Following developments in non-linear leastpth optimization by the authors it is possible to derive two new methods of non-linear minimax optimization. Unlike the Polya algorithm in which a sequence of leastpth optimizations asp→∞ is taken our methods do not require the value ofpto tend to infinity. Instead we construct a sequence of leastpth optimization problems with a finite value ofp. It is shown that this sequence will converge to a minimax solution. Two interesting minimax problems were constructed which illustrate some of the theoretical ideas. Further numerical evidence is presented on the modelling of a fourth-order system by a second-order model with values ofpvarying between 2 and 10 000.