We study optimizationproblems 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, c...
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We study optimizationproblems 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.
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 thi...
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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.
In this study, a gH-penalty method is developed to obtain efficient solutions to constrained optimizationproblems with interval-valued functions. The algorithmic implementation of the proposed method is illustrated. ...
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In this study, a gH-penalty method is developed to obtain efficient solutions to constrained optimizationproblems 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.
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...
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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.
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....
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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.
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 conv...
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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 intervaloptimization problem. We also provide illustrative examples to support the study. (C) 2020 Elsevier Inc. All rights reserved.
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