Abstract
We study the approximation of univariate and multivariate set-valued functions (SVFs) by the adaptation to SVFs of positive sample-based approximation operators for real-valued functions. To this end, we introduce a new weighted average of several sets and study its properties. The approximation results are obtained in the space of Lebesgue measurable sets with the symmetric difference metric. In particular, we apply the new average of sets to adapt to SVFs the classical Bernstein approximation operators, and show that these operators approximate continuous SVFs. The rate of approximation of Holder continuous SVFs by the adapted Bernstein operators is studied and shown to be asymptotically equal to the one for real-valued functions. Finally, the results obtained in the metric space of sets are generalized to metric spaces endowed with an average satisfying certain properties.
Highlights
Set-valued functions (SVFs) have various applications in optimization, control theory, mathematical economics and other areas
The approximation of set-valued functions (SVFs) from a finite number of samples has been the subject of several recent research works ([3],[12],[13],[19]) and reviews ([11],[24])
In order to adapt to SVFs sample-based approximation methods known for real-valued functions, it is required to define linear combinations of two or more sets
Summary
Set-valued functions (SVFs) have various applications in optimization, control theory, mathematical economics and other areas. In case of data sampled from a SVF mapping real-numbers to convex sets, methods based on the classical Minkowski sum of sets can be used for the approximation [8, 27]. Approximation of set-valued functions mapping real-numbers to general sets is a more challenging task In this case, methods based on Minkowski sum of sets fail to approximate the sampled function [27, 10], and other weighted averages of sets are needed. We consider the adaptation of positive sample-based approximation operators to univariate and multivariate SVFs. The adaptation is based on a new weighted average of several sets, termed the partition average, which is studied in details.
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