Generation of α‐dense curves and application to global optimization

Abstract

The reducing transformation and global optimization technique called Alienor has been developed in the 1980s by Cherruault and Guillez. These methods are based on the approximating properties of α ‐dense curves. The aim of this work is to give a very large class of functions generating α ‐dense curves in a hyper‐rectangle of Rn.

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

“…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].…”

“…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].…”

“…The curves used in the dimensionality reduction method belong to two different classes. 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.…”

“…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.…”

“…This due to the fact that f (t) is the restriction of F (x) to the α-dense curve in D. Thus, we can solve the problem (4) by applying a more efficient and performing algorithm proposed for minimizing functions depending on a single variable and satisfying the Hölder condition (2). It should be mentioned that recently, the dimensionality reduction approach, coupled with some one-dimensional global optimization methods has proved its effectiveness in solving various multi-dimensional problems where the objective function is Lipschitzian and the feasible set is a relatively small hyper-rectangle [1,17,23,24]. The paper is organized as follows.…”