The existence of α‐dense curves with minimal length in a metric space

Abstract

Some results concerning the existence of -dense curves with minimal length are given. This type of curves used in the reducing transformation called Alienor was invented by Cherruault and Guillez. They have been applied to global optimization in the following way: a multivariable optimization problem is transformed in an optimization problem depending on a single variable. Then this idea was extended by Cherruault and his team for obtaining general classes of reducing transformations having minimal properties (length of the -dense curves, minimization of the calculus time, etc.).

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

“…Butz, Strongin and Sergeyev et al [2,19,20], use numerical approximations of Peano type curves whereas Cherruault and their collaborators [24], use α-dense ones. The approximations of Peano curves are obviously α-dense but the curves used by Cherruault et al [25] do not necessarily converge to Peano type curves. Although both classes of curves have the common property to approach as much as wanted all the points of a hyper-rectangle of R n , there are important differences: they do not possess the same regularity, they differ also concerning repartition uniformity (on the hyper-rectangle) and asymptotic behavior.…”

“…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]. 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].…”

“…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].…”

“…to the one-dimensional unconstrained one. As for all α > 0, the set of α-dense curves in a compact D of R n , contains a curve with minimal length, see Ziadi et al [25], it is therefore, very interest to find other curves with small length in order to improve the (DR) method.…”

On the Determination of Solutions Sets Regarding Highly Non-Linear Equations Systems

“…Hereafter we would like to approximate the set S. Our main idea for achieving this scheme comes from the reduction transformation described in [10] that consists in considering an α-dense curve of R n , that is…”