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.

中文翻译：

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在非凸集中生成$ \ alpha $-密集曲线，以解决一类非平滑约束全局优化

本文研究了降维方法，以研究多维约束全局优化问题，其中目标函数在$ \ mathbb {R} ^ {n} $和H \“ { lderian。基本原理是明确提供参数表示形式$ x_ {i} = \ ell _ {i}（t），1 \ leq i \ leq n $ of $ \ alpha $-密度曲线$ \ ell_ {紧凑型$ D $中的\ alpha} $，$ \ mathbb {R} $的间隔$ \ mathbb {I} $中的$ t $，这允许将初始问题转换为一维H \“ {o } lder不受限制。因此，在函数取决于单个变量的情况下，我们可以通过使用有效的算法来解决该问题。给出了曲线$ \ ell _ {\\ alpha} $的参数$ \ alpha $与获得最优解的精度之间的关系。在非凸可行区域$ D $中构造了一些具体的$ \ alpha $密集曲线。数值结果表明，该方法是有效的。