| # Optimality Conditions Strong Duality > Slater Condition | Optimality conditions | Sensitivity | Applications Dual optimum attainment Primal optimum attainment Optimality conditions Special cases ## Sufficient condition for dual optimum attainment Slater condition, namely strict feasibility of the primal, ensures that the dual problem is attained. ## Primal optimum attainment Likewise, if in addition the dual problem is strictly feasible, that is if: then strong duality holds, and both problems are attained, that is: there exist such that is feasible for the primal problem; are feasible for the dual problem: , and . Example: ## Complementary slackness Assume that strong duality holds, and both primal and dual problems are attained, by and respectively. Then we have where the first inequality is by definition of the dual function as a minimum over , and the second from the fact that is feasible. Hence the sum in the above is zero. Since every term in that sum is non-positive, each term is zero: The above are called the complementary slackness conditions. They imply that if the -th constraint is strictly satisfied, then the corresponding dual variable is zero. Conversely, if then we must have . ## Optimality conditions The following conditions: Primal feasibility: Dual feasibility: Lagrangian stationarity: (in the case when every function involved is differentiable) Complementary slackness are called the Karush-Kuhn-Tucker (KKT) conditions. If the problem is convex, and satisfies Slater’s condition, then a primal point is optimal if and only if there exist such that the KKT conditions are satisfied. Conversely, the above conditions guarantee that strong duality holds, and are optimal.