An Algorithm for Best Generalised Rational Approximation of Continuous Functions

In this paper we introduce an algorithm for solving variational inequality problems when the operator is pseudomonotone and point-to-set (therefore not relying on continuity assumptions). Our motivation is the development of a method for solving optimisation problems appearing in Chebyshev rational and generalised rational approximation problems, where the approximations are constructed as ratios of linear forms (linear combinations of basis functions). The coefficients of the linear forms are subject to optimisation and the basis functions are continuous functions. It is known that the objective functions in generalised rational approximation problems are quasiconvex. In this paper we prove a stronger result, the objective functions are pseudoconvex in the sense of Penot and Quang. Then we develop numerical methods, that are efficient for a wide range of pseudoconvex functions and test them on generalised rational approximation problems.