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Leveraging Space-Filling Curves and the DIRECT Algorithm. A Novel Approach to Derivative-Free Multi-Dimensional Global Optimization - Couverture souple
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Bachelor Thesis from the year 2018 in the subject Mathematics - Applied Mathematics, Nanyang Technological University, language: English, abstract: Imagine unlocking solutions to the most complex, multi-dimensional optimization problems with unprecedented efficiency. This groundbreaking work introduces a novel adaptation of the Dividing Rectangles (DIRECT) algorithm, ingeniously enhanced through the application of space-filling curves. Delve into a revolutionary approach that transforms intricate, high-dimensional challenges into simpler, one-dimensional equivalents, effectively sidestepping the notorious computational bottlenecks that plague traditional methods. Explore the depths of Lipschitzian optimization and witness the elegant integration of the Hölder condition, a cornerstone in the quest for global optima. Uncover the secrets behind the construction of the Hilbert curve and its transformative role in mapping points for optimized sampling. This research meticulously details the implementation of both the Hilbert Space Filling Curve module and the modified DIRECT module, providing a comprehensive blueprint for practical application. Through rigorous simulations and detailed analysis, witness firsthand the enhanced performance of the modified DIRECT algorithm, surpassing its standard counterpart in both speed and accuracy. Discover how this innovative technique achieves significant dimensionality reduction, opening new avenues for solving previously intractable problems in diverse fields ranging from engineering to finance. This study not only presents a theoretical framework but also provides concrete evidence of its effectiveness, paving the way for future advancements in global optimization and algorithm efficiency. This report offers a beacon of innovation in the realm of computational complexity, providing a pathway to more efficient and effective solutions in multi-dimensional optimization. The insights presented within will empower researchers and pract
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Leveraging Space-Filling Curves and the DIRECT Algorithm. A Novel Approach to Derivative-Free Multi-Dimensional Global Optimization
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Taschenbuch. Etat : Neu. Druck auf Anfrage Neuware - Printed after ordering - Bachelor Thesis from the year 2018 in the subject Mathematics - Applied Mathematics, Nanyang Technological University, language: English, abstract: This study aims to focus on a new approach to the Dividing Rectangles or DIRECT algorithm, which is used to solve multi-dimensional global optimization problems. It also delves into the resolution of DIRECT's combinatorial complexity in higher dimensions by transforming the problem domain into its one-dimensional equivalent.Many real-world problems involve multivariate global optimization which can be difficult to solve. In this report, a new approach to the Dividing Rectangles or DIRECT algorithm for solving multi-dimensional global optimization problems with bounds and a real-valued objective function, is discussed. DIRECT is a variation of the standard Lipschitzian optimization omitting the requirement of having to specify a Lipschitz constant; by viewing the Lipschitzian constant as a weighting parameter for indicating the emphasis to be placed on global versus local search. Typically, this constant is not so small in standard Lipschitz approaches, since the constant needs be at least as large as the maximum rate of change of the objective function; which forces a higher emphasis on global search and thus, results in a slower convergence. However, DIRECT enables operation at both global and local level by concurrently searching using all possible constants. The global part of the algorithm figures out the basin of convergence of the optimum, which the local part of the algorithm can suitably exploit. This justifies the fast convergence of DIRECT in computing the approximate minimum with the guaranteed precision. One major drawback of DIRECT is its combinatorial complexity in higher dimensions where DIRECT often takes many more function evaluations to find a good approximation of the global minimum. One method to resolve this difficulty is to transform the problem domain into its one-dimensional equivalent. This approach is demonstrated in this report using space filling curves, to reduce the multiextremal optimization problem to the minimization of a univariate function. In this case, the Hölder continuity of space filling curves has been exploited to solve global optimization problems. N° de réf. du vendeur 9783346913760
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Leveraging Space-Filling Curves and the DIRECT Algorithm. A Novel Approach to Derivative-Free Multi-Dimensional Global Optimization
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Taschenbuch. Etat : Neu. This item is printed on demand - it takes 3-4 days longer - Neuware -Bachelor Thesis from the year 2018 in the subject Mathematics - Applied Mathematics, Nanyang Technological University, language: English, abstract: This study aims to focus on a new approach to the Dividing Rectangles or DIRECT algorithm, which is used to solve multi-dimensional global optimization problems. It also delves into the resolution of DIRECT's combinatorial complexity in higher dimensions by transforming the problem domain into its one-dimensional equivalent.Many real-world problems involve multivariate global optimization which can be difficult to solve. In this report, a new approach to the Dividing Rectangles or DIRECT algorithm for solving multi-dimensional global optimization problems with bounds and a real-valued objective function, is discussed. DIRECT is a variation of the standard Lipschitzian optimization omitting the requirement of having to specify a Lipschitz constant; by viewing the Lipschitzian constant as a weighting parameter for indicating the emphasis to be placed on global versus local search. Typically, this constant is not so small in standard Lipschitz approaches, since the constant needs be at least as large as the maximum rate of change of the objective function; which forces a higher emphasis on global search and thus, results in a slower convergence. However, DIRECT enables operation at both global and local level by concurrently searching using all possible constants. The global part of the algorithm figures out the basin of convergence of the optimum, which the local part of the algorithm can suitably exploit. This justifies the fast convergence of DIRECT in computing the approximate minimum with the guaranteed precision. One major drawback of DIRECT is its combinatorial complexity in higher dimensions where DIRECT often takes many more function evaluations to find a good approximation of the global minimum. One method to resolve this difficulty is to transform the problem domain into its one-dimensional equivalent. This approach is demonstrated in this report using space filling curves, to reduce the multiextremal optimization problem to the minimization of a univariate function. In this case, the Hölder continuity of space filling curves has been exploited to solve global optimization problems. 48 pp. Englisch. N° de réf. du vendeur 9783346913760
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Leveraging Space-Filling Curves and the DIRECT Algorithm. A Novel Approach to Derivative-Free Multi-Dimensional Global Optimization
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Taschenbuch. Etat : Neu. Neuware -Bachelor Thesis from the year 2018 in the subject Mathematics - Applied Mathematics, Nanyang Technological University, language: English, abstract: This study aims to focus on a new approach to the Dividing Rectangles or DIRECT algorithm, which is used to solve multi-dimensional global optimization problems. It also delves into the resolution of DIRECT's combinatorial complexity in higher dimensions by transforming the problem domain into its one-dimensional equivalent.Many real-world problems involve multivariate global optimization which can be difficult to solve. In this report, a new approach to the Dividing Rectangles or DIRECT algorithm for solving multi-dimensional global optimization problems with bounds and a real-valued objective function, is discussed. DIRECT is a variation of the standard Lipschitzian optimization omitting the requirement of having to specify a Lipschitz constant; by viewing the Lipschitzian constant as a weighting parameter for indicating the emphasis to be placed on global versus local search. Typically, this constant is not so small in standard Lipschitz approaches, since the constant needs be at least as large as the maximum rate of change of the objective function; which forces a higher emphasis on global search and thus, results in a slower convergence. However, DIRECT enables operation at both global and local level by concurrently searching using all possible constants. The global part of the algorithm figures out the basin of convergence of the optimum, which the local part of the algorithm can suitably exploit. This justifies the fast convergence of DIRECT in computing the approximate minimum with the guaranteed precision. One major drawback of DIRECT is its combinatorial complexity in higher dimensions where DIRECT often takes many more function evaluations to find a good approximation of the global minimum. One method to resolve this difficulty is to transform the problem domain into its one-dimensional equivalent. This approach is demonstrated in this report using space filling curves, to reduce the multiextremal optimization problem to the minimization of a univariate function. In this case, the Hölder continuity of space filling curves has been exploited to solve global optimization problems. 48 pp. Englisch. N° de réf. du vendeur 9783346913760
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Leveraging Space-Filling Curves and the DIRECT Algorithm. A Novel Approach to Derivative-Free Multi-Dimensional Global Optimization
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Taschenbuch. Etat : Neu. Leveraging Space-Filling Curves and the DIRECT Algorithm. A Novel Approach to Derivative-Free Multi-Dimensional Global Optimization | Aditi Dutta | Taschenbuch | Englisch | 2023 | GRIN Verlag | EAN 9783346913760 | Verantwortliche Person für die EU: preigu GmbH & Co. KG, Lengericher Landstr. 19, 49078 Osnabrück, mail[at]preigu[dot]de | Anbieter: preigu. N° de réf. du vendeur 127360223
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