An algebra for piecewise-linear minimax problems

Abstract A piecewise-linear function whose definition involves the operator max and min may be reformulated as a ‘sum-of-partial-fractions’ by use of an algebraic structure J and so may be ‘rationalized’ to become a ‘quotient-of-polynomials’ in the notation of J We show that these ‘partial fractions’ and ‘polynomials’ have algebraic properties closely analogous to those of their counterparts in traditional elementary algebra: in particular an analogue of the fundamental theorem of algebra holds. These formal properties lead to straightforward procedures for finding maxima and minima of such functions.