The extension of the linear inequality method for generalized rational Chebyshev approximation to approximation by general quasilinear functions

Abstract

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.

Keywords:

• Generalized rational approximation
• Chebyshev approximation
• quasiconvex functions
• linear inequality method
• bisection method

AMS CLASSIFICATIONS:

• 90C25
• 90C26
• 90C90
• 90C47
• 65D15
• 65K10

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This research was supported by the Australian Research Council, Solving hard Chebyshev approximation problems through nonsmooth analysis [Discovery Project DP180100602].