KKTfeltételek

KKTfeltételek, often translated as Karush-Kuhn-Tucker conditions, are a set of first-order necessary conditions for a solution in nonlinear programming to be optimal. They apply to optimization problems with inequality constraints. These conditions generalize the method of Lagrange multipliers, which is used for equality constraints.

The KKT conditions state that for a constrained optimization problem, if a point is an optimal solution, then there exist multipliers for each constraint such that certain equations and inequalities are satisfied. These conditions involve the gradients of the objective function and the constraint functions. Specifically, the gradient of the objective function must be a linear combination of the gradients of the active constraints at the optimal point, with the coefficients being the Lagrange multipliers.

The conditions also include requirements for the multipliers themselves. For minimization problems with inequality constraints of the form g(x) <= 0, the multipliers associated with these constraints must be non-negative. Furthermore, for each inequality constraint, either the constraint is inactive (g(x) < 0), in which case its corresponding multiplier is zero, or the constraint is active (g(x) = 0), in which case its multiplier can be positive. This latter property is known as complementary slackness.

While KKT conditions are necessary for optimality, they are not always sufficient. Sufficiency is guaranteed under certain convexity assumptions. If the objective function is convex and the feasible region is a convex set (which is ensured if all inequality constraints are convex), then satisfying the KKT conditions implies that the point is a global optimum. These conditions are fundamental in the theory and practice of constrained optimization.