informat Technol Dept Math Sect 62 Noida 201309 India Indian Inst technol BHU Dept Math Sci Varanasi 221005 India Aligarh Muslim Univ Dept Math Aligarh 202002 India Chongqing Univ Technol Coll Sci Chongqing 400054 Peoples R China

In this article, we study the notion of gH-Hadamard derivative for interval-valued functions (IVFs) and apply it to solve interval optimization problems (IOPs). It is shown that the existence of gH-Hadamard derivative implies the existence of gH-Frechet derivative and vise-versa. Further, it is proved that the existence of gH-Hadamard derivative implies the existence of gH-continuity of IVFs. We found that the composition of a Hadamard differentiable real-valued function and a gH-Hadamard differentiable IVF is gH-Hadamard differentiable. Further, for finite comparable IVF, we prove that the gH-Hadamard derivative of the maximum of all finite comparable IVFs is the maximum of their gH-Hadamard derivative. The proposed derivative is observed to be useful to check the convexity of an IVF and to characterize efficient points of an optimization problem with IVF. For a convex IVF, we prove that if at a point the gH-Hadamard derivative does not dominate to zero, then the point is an efficient point. Further, it is proved that at an efficient point, the gH-Hadamard derivative does not dominate zero and also contains zero. For constraint IOPs, we prove an extended Karush-Kuhn-Tucker condition using the proposed derivative. The entire study is supported by suitable examples.

关键词： interval-valued functions interval optimization problems Efficient solutions gH-Hadamard derivative gH-Frechet derivative

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Generalized-Hukuhara subgradient and its application in optimization problem with interval-valued functions

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SADHANA-ACADEMY PROCEEDINGS IN ENGINEERING SCIENCES 2022年第2期47卷 1-16页

作者： Ghosh, Debdas Debnath, Amit Kumar Chauhan, Ram Surat Mesiar, Radko Indian Inst Technol BHU Dept Math Sci Varanasi 221005 Uttar Pradesh India Slovak Univ Technol Bratislava Fac Civil Engn Radlinskeho 11 Bratislava 81005 Slovakia Palacky Univ Olomouc Dept Algebra & Geometry Fac Sci 17 Listopadu 12 Olomouc 77146 Czech Republic

In this article, the concepts of gH-subgradient and gH-subdifferential of interval-valued functions are illustrated. Several important characteristics of the gH-subdifferential of a convex interval-valued function, e.g., closeness, boundedness, chain rule, etc. are studied. Alongside, we prove that gH-subdifferential of a gH-differentiable convex interval-valued function contains only the gH-gradient. It is observed that the directional gH-derivative of a convex interval-valued function is the maximum of all the products between gH-subgradients and the direction. Importantly, we prove that a convex interval-valued function is gH-Lipschitz continuous if it has gH-subgradients at each point in its domain. Furthermore, relations between efficient solutions of an optimization problem with interval-valued function and its gH-subgradients are derived.

关键词： Convex programming gH-subgradient gH-subdifferential interval optimization problems

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A variable and a fixed ordering of intervals and their application in optimization with interval-valued functions

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INTERNATIONAL JOURNAL OF APPROXIMATE REASONING 2020年 121卷 187-205页

作者： Ghosh, Debdas Debnath, Amit Kumar Pedrycz, Witold Indian Inst Technol BHU Varanasi Dept Math Sci Varanasi 221005 Uttar Pradesh India Univ Alberta Dept Elect & Comp Engn Edmonton AB T6R 2V4 Canada King Abdulaziz Univ Fac Engn Dept Elect & Comp Engn Jeddah 21589 Saudi Arabia Polish Acad Sci Syst Res Inst PL-01447 Warsaw Poland

In this study, we introduce and analyze the concepts of a fixed ordering structure and a variable ordering structure on intervals. The fixed ordering structures on intervals are defined with the help of a pointed convex cone of intervals. A variable ordering is defined by a set-valued map whose values are convex cones of intervals. In the sequel, a few properties of a cone of intervals are derived. It is shown that a binary relation, defined by a convex cone of intervals, is a partial order relation on intervals;further, the relation is antisymmetric if the convex cone of intervals is pointed. Several results under which a variable ordering map of intervals satisfies the conditions of a partial ordering relation of intervals are provided. The introduced fixed and variable ordering of intervals are applied to define and characterize optimal elements of an optimization problem with interval-valued functions. Finally, we propose a numerical technique and present its algorithmic implementation to obtain the set of optimal elements of an interval optimization problem. We also provide illustrative examples to support the study. (C) 2020 Elsevier Inc. All rights reserved.

关键词： Partial order relation of intervals Cone of intervals Variable ordering structure on intervals interval-valued functions interval optimization problems Nondominated solutions

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Contributions in interval optimization and interval optimal control

Contributions in interval optimization and interval optimal ...

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作者： Villanueva, Fabiola Roxana Universidade Estadual Paulista

Neste trabalho, primeiramente, serão apresentados problemas de otimização nos quais a função objetivo é de múltiplas variáveis e de valor intervalar e as restrições de... 详细信息

Neste trabalho, primeiramente, serão apresentados problemas de otimização nos quais a função objetivo é de múltiplas variáveis e de valor intervalar e as restrições de desigualdade são dadas por funcionais clássicos, isto é, de valor real. Serão dadas as condições de otimalidade usando a E−diferenciabilidade e, depois, a gH−diferenciabilidade total das funções com valor intervalar de várias variáveis. As condições necessárias de otimalidade usando a gH−diferenciabilidade total são do tipo KKT e as suficientes são do tipo de convexidade generalizada. Em seguida, serão estabelecidos problemas de controle ótimo nos quais a funçãao objetivo também é com valor intervalar de múltiplas variáveis e as restrições estão na forma de desigualdades e igualdades clássicas. Serão fornecidas as condições de otimalidade usando o conceito de Lipschitz para funções intervalares de várias variáveis e, logo, a gH−diferenciabilidade total das funções com valor intervalar de várias variáveis. As condições necessárias de otimalidade, usando a gH−diferenciabilidade total, estão na forma do célebre Princípio do Máximo de Pontryagin, mas desta vez na versão *** this work, firstly, it will be presented optimization problems in which the objective function is interval−valued of multiple variables and the inequality constraints are given by classical functionals, that is, real−valued ones. It will be given the optimality conditions using the E−differentiability and then the total gH−differentiability of interval−valued functions of several variables. The necessary optimality conditions using the total gH−differentiability are of KKT−type and the sufficient ones are of generalized convexity type. Next, it will be established optimal control problems in which the objective function is also interval−valued of multiple variables and the constraints are in the form of classical inequalities and equalities. It will be furnished the optimality conditions using the Lipschitz concept for interv

关键词： Problemas de otimização intervalar Condições de tipo Karush-Kuhn-Tucker Condições suficientes Problemas de controle ótimo intervalar Principio do Máximo de Pontryagin local intervalar interval optimization problems Karush-Kuhn-Tucker-type conditions

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