Global optimality conditions for discrete and nonconvex optimization-with applications to Lagrangian heuristics and column generation
2006 (English)In: Operations Research, ISSN 0030-364X, Vol. 54, no 3, 436-453 p.Article in journal (Refereed) Published
The well-known and established global optimality conditions based on the Lagrangian formulation of an optimization problem are consistent if and only if the duality gap is zero. We develop a set of global optimality conditions that are structurally similar but are consistent for any size of the duality gap. This system characterizes a primal-dual optimal solution by means of primal and dual feasibility, primal Lagrangian ε-optimality, and, in the presence of inequality constraints, a relaxed complementarity condition analogously called δ-complementarity. The total size ε + δ of those two perturbations equals the size of the duality gap at an optimal solution. Further, the characterization is equivalent to a near-saddle point condition which generalizes the classic saddle point characterization of a primal-dual optimal solution in convex programming. The system developed can be used to explain, to a large degree, when and why Lagrangian heuristics for discrete optimization are successful in reaching near-optimal solutions. Further, experiments on a set-covering problem illustrate how the new optimality conditions can be utilized as a foundation for the construction of new Lagrangian heuristics. Finally, we outline possible uses of the optimality conditions in column generation algorithms and in the construction of core problems. © 2006 INFORMS.
Place, publisher, year, edition, pages
2006. Vol. 54, no 3, 436-453 p.
IdentifiersURN: urn:nbn:se:liu:diva-36281DOI: 10.1287/opre.1060.0292Local ID: 30845OAI: oai:DiVA.org:liu-36281DiVA: diva2:257129