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 optimization. Although this correspondence is not surprising, it naturally connects rational and generalized 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. 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 optimization problems.

在本文中，我们证明了为有理切比雪夫近似开发的众所周知的线性不等式方法等效于拟凸优化中使用的二分法的应用。虽然这种对应并不奇怪，但它自然地将有理和广义​​有理切比雪夫逼近问题与准凸函数领域的现代发展联系起来，因此为解决这个问题提供了更多的理论和计算工具。本文的第二个重要贡献是将线性不等式方法扩展到更广泛的切比雪夫逼近问题，其中相应的目标函数仍然是准凸的。在这个更广泛的函数类别中，不再要求不等式是线性的：每个不等式定义一个凸集就足够了，计算挑战在于解决相应的凸可行性问题。因此，我们提出了一种更系统和更通用的方法来处理切比雪夫近似问题。特别是，我们正在研究近似值是关于其参数的拟线性函数的问题，这些参数也是相应优化问题中的决策变量。