# Complementary slackness theorem

**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 *k*th relation of either system (the corresponding slack variable is positive), then the *k*th variable of its dual is zero; if the *k*th variable is positive in either system, the *k*th 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 *k*th relation of the dual is an inequality, then the *k*th variable of the primal is zero; if the *k*th variable of the primal is positive, then the *k*th relation of the dual is equality. This theorem just states the optimality conditions of the simplex method. See...