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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Vydáno v:Operations research Ročník 54; číslo 3; s. 436 - 453
Hlavní autoři: Larsson, Torbjorn, Patriksson, Michael
Médium: Journal Article
Jazyk:angličtina
Vydáno: Linthicum, MD INFORMS 01.05.2006
Institute for Operations Research and the Management Sciences
Témata:
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
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Popis
Shrnutí: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.
Bibliografie:SourceType-Scholarly Journals-1
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ISSN:0030-364X
1526-5463
1526-5463
DOI:10.1287/opre.1060.0292