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

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.