We found a match
Your institution may have access to this item. Find your institution then sign in to continue.
- Title
Global Optimality Conditions for Discrete and Nonconvex Optimization--With Applications to Lagrangian Heuristics and Column Generation.
- Authors
Larsson, Torbjörn; Patriksson, Michael
- Abstract
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.
- Subjects
PROBLEM solving; MANAGEMENT science; MATHEMATICAL optimization; FEASIBILITY studies; ALGORITHMS; CONVEX programming
- Publication
Operations Research, 2006, Vol 54, Issue 3, p436
- ISSN
0030-364X
- Publication type
Article
- DOI
10.1287/opre.1060.0292