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

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 $x_{i}=ell _{i}(t),1leq ileq n$ 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.</p