Global optimization: A new variant of the Alienor method

Ziadi, Abdelkader; Cherruault, Y.; Mora, Gaspar

doi:10.1016/s0898-1221(01)85006-9

Cited by 23 publications

(30 citation statements)

References 4 publications

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“…The dimensionality reduction approach (DR) that we will present in our work, is different with the approach given above. It has been proposed and developed by Cherruault et al [1,5,24,26] and based on the approximation of an n variables of the objective function by a single variable function by using a continuous parametric curve α , defined from a real interval I onto the n-dimensional compact D and satisfying a property described by the definition given below. Such a function α is said to be a space-densifying-curve in D [23,25].…”

Section: The Dimensionality Reduction Methodsmentioning

confidence: 99%

“…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%

See 2 more Smart Citations

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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Section: The Dimensionality Reduction Methodsmentioning

confidence: 99%

Section: The Dimensionality Reduction Methodsmentioning

confidence: 99%

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

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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“…To illustrate this method, let us recall some known results (Cherruault, 1999;Ziadi et al, 2001;Mora et al, 2000).…”

Section: Alienor Methodsmentioning

confidence: 99%

Optimal control of chemotherapy of HIV model using the coupled Adomian/Alienor methods

2005

Self Cite

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PurposeThis paper aims at applying the proven Adomian and Alienor methods to solve the problem of optimal chemotherapy of HIV model.Design/methodology/approachThe combination of the Adomian decomposition method and the Alienor reduction method allows us to solve the control problem as if it were a classical one‐dimensional minimization problem. The methodology is applied to a HIV model and simulation results given.FindingsA general abstract framework for the control of a non‐linear evolution system has been developed. It was shown that it is possible to control a system by using the powerful techniques of Adomian and Alienor, and produce results comparable with those obtained by classical methods where other cost functions are used.Research limitations/implicationsThe benefit of this work is based on the CD4 healthy cells being maximized and the cost based on a drug dose being minimized. The new methodology could be used after further research for solving many control problems in biology and in other areas such as those involving industrial processes.Practical implicationsNew methodology is cost‐effective in controlling the drug dose affecting rate of infection of cells by HIV virus.Originality/valueNew application of proven methodology will solve many control problems in biocybernetics/biomedicine/industry and other fields.

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“…When information derivative is not required, many authors have used the regularity of the objective function to elaborate algorithms giving the optimum [3] [4]. schitzian.…”

Section: Introductionmentioning

confidence: 99%

Global Optimization of Multivariate Holderian Functions Using Overestimators

Yahyaoui¹,

Ammar²

2017

OALib

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This paper deals with the global optimization of several variables Holderian functions. An algorithm using a sequence of overestimators of a single variable objective function was developed converging to the maximum. Then by the use of α-dense curves, we show how to implement this algorithm in a multidimensional optimization problem. Finally, we validate the algorithm by testing it on some test functions.

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Global optimization: A new variant of the Alienor method

Cited by 23 publications

References 4 publications

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

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

Optimal control of chemotherapy of HIV model using the coupled Adomian/Alienor methods

Global Optimization of Multivariate Holderian Functions Using Overestimators

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