Original scientific paper
https://doi.org/10.17535/crorr.2019.0024
Generating $\alpha $-dense curves in non-convex sets to solve a class of non-smooth constrained global optimization
Mohamed Rahal
orcid.org/0000-0003-4656-3426
; Department of Mathematics, Ferhat Abbas Setif 1, University of Setif
Ziadi Abdelkader
; Department of Mathematics, Ferhat Abbas Setif 1, University of Setif
Ellaia Rachid
; Laboratory of Study and Research for Applied Mathematics, Mohammadia School of Engineers
Abstract
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 $\mathbb{R}^{n}$ and H\"{o}lderian. The fundamental principle is to provide explicitly a parametric representation $x_{i}=\ell _{i}(t),1\leq i\leq n$ of $\alpha $-dense curve $\ell_{\alpha }$ in the compact $D$, for $t$ in an interval $\mathbb{I}$ of $\mathbb{R}$, which allows to convert the initial problem to a one dimensional H\"{o}lder 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 $\alpha $ of the curve $\ell _{\alpha }$ and the accuracy of attaining the optimal solution is given. Some concrete $\alpha $ dense curves in a non-convex feasible region $D$ are constructed. The numerical results show that the proposed approach is efficient.
Keywords
$α$-dense curves; constrained global optimization; non-smooth non-convex functions; H\"{o}lder condition; Piyavskii’s algorithm
Hrčak ID:
229888
URI
Publication date:
13.12.2019.
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