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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.

The necessary conditions are sufficient for optimality if the objective function f {\displaystyle f} of a maximization problem is a differentiable concave function, the inequality constraints g j {\displaystyle g_{j}} are differentiable convex functions, the equality constraints h i {\displaystyle h_{i}} are affine functions, and Slater's condition holds. Similarly, if the objective function f {\displaystyle f} of a minimization problem is a differentiable convex function, the necessary conditions are also sufficient for optimality.

It was shown by Martin in 1985 that the broader class of functions in which KKT conditions guarantees global optimality are the so-called Type 1 invex functions.