Continuous global optimization through the generation of parametric curves

Ziadi, Raouf; Bencherif-Madani, Abdelatif; Ellaia, Rachid

doi:10.1016/j.amc.2016.01.067

Cited by 7 publications

(6 citation statements)

References 28 publications

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“…Other examples of α-dense curves, as well as their applications, can be found in Refs. [4,9,22,24,25,37]. As expected not every subset of a metric space is densifiable, as we show in the next example.…”

Section: α-Dense Curves and Attractor Setssupporting

confidence: 75%

Approximating the Attractor Set of Iterated Function Systems of Order m by $$\alpha $$-Dense Curves

García

2020

Mediterr. J. Math.

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We construct a sequence of finite sets which approximates, with arbitrarily small and controlled error, the attractor set of a (finite or countable) given iterated function system of order m. Our main tool are the so-called α-dense curves which are from the Hausdorff-Pompeiu distance point of view, a generalization of the space-filling curves. Also we prove that, under suitable conditions, our result can be used to approximate infinite-dimensional fractals generated by integral equations of Volterra type.

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Section: α-Dense Curves and Attractor Setssupporting

confidence: 75%

Approximating the Attractor Set of Iterated Function Systems of Order m by $$\alpha $$-Dense Curves

García

2020

Mediterr. J. Math.

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“…Such a function α is said to be a space-densifying-curve in D [23,25]. However, the introduction of these α-dense curves has allowed analytical argumentation of certain dimensionality reduction methods in global optimization techniques [5,17,24,26,27].…”

Section: The Dimensionality Reduction Methodsmentioning

confidence: 99%

“…Some interesting results concerning the existence of α-dense curves with minimal length of an α-densifiable compact in a general metric space were given by Ziadi et al [25]. In the case where the feasible set is a hyper-rectangle Ω of R n , some ways of constructing the α-dense curves in Ω have been given [14,24,26,27]. In this subsection we shall give the relationship existing between the components i (i = 1, ..., n) of the parametrization in order to generate a new class of α-dense curves in a compact D ⊂ R n whose boundary is defined by a non-smooth functions.…”

Section: Generation Of α-Dense Curves In a Compact Non-convex Setsmentioning

confidence: 99%

“…More details are discussed in [17,19,20,21]. Another idea that can be applied for some covering global optimization methods which use the constant L, is to construct an appropriate increasing sequence of a real positive constants (L k ), k ∈ N which satisfies the following conditions: L k → ∞ for k → ∞ and for a given accuracy ε > 0, there exists j ∈ N, such that L ≤ L j [27]. Consequently, the inequality (8) is holds with the constant L j for which the convergence to the global minimum is guaranteed [17].…”

Section: Constructing Piecewise Affine Under-estimator Functionsmentioning

confidence: 99%

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Generating $\alpha $-dense curves in non-convex sets to solve a class of non-smooth constrained global optimization

Rahal

Ziadi

Ellaia

2019

Cro. Oper. Res. Rev.

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This paper deals with the dimensionality reduction approach to study multi-dimensional constrained global optimization problems where the objective function is non-differentiable over a general compact set D of R n and Hölderian. The fundamental principle is to provide explicitly a parametric representation xi = i(t), 1 ≤ i ≤ n of α-dense curve α in the compact D, for t in an interval I of R, which allows to convert the initial problem to a one-dimensional Hölder unconstrained one. Thus, we can solve the problem by using an efficient algorithm available in the case of functions depending on a single variable. A relation between the parameter α of the curve α and the accuracy of attaining the optimal solution is given. Some concrete α-dense curves in a non-convex feasible region D are constructed. The numerical results show that the proposed approach is efficient.

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“…The cosines curve. For each positive integer k, the mapping γ k : Other examples of α-dense curves and their applications, can be found in [11,12,16,17,27] and references therein.…”

Section: Example 22mentioning

confidence: 99%

Approximating multiple integrals of continuous functions by $$\delta $$-uniform curves

García

Mora

2021

Ann Univ Ferrara

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Continuous global optimization through the generation of parametric curves

Cited by 7 publications

References 28 publications

Approximating the Attractor Set of Iterated Function Systems of Order m by $$\alpha $$-Dense Curves

Approximating the Attractor Set of Iterated Function Systems of Order m by $$\alpha $$-Dense Curves

Generating $\alpha $-dense curves in non-convex sets to solve a class of non-smooth constrained global optimization

Approximating multiple integrals of continuous functions by $$\delta $$-uniform curves

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