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Authors:Torbjörn Larsson: Linköping University, Sweden
Michael Patriksson: Chalmers University of Technology, Sweden
Publication title:Global Optimality Conditions for Discrete and Nonconvex Optimization; With Applications to Lagrangian Heuristics and Column Generation
Conference:Nordic MPS 2004. The Ninth Meeting of the Nordic Section of the Mathematical Programming Society
Publication type: Abstract
Issue:014
Article No.:021
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 epsilonoptimality; and; in the presence of inequality constraints; delta-complementarity; that is; a relaxed complementarity condition. The total size epsilon + delta of those two perturbations equals the size of the duality gap at an optimal solution. The characterization is further 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 Lagrangian heuristics. Finally; we outline possible uses of the optimality conditions in column generation algorithms and in the construction of core problems; and illustrate our findings on instances of the generalized assignment problem.
Language:English
Year:2004
No. of pages:1
Series:Linköping Electronic Conference Proceedings
ISSN (print):1650-3686
ISSN (online):1650-3740
File:http://www.ep.liu.se/ecp/014/021/ecp014021.pdf
Available:2004-12-28
Publisher:Linköping University Electronic Press; Linköpings universitet

REFERENCE TO THIS PAGE
Torbjörn Larsson, Michael Patriksson (2004). Global Optimality Conditions for Discrete and Nonconvex Optimization; With Applications to Lagrangian Heuristics and Column Generation, Nordic MPS 2004. The Ninth Meeting of the Nordic Section of the Mathematical Programming Society http://www.ep.liu.se/ecp_article/index.en.aspx?issue=014;article=021 (accessed 12/23/2014)