  In this paper we develop an optimisation based approach to multivariate
Chebyshev approximation on a finite grid. We consider two models: multivariate
polynomial approximation and multivariate generalised rational approximation.
In the second case the approximations are ratios of linear forms and the basis
functions are not limited to monomials. It is already known that in the case of
multivariate polynomial approximation on a finite grid the corresponding
optimisation problems can be reduced to solving a linear programming problem,
while the area of multivariate rational approximation is not so well
understood.In this paper we demonstrate that in the case of multivariate
generalised rational approximation the corresponding optimisation problems are
quasiconvex. This statement remains true even when the basis functions are not
limited to monomials. Then we apply a bisection method, which is a general
method for quasiconvex optimisation. This method converges to an optimal
solution with given precision. We demonstrate that the convex feasibility
problems appearing in the bisection method can be solved using linear
programming. Finally, we compare the deviation error and computational time for
multivariate polynomial and generalised rational approximation with the same
number of decision variables.
