Layering strategies for creating exploitable structure in linear and integer programs

Abstract

The strategy of subdividing optimization problems into layers by splitting variables into multiple copies has proved useful as a method for inducing exploitable structure in a variety of applications, particularly those involving embedded pure and generalized networks. A framework is proposed in this paper which leads to new relaxation and restriction methods for linear and integer programming based on our extension of this strategy. This framework underscores the use of constructions that lead to stronger relaxations and more flexible strategies than previous applications. Our results establish the equivalence of all layered Lagrangeans formed by parameterizing the equal value requirement of copied variables for different choices of the principal layers. It is further shown that these Lagrangeans dominate traditional Lagrangeans based on incorporating non-principal layers into the objective function. In addition a means for exploiting the layered Lagrangeans is provided by generating subgradients based on a simple averaging calculation. Finally, we show how this new layering strategy can be augmented by an integrated relaxation/ restriction procedure, and indicate variations that can be employed to particular advantage in a parallel processing environment. Preliminary computational results on fifteen real world zero-one personnel assignment problems, comparing two layering approaches with five procedures previously found best for those problems, are encouraging. One of the layering strategies tested dominated all non-layering procedures in terms of both quality and solution time.

Lagrangean decomposition for integer programming : theory and applications

“…• This resuit was also proved independently by [Glover and Klingman (1984)] and [Jörnsten, Nàsberg and Smeds (1985)]. …”

En hommage à Joseph-Louis Lagrange et à Pierre Huard

“…I personally think that that name describes only the first stage of the process, and hides the ensuing dualization, but I also must admit that Lagrangean Decomposition isn't a perfect name either, first because LR may yield decomposed subproblems even without the copying trick, for instance, when one dualizes all linking constraints, and second because one could make copies of variables and still not get a decomposable problem, as in the paper by Glover and Klingman (1988) that appeared in Mathematical Programming at roughly the same time as ours.…”

Netform Modeling and Applications

“…A growing body of research is dedicated to detecting network structures that may be hidden inside more complex models [Bixby 1984;Bixby and Wagner 1985;Brown, McBride and Wood 1985;Truemper 1983]. This has led more recently to research into creating these and other netform structures in problem settings where they may not otherwise be found [Glover and Klingman 1988a].…”