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In this paper we demonstrate that a well known linear inequality methoddeveloped for rational Chebyshev approximation is equivalent to the applicationof the bisection method used in quasiconvex optimisation. Although thiscorrespondence is not surprising, it naturally connects rational andgeneralised rational Chebyshev approximation problems with modern developmentsin the area of quasiconvex functions and therefore offers more theoretical andcomputational tools for solving this problem. %In particular, this observationleads a straightforward extension of the linear inequality method togeneralised rational approximation in the sense of Loeb (ratio of two linearforms). The second important contribution of this paper is the extension of thelinear inequality method to a broader class of Chebyshev approximationproblems, where the corresponding objective functions remain quasiconvex. Inthis broader class of functions, the inequalities are no longer required to belinear: it is enough for each inequality to define a convex set and thecomputational challenge is in solving the corresponding convex feasibilityproblems. Therefore, we propose a more systematic and general approach fortreating Chebyshev approximation problems. In particular, we are looking at theproblems where the approximations are quasilinear functions with respect totheir parameters, that are also the decision variables in the correspondingoptimisation problems.

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