Max-linear Programs

Abstract

A max-linear program is the task to minimize or maximize a max-linear function subject to one or two-sided max-linear equations. Since one-sided max-linear systems are substantially easier to solve than the two-sided, these two cases are dealt with separately. Unlike in conventional linear programming, in max-linear programming there is no obvious simple conversion of minimization to maximization or vice versa and both problems need to be considered.

In this chapter upper and lower bounds of the objective function are found and theorems on the attainment of the maximum and minimum are proved. Then bisection methods for localizing the optimal values with a given precision (both for maximization and minimization) are presented. They are based on the result that an interval containing all objective function values can be found in polynomial time. These methods are exact and pseudopolynomial when applied to programs with integer entries. The theory is illustrated by detailed numerical examples.