Global Optimality Conditions for Discrete and Nonconvex Optimization--With Applications to Lagrangian Heuristics and Column Generation

• Published in: Operations research Vol. 54; no. 3; pp. 436 - 453
• Main Authors: Larsson, Torbjorn, Patriksson, Michael
• Format: Journal Article
• Language: English
• Published: Linthicum, MD INFORMS 01.05.2006; Institute for Operations Research and the Management Sciences
• Subjects: Algorithms; Applied sciences; column generation algorithms; Combinatorics; Computer programming; Conservatism; core problems; Exact sciences and technology; global optimality conditions; Heuristic; Heuristics; integer; Integer programming; Integers; Lagrangian function; Lagrangian functions; Lagrangian heuristics; Linear programming; Management; MATEMATIK; Mathematical vectors; MATHEMATICS; Operational research. Management science; Optimal solutions; Optimization; programming; Studies; theory; California; Saddle point method; Non convex programming; Covering problem; Complementarity problem; integer; Convex programming; Column generation; Classification; Inequality constraint; Discrete programming; Set covering; programming; Lagrangian; Subgradient optimization; algorithms: Subject classifications: programming; Coverage; Saddle point; theory; global optimality condi Lagrangian heuristics; column generation algorithms; core problems. Optimization; theory: global optimality conditions; programming; Condition of use; Heuristic method; Primal dual method; Optimality condition; Optimality criterion; Feasibility
• ISSN: 0030-364X; 1526-5463; 1526-5463
• DOI: 10.1287/opre.1060.0292

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 -optimality, and, in the presence of inequality constraints, a relaxed complementarity condition analogously called -complementarity. The total size + 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.