In mathematical optimization, the Karush–Kuhn–Tucker (KKT) conditions, also known as the Kuhn–Tucker conditions, are first derivative tests (sometimes called first-order necessary conditions) for a solution in nonlinear programming to be optimal, provided that some regularity conditions are satisfied.
Allowing inequality constraints, the KKT approach to nonlinear programming generalizes the method of Lagrange multipliers, which allows only equality constraints. Similar to the Lagrange approach, the constrained maximization (minimization) problem is rewritten as a Lagrange function whose optimal point is a saddle point, i.e. a global maximum (minimum) over the domain of the choice variables and a global minimum (maximum) over the multipliers, which is why the Karush–Kuhn–Tucker theorem is sometimes referred to as the saddle-point theorem.
The KKT conditions were originally named after Harold W. Kuhn and Albert W. Tucker, who first published the conditions in 1951. Later scholars discovered that the necessary conditions for this problem had been stated by William Karush in his master's thesis in 1939.
Nonlinear optimization problemEdit
Consider the following nonlinear minimization or maximization problem:
where is the optimization variable chosen from a convex subset of , is the objective or utility function, are the inequality constraint functions and are the equality constraint functions. The numbers of inequalities and equalities are denoted by and respectively. Corresponding to the constrained optimization problem one can form the Lagrangian function
where , . The Karush–Kuhn–Tucker theorem then states the following.
Theorem. If is a saddle point of in , , then is an optimal vector for the above optimization problem. Suppose that and , , are convex in and that there exists such that . Then with an optimal vector for the above optimization problem there is associated a non-negative vector such that is a saddle point of .
The system of equations and inequalities corresponding to the KKT conditions is usually not solved directly, except in the few special cases where a closed-form solution can be derived analytically. In general, many optimization algorithms can be interpreted as methods for numerically solving the KKT system of equations and inequalities.
Suppose that the objective function and the constraint functions and are continuously differentiable at a point . If is a local optimum and the optimization problem satisfies some regularity conditions (see below), then there exist constants and , called KKT multipliers, such that the following four groups of conditions hold:
- For minimizing :
- For maximizing :
- Primal feasibility
- Dual feasibility
- Complementary slackness
The last condition is sometimes written in the equivalent form:
In the particular case , i.e., when there are no inequality constraints, the KKT conditions turn into the Lagrange conditions, and the KKT multipliers are called Lagrange multipliers.
The necessary conditions can be written with Jacobian matrices of the constraint functions. Let be defined as and let be defined as . Let and . Then the necessary conditions can be written as:
- For maximizing :
- For minimizing :
- Primal feasibility
- Dual feasibility
- Complementary slackness
Regularity conditions (or constraint qualifications)Edit
One can ask whether a minimizer point of the original, constrained optimization problem (assuming one exists) has to satisfy the above KKT conditions. This is similar to asking under what conditions the minimizer of a function in an unconstrained problem has to satisfy the condition . For the constrained case, the situation is more complicated, and one can state a variety of (increasingly complicated) "regularity" conditions under which a constrained minimizer also satisfies the KKT conditions. Some common examples for conditions that guarantee this are tabulated in the following, with the LICQ the most frequently used one:
|Linearity constraint qualification||LCQ||If and are affine functions, then no other condition is needed.|
|Linear independence constraint qualification||LICQ||The gradients of the active inequality constraints and the gradients of the equality constraints are linearly independent at .|
|Mangasarian-Fromovitz constraint qualification||MFCQ||The gradients of the equality constraints are linearly independent at and there exists a vector such that for all active inequality constraints and for all equality constraints.|
|Constant rank constraint qualification||CRCQ||For each subset of the gradients of the active inequality constraints and the gradients of the equality constraints the rank at a vicinity of is constant.|
|Constant positive linear dependence constraint qualification||CPLD||For each subset of gradients of active inequality constraints and gradients of equality constraints, if the subset of vectors is linearly dependent at with non-negative scalars associated with the inequality constraints, then it remains linearly dependent in a neighborhood of .|
|Quasi-normality constraint qualification||QNCQ||If the gradients of the active inequality constraints and the gradients of the equality constraints are linearly dependent at with associated multipliers for equalities and for inequalities, then there is no sequence such that and|
|Slater's condition||SC||For a convex problem (i.e., assuming minimization, are convex and is affine), there exists a point such that and|
It can be shown that
- LICQ ⇒ MFCQ ⇒ CPLD ⇒ QNCQ
- LICQ ⇒ CRCQ ⇒ CPLD ⇒ QNCQ
(and the converses are not true), although MFCQ is not equivalent to CRCQ. In practice weaker constraint qualifications are preferred since they apply to a broader selection of problems.
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 of a maximization problem is a concave function, the inequality constraints are continuously differentiable convex functions and the equality constraints are affine functions. Similarly, if the objective function of a minimization problem is a convex function, the necessary conditions are also sufficient for optimality.
Second-order sufficient conditionsEdit
For smooth, non-linear optimization problems, a second order sufficient condition is given as follows.
The solution found in the above section is a constrained local minimum if for the Lagrangian,
where is a vector satisfying the following,
where only those active inequality constraints corresponding to strict complementarity (i.e. where ) are applied. The solution is a strict constrained local minimum in the case the inequality is also strict.
If , the third order Taylor expansion of the Lagrangian should be used to verify if is a local minimum. The minimization of is a good counter-example, see also Peano surface.
Often in mathematical economics the KKT approach is used in theoretical models in order to obtain qualitative results. For example, consider a firm that maximizes its sales revenue subject to a minimum profit constraint. Letting be the quantity of output produced (to be chosen), be sales revenue with a positive first derivative and with a zero value at zero output, be production costs with a positive first derivative and with a non-negative value at zero output, and be the positive minimal acceptable level of profit, then the problem is a meaningful one if the revenue function levels off so it eventually is less steep than the cost function. The problem expressed in the previously given minimization form is
- subject to
and the KKT conditions are
Since would violate the minimum profit constraint, we have and hence the third condition implies that the first condition holds with equality. Solving that equality gives
Because it was given that and are strictly positive, this inequality along with the non-negativity condition on guarantees that is positive and so the revenue-maximizing firm operates at a level of output at which marginal revenue is less than marginal cost — a result that is of interest because it contrasts with the behavior of a profit maximizing firm, which operates at a level at which they are equal.
If we reconsider the optimization problem as a maximization problem with constant inequality constraints:
The value function is defined as
so the domain of is
Given this definition, each coefficient is the rate at which the value function increases as increases. Thus if each is interpreted as a resource constraint, the coefficients tell you how much increasing a resource will increase the optimum value of our function . This interpretation is especially important in economics and is used, for instance, in utility maximization problems.
With an extra multiplier , which may be zero (as long as ), in front of the KKT stationarity conditions turn into
which are called the Fritz John conditions. This optimality conditions holds without constraint qualifications and it is equivalent to the optimality condition KKT or (not-MFCQ).
The KKT conditions belong to a wider class of the first-order necessary conditions (FONC), which allow for non-smooth functions using subderivatives.
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