关于在精度约束下逼近非线性函数所需的最小线性分段数
On the Minimum Number of Linear Pieces Required to Approximate Nonlinear Functions under an Accuracy Constraint
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中文总结 AI 辅助
本文研究走廊拟合问题，即在给定误差界限下用最少分段线性函数逼近非线性函数，提出基于定义域离散化的四种松弛方法，给出R^2实例的新上界和首个下界，解决超三分之一文献实例。
中文摘要 AI 辅助
在混合整数非线性问题中，用分段线性函数逼近非线性函数是一种常用工具。通常，通过用分段线性函数替换非线性项，可以将问题转化为混合整数线性问题，这可能大大简化求解。然而，使用近似函数可能产生对原始问题不可行或远离最优的解。为了控制这些误差，有必要对函数逼近过程中产生的误差进行界定。此外，获得分段数较少的分段线性函数通常会使混合整数线性问题更容易求解。这促使我们研究走廊拟合问题。该问题旨在构建一个分段线性函数，其分段数最少，在给定定义域上每个点的逼近误差界限下逼近一个非线性函数。走廊拟合问题主要针对单变量函数或通过启发式方法处理多变量函数。值得注意的是，对于后一种情况，目前尚不清楚有精确算法或已建立的松弛方法。在这项工作中，我们探讨了这一方面，并基于定义域的离散化提出了走廊拟合问题在R^m中可利用的松弛方法。我们证明了定义域的离散化会诱导出超图着色问题的结构。我们定义了四种利用该超图着色问题的松弛方法。我们为R^2中的经典实例集提供了新的最佳上界，并推导了这些实例的首个下界，从而解决了文献中超过三分之一的实例。
英文摘要
The approximation of nonlinear functions by piecewise linear functions is a tool commonly used when dealing with mixed-integer nonlinear problems. Typically, by replacing nonlinearities by piecewise linear functions one can transform the problem into a mixed-integer linear problem, which may be substantially easier to solve. However, using approximate functions can produce solutions that are infeasible for the original problem or far from optimal. To control these errors it is useful to bound the error created during the function approximation process. Moreover, obtaining a piecewise linear function with few pieces usually results in an easier to solve mixed-integer linear problem. This leads us to study the Corridor Fitting Problem. It consists in building a piecewise linear function with the minimum number of pieces which approximates a nonlinear function given a bound on the approximation error on each point of the domain. The Corridor Fitting Problem has primarily been addressed for univariate functions or via heuristic approaches for multivariate functions. Notably, for the latter setting, no exact algorithms or established relaxations are currently known. In this work, we explore this aspect and propose exploitable relaxations of the Corridor Fitting Problem in Rm based on a discretization of the domain. We show that a structure of hypergraph coloring problem is induced by the discretization of the domain. We define four relaxations making use of this hypergraph coloring problem. We provide new best upper bounds for the classical instance set in R2 and we derive the first lower bounds for these instances, closing more than a third of the instances from the literature.
发表机构
- Trier University（特里尔大学）
- Université de Toulouse, INP, LAAS-CNRS（图卢兹大学，国立高等理工学院，法国国家科学研究中心实验室）
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