Encyclopedia of Operations Research and Management Science

2001 Edition
| Editors: Saul I. Gass, Carl M. Harris

Complementary slackness theorem

Reference work entry
DOI: https://doi.org/10.1007/1-4020-0611-X_140
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For the symmetric form of the primal and dual problems the following theorem holds: For optimal feasible solutions of the primal and dual (symmetric) systems, whenever inequality occurs in the kth relation of either system (the corresponding slack variable is positive), then the kth variable of its dual is zero; if the kth variable is positive in either system, the kth relation of its dual is equality (the corresponding slack variable is zero). Feasible solutions to the primal and dual problems that satisfy the complementary slackness conditions are also optimal solutions. A similar theorem holds for the unsymmetric primal-dual problems: For optimal feasible solutions of the primal and dual (unsymmetric) systems, whenever the kth relation of the dual is an inequality, then the kth variable of the primal is zero; if the kth variable of the primal is positive, then the kth relation of the dual is equality. This theorem just states the optimality conditions of the simplex method. See...

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© Kluwer Academic Publishers 2001

Authors and Affiliations

  1. 1.Robert H. Smith School of BusinessUniversity of MarylandCollege PartUSA
  2. 2.School of Information Technology & EngineeringGeorge Mason UniversityFairfaxUSA