We develop a numerical method for the computation of a minimal convex and compact set, $\mathcal{B}\subset\mathbb{R}^N$, in the sense of mean width. This minimisation is constrained by the requirement that $\max_{b\in\mathcal{B}}\langle b , u\rangle\geq C(u)$ for all unit vectors $u\in S^{N-1}$ given some Lipschitz function $C$. This problem arises in the construction of environmental contours under the assumption of convex failure sets. Environmental contours offer descriptions of extreme environmental conditions commonly applied for reliability analysis in the early design phase of marine structures. Usually, they are applied in order to reduce the number of computationally expensive response analyses needed for reliability estimation. We solve this problem by reformulating it as a linear programming problem. Rigorous convergence analysis is performed, both in terms of convergence of mean widths and in the sense of the Hausdorff metric. Additionally, numerical examples are provided to illustrate the presented methods.

翻译：本文开发了一种数值方法，用于在均宽意义下计算最小凸紧集$\mathcal{B}\subset\mathbb{R}^N$。该最小化问题受以下约束：对于所有单位向量$u\in S^{N-1}$（给定某个利普希茨函数$C$），满足$\max_{b\in\mathcal{B}}\langle b , u\rangle\geq C(u)$。该问题源于凸失效集假设下的环境轮廓构建。环境轮廓描述了极端环境条件，常用于海洋结构早期设计阶段的可靠性分析，其目的是减少可靠性评估中计算成本高昂的响应分析次数。我们通过将问题重构为线性规划问题进行求解，并在均宽收敛性和豪斯多夫度量意义下进行了严格的收敛性分析。此外，通过数值算例验证了所提方法的有效性。