What are the KKT conditions?
The Karush-Kuhn-Tucker (KKT) conditions are a set of necessary conditions for a local optimum in nonlinear programming problems. They are named after the mathematicians Harry M. Karush, Tibor Kuhn, and Martin Tucker, who independently derived them in the 1950s. These conditions are crucial in the field of optimization, as they provide a way to determine whether a given point is a local optimum and to derive necessary conditions for optimality in constrained optimization problems.
The KKT conditions consist of four main parts: the primal feasibility condition, the dual feasibility condition, the complementary slackness condition, and the stationarity condition. Each of these conditions plays a vital role in ensuring that a point is a local optimum.
Primal Feasibility Condition
The primal feasibility condition states that at a local optimum, the primal variables must satisfy all the constraints of the problem. In other words, the point must be feasible with respect to the original problem. This condition ensures that the solution is within the feasible region of the problem.
Dual Feasibility Condition
The dual feasibility condition requires that the Lagrange multipliers associated with the constraints must be non-negative. This condition ensures that the dual problem is feasible, which is essential for the validity of the KKT conditions.
Complementary Slackness Condition
The complementary slackness condition is perhaps the most significant of the KKT conditions. It states that at a local optimum, the product of each Lagrange multiplier and its corresponding constraint must be zero. This condition implies that either the constraint is active (i.e., the primal variable is at its boundary value) or the Lagrange multiplier is zero (i.e., the constraint is inactive). This condition ensures that the primal and dual problems are complementary, meaning that they provide information about each other.
Stationarity Condition
The stationarity condition requires that the gradient of the Lagrangian function with respect to the primal variables must be zero at a local optimum. This condition ensures that the point is a stationary point, meaning that there is no direction in which the objective function can be improved while still satisfying the constraints.
In summary, the KKT conditions are a powerful tool for analyzing and solving constrained optimization problems. By satisfying these conditions, we can determine whether a given point is a local optimum and derive necessary conditions for optimality. These conditions have wide applications in various fields, including engineering, economics, and finance.
