Karush–Kuhn–Tucker conditions
Regularity conditions (or constraint qualifications)
In practice weaker constraint qualifications are preferred since they apply to a broader selection of problems.
Sufficient conditions
In some cases, the necessary conditions are also sufficient for optimality. In general, the necessary conditions are not sufficient for optimality and additional information is required, such as the Second Order Sufficient Conditions (SOSC). For smooth functions, SOSC involve the second derivatives, which explains its name.
Value function
If we reconsider the optimization problem as a maximization problem with constant inequality constraints:
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