On mixed-integer zero–one representations for separable lower-semicontinuous piecewise-linear functions

In a recent paper, Padberg has provided some insights into constructing locally ideal formulations for continuous piecewise-linear approximations to separable nonlinear programs. He shows that in contrast with such representations, the standard text-book modeling strategy is weak with respect to its linear programming relaxation. We propose a new pedagogically simpler modification of the latter formulation that constructs its convex hull representation, thereby rendering it locally ideal. Moreover, this modeling strategy imparts a totally unimodular structure to the formulation, it readily extends to representing separable lower-semicontinuous piecewise-linear functions, and it also facilitates a reduced locally ideal representation based on a piecewise-linear convex decomposition of the function. In the special case of continuous piecewise-linear functions, we exhibit a nonsingular linear transformation that equivalently converts the proposed model into Padberg's formulation.