论文标题
线性不等式方法的扩展，用于通过一般准线性函数近似近似的广义理性Chebyshev近似
The extension of linear inequality method for generalised rational Chebyshev approximation to approximation by general quasilinear functions
论文作者
论文摘要
在本文中，我们证明了一种用于理性Chebyshev近似开发的众所周知的线性不等式方法等于在准核孔优化中使用的一分为二方法。尽管这种对应关系不足为奇，但它自然地将合理和广义的理性Chebyshev近似问题与Quasiconvex函数领域的现代发展联系起来，因此提供了更多理论和计算工具来解决此问题。尤其是％，该观察结果导致线性不等式方法的直接扩展在loeb的意义上（两种线性形式的比率）。本文的第二个重要贡献是将线性不平等方法扩展到更广泛的Chebyshev近似问题，其中相应的目标函数仍然是Quasiconvex。在这种更广泛的函数类别中，不再需要线性不平等：足以让每个不平等定义一个凸集，而计算挑战在于解决相应的凸出可行性问题。因此，我们提出了一种更系统，更一般的方法来治疗Chebyshev近似问题。特别是，我们正在研究近似值是准线性函数相对于其参数的问题，这些函数也是相应优化问题中的决策变量。
In this paper we demonstrate that a well known linear inequality method developed for rational Chebyshev approximation is equivalent to the application of the bisection method used in quasiconvex optimisation. Although this correspondence is not surprising, it naturally connects rational and generalised rational Chebyshev approximation problems with modern developments in the area of quasiconvex functions and therefore offers more theoretical and computational tools for solving this problem. %In particular, this observation leads a straightforward extension of the linear inequality method to generalised rational approximation in the sense of Loeb (ratio of two linear forms). The second important contribution of this paper is the extension of the linear inequality method to a broader class of Chebyshev approximation problems, where the corresponding objective functions remain quasiconvex. In this broader class of functions, the inequalities are no longer required to be linear: it is enough for each inequality to define a convex set and the computational challenge is in solving the corresponding convex feasibility problems. Therefore, we propose a more systematic and general approach for treating Chebyshev approximation problems. In particular, we are looking at the problems where the approximations are quasilinear functions with respect to their parameters, that are also the decision variables in the corresponding optimisation problems.
