Decomposition Methods for Combinatorial Optimization

Abstract: This thesis aims at research in the field of combinatorial optimization. Problems within this field often posses special structures allowing them to be decomposed into more easily solved subproblems, which can be exploited in solution methods. These structures appear frequently in applications. We contribute with both re-search on the development of decomposition principles and on applications. The thesis consists of an introduction and three papers. In Paper I, we develop a Lagrangian meta-heuristic principle, which is founded on a primal-dual global optimality condition for discrete and non-convex optimization problems. This condition characterizes (near-)optimal solutions in terms of near-optimality and near-complementarity measures for Lagrangian relaxed solutions. The meta-heuristic principle amounts to constructing a weighted combination of these measures, thus creating a parametric auxiliary objective function (which is a close relative to a Lagrangian function), and embedding a Lagrangian heuristic in a search procedure in the space of the weight parameters. We illustrate and assess the Lagrangian meta-heuristic principle by applying it to the generalized assignment problem and to the set covering problem. Our computational experience shows that the meta-heuristic extension of a standard Lagrangian heuristic principle can significantly improve upon the solution quality. In Paper II, we study the duality gap for set covering problems. Such problems sometimes have large duality gaps, which make them computationally challenging. The duality gap is dissected with the purpose of understanding its relationship to problem characteristics, such as problem shape and density. The means for doing this is the above-mentioned optimality condition, which is used to decompose the duality gap into terms describing near-optimality in a Lagrangian relaxation and near-complementarity in the relaxed constraints. We analyse these terms for numerous problem instances, including some large real-life instances, and conclude that when the duality gap is large, the near-complementarity term is typically large and the near-optimality term small. The large violation of complementarity is due to extensive over-coverage. Our observations have implications for the design of solution methods, especially for the design of core problems. In Paper III, we study a bi-objective covering problem stemming from a real-world application concerning the design of camera surveillance systems for large-scale outdoor areas. It is prohibitively costly to surveil the entire area, and therefore relevant to be able to present a decision-maker with trade-offs between total cost and the portion of the area that is surveilled. The problem is stated as a set covering problem with two objectives, describing cost and portion of covering constraints that are fulfilled, respectively. Finding the Pareto frontier for these objectives is very computationally demanding and we therefore develop a method for finding a good approximate frontier in a reasonable computing time. The method is based on the ε−constraint reformulation, an established heuristic for set covering problems, and subgradient optimization.