Global Optimality Conditions for Discrete and Nonconvex Optimization--With Applications to Lagrangian Heuristics and Column Generation
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 dual...
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| Veröffentlicht in: | Operations research Jg. 54; H. 3; S. 436 - 453 |
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| Format: | Journal Article |
| Sprache: | Englisch |
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Linthicum, MD
INFORMS
01.05.2006
Institute for Operations Research and the Management Sciences |
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| ISSN: | 0030-364X, 1526-5463, 1526-5463 |
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| 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 primaldual 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 primaldual 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. |
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| AbstractList | 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 primaldual 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 primaldual 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. 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. 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 e-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. [PUBLICATION 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 -optimality, and, in the presence of inequality constraints, a relaxed complementarity condition analogously called δ-complementarity. The total size + 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. 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. |
| Audience | Trade |
| Author | Larsson, Torbjorn Patriksson, Michael |
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| Keywords | 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 |
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| SubjectTerms | 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 |
| Title | Global Optimality Conditions for Discrete and Nonconvex Optimization--With Applications to Lagrangian Heuristics and Column Generation |
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| Volume | 54 |
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