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Computation of the output of a function with fuzzy inputs based on a low-rank tensor approximation

机译：基于低秩张量逼近计算具有模糊输入的函数的输出

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摘要

We apply a derivative-free optimization method based on novel low-rank tensor methods to the problem of propagating fuzzy uncertainty through a continuous real-valued function. Adhering to Zadeh's extension principle, such a problem can be reformulated as a sequence of optimization problems over nested search spaces. The optimization method we use is based on a low-rank tensor approximation of the function sampled on a grid and a search for the minimal and maximal entries in this low-rank tensor. In contrast to classical fuzzy uncertainty propagation methods, such as the vertex method and the transformation method, the method we propose does not exhibit an inherent exponential scaling for increasing dimension of the search space. Obviously, no derivative-free optimization algorithm can exist which shows sub-exponential scaling with the dimension for all possible continuous functions. The algorithm that we present here, however, can exploit a specific type of structure and regularity (beyond continuity) that is often present in real-world optimization problems. We illustrate this with some high-dimensional numerical examples where the presented method clearly outperforms some established derivative-free optimization codes. (C) 2016 Elsevier B.V. All rights reserved.
机译：我们将基于新颖的低秩张量方法的无导数优化方法应用于通过连续实值函数传播模糊不确定性的问题。遵循Zadeh的扩展原理，可以将此问题重新表述为嵌套搜索空间上的一系列优化问题。我们使用的优化方法基于在网格上采样的函数的低秩张量逼近，并搜索该低秩张量中的最小和最大项。与经典的模糊不确定性传播方法（例如顶点方法和变换方法）相比，我们提出的方法没有表现出固有的指数缩放比例，无法增大搜索空间的尺寸。显然，不存在没有导数的优化算法，该算法会显示所有可能的连续函数的维数与子指数成比例的缩放。但是，我们在此处介绍的算法可以利用特定类型的结构和规则性（超越连续性），而这种类型通常在现实世界中的优化问题中存在。我们用一些高维数值示例对此进行了说明，其中所提出的方法明显优于某些已建立的无导数优化代码。 （C）2016 Elsevier B.V.保留所有权利。

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