Interval Global Optimization

Ratschek, Helmut; Rokne, Jon G.

doi:10.1007/978-0-387-74759-0_306

Cited by 6 publications

(4 citation statements)

References 62 publications

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“…Interval arithmetic was standardized by the IEEE in 2015 [223]. Global optimization problems can be solved using the interval branch-and-bound method which iteratively splits the search space, and removes its parts that do not contain a global solution; multiple splitting schemes have been proposed in the literature [224]. In this case, application of interval arithmetic allows to guarantee, if a solution exists within a region of interest.…”

Section: Interval Methodsmentioning

confidence: 99%

Survey of Methods for Solving Systems of Nonlinear Equations, Part I: Root-finding Approaches

Kotsireas¹,

Pardalos²,

Semenov³

et al. 2022

Preprint

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This paper presents a comprehensive survey of methods which can be utilized to search for solutions to systems of nonlinear equations (SNEs). Our objectives with this survey are to synthesize pertinent literature in this field by presenting a thorough description and analysis of the known methods capable of finding one or many solutions to SNEs, and to assist interested readers seeking to identify solution techniques which are well suited for solving the various classes of SNEs which one may encounter in real world applications.To accomplish these objectives, we present a multi-part survey. In part one, we focus on root-finding approaches which can be used to search for solutions to a SNE without transforming it into an optimization problem. In part two, we will introduce the various transformations which have been utilized to transform a SNE into an optimization problem, and we discuss optimization algorithms which can then be used to search for solutions. In part three, we will present a robust quantitative comparative analysis of methods capable of searching for solutions to SNEs.

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Section: Interval Methodsmentioning

confidence: 99%

Survey of Methods for Solving Systems of Nonlinear Equations, Part I: Root-finding Approaches

Kotsireas¹,

Pardalos²,

Semenov³

et al. 2022

Preprint

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show abstract

“…The existing optimization algorithms, such as gradient-based method, direct search method, intelligent method, and so on, can be flexibly selected to update the design variables for different kinds of engineering problems. [45][46][47] However, it should be noted that the computational burden caused by the nested loop will be huge, especially for the problems without explicit response functions. In order to improve the computational cost of parameter identification, a more efficient numerical method will be introduced in the next section for interval temperature response prediction.…”

Section: Nested-loop Optimization For Interval Parameter Identificationmentioning

confidence: 99%

“…From the flowchart, as shown in Figure , it easily can be seen that the interval parameter identification framework is a nested‐loop optimization model where the inner loop (listed in Steps 2 to 4) is employed to predict the bounds of computational response interval at feature points, and the outer loop (listed in Steps 1 to 7) is executed to capture the optimal bounds of interval input parameters. The existing optimization algorithms, such as gradient‐based method, direct search method, intelligent method, and so on, can be flexibly selected to update the design variables for different kinds of engineering problems –. However, it should be noted that the computational burden caused by the nested loop will be huge, especially for the problems without explicit response functions.…”

Section: Interval‐based Parameter Identification Frameworkmentioning

confidence: 99%

Novel interval theory‐based parameter identification method for engineering heat transfer systems with epistemic uncertainty

Wang

Matthies

2018

Numerical Meth Engineering

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Summary The parameter identification problem with epistemic uncertainty, where only a small amount of experimental information is available, is a challenging issue in engineering. To overcome the drawback of traditional probabilistic methods in dealing with limited data, this paper proposes a novel interval theory‐based inverse analysis method. First, the interval variables are introduced to represent the input uncertainties, whose lower and upper bounds are to be identified. Subsequently, an unbiased estimation method is presented to quantify the experimental response interval from limited measurements. Meanwhile, a quantitative metric is defined to characterize the relative errors between computational and experimental response intervals by which the interval parameter identification can be constructed as a nested‐loop optimization procedure. To improve the computational efficiency of response prediction with respect to various interval variables, a universal surrogate model is established in the support box via Legendre polynomial chaos expansion, where the expansion coefficients can be evaluated by a collocation method under Clenshaw‐Curtis points and Smolyak algorithm. Eventually, a heat conduction example is provided to verify the feasibility of proposed method, especially in the case with noise‐contaminated temperature measurements.

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“…It delves into both single-variable and multi-variable scenarios, providing explanations and illustrations to demonstrate the application of the Interval Newt0n Method in tackling these optimization problems. Helmut Ratschek [4]explored interval methods in global optimization, providing solutions for diverse optimization scenarios, including unc0nstrained, constrained, and n0nsmooth optimization. It underscores the importance of bisection techniques in interval-based global optimization algorithms, where the problem domain is recursively divided and also acknowledges advancements in bisection strategies over the last decade.…”

Section: Introductionmentioning

confidence: 99%

Solving Intuitionistic Fuzzy Unconstrained Optimization Problems Using Interval Newton’s Method

S. Shilpa Ivin Emimal

2024

cana

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This research work studies the optimization of intuitionistic fuzzy valued functions in unconstrained problems. The Interval Newton's Method using an Intuitionistic approach addresses both single and multivariable optimization problems. The study incorporates a mathematical comprehension of interval intuitionistic fuzzy valued problems, as well as real-world examples to demonstrate their effectiveness. Furthermore, MATLAB code is provided to demonstrate the implementation of the Interval Newton's Method.

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Interval Global Optimization

Cited by 6 publications

References 62 publications

Survey of Methods for Solving Systems of Nonlinear Equations, Part I: Root-finding Approaches

Survey of Methods for Solving Systems of Nonlinear Equations, Part I: Root-finding Approaches

Novel interval theory‐based parameter identification method for engineering heat transfer systems with epistemic uncertainty

Solving Intuitionistic Fuzzy Unconstrained Optimization Problems Using Interval Newton’s Method

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