The problem under consideration is that of minimizing the objective function $$F(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{x} ) = \mathop {max}\limits_{l \leqslant j \leqslant m} f_j (\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{x} )$$ where {fj} is a set of m nonlinear, differentiable functions of n variables x={x1,x2,...,xn}T. This problem can be solved by a method that uses linear approximations to the functions fj, and normally this method will have a quadratic final rate of convergence. However, if some regularity condition is not fulfilled at the solution then the final rate of convergence may be very slow. In this case second order information is required in order to obtain a fast final convergence. We present a method which combines the two types of algorithms. If an irregularity is detected a switch is made from the first order method to a method which is based on approximations of the second order information using only first derivatives. It has been proved that the combined method has sure convergence properties, and that normally the final rate of convergence will be either quadratic or superlinear.