Transforming Mathematical Models Using Declarative Reformulation Rules

Abstract

Reformulation is one of the most useful and widespread activities in mathematical modeling, in that finding a "good" formulation is a fundamental step in being able so solve a given problem. Currently, this is almost exclusively a human activity, with next to no support from modeling and solution tools. In this paper we show how the reformulation system defined in [15] allows to automatize the task of exploring the formulation space of a problem, using a specific example (the Hyperplane Clustering Problem). This nonlinear problem admits a large number of both linear and nonlinear formulations, which can all be generated by defining a relatively small set of general Atomic Reformulation Rules (ARR). These rules are not problem-specific, and could be used to reformulate many other problems, thus showing that a general-purpose reformulation system based on the ideas developed in [15] could be feasible.

A data driven Dantzig–Wolfe decomposition framework

“…A second line of research attempts to relief the general purpose solver user from the explicit choice of a decomposition scheme, looking for a suitable one either on the basis of model metadata [14,15], like annotations on the type of constraints which are encoded, or by the explicit search for specific structures in it (e.g. partitioning constraints, network flows, temporal layers) via detectors [13].…”