KKTongelma

The KKTongelma, also known as the Karush-Kuhn-Tucker conditions, are a set of first-order necessary conditions for a solution in mathematical optimization to be optimal, provided that the constraint qualification holds. These conditions are named after Harold W. Kuhn and Albert W. Tucker, who published them in 1951. The KKT conditions generalize the method of Lagrange multipliers to problems with inequality constraints.

For a constrained optimization problem with differentiable objective and constraint functions, the KKT conditions provide a set of equations and inequalities that must be satisfied at an optimal point. These conditions involve the gradients of the objective function and the constraint functions, as well as dual variables, often called Lagrange multipliers. The dual variables associated with inequality constraints have specific sign requirements and are zero if the constraint is not binding at the optimum.

The KKT conditions are not always sufficient for optimality. In cases where the objective function is convex and the feasible region is convex (i.e., the problem is a convex optimization problem), the KKT conditions are also sufficient for optimality. For non-convex problems, satisfying the KKT conditions does not guarantee that a point is a global optimum, but it does indicate a candidate for a local optimum. The study and application of the KKT conditions are fundamental in various fields of applied mathematics, economics, and engineering.