Karush–Kuhn–Tucker conditions

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 global maximum or minimum over the domain of the choice variables and a global minimum (maximum) over the multipliers. The Karush–Kuhn–Tucker theorem is sometimes referred to as the saddle-point theorem.[1]

The KKT conditions were originally named after Harold W. Kuhn and Albert W. Tucker, who first published the conditions in 1951.[2] Later scholars discovered that the necessary conditions for this problem had been stated by William Karush in his master's thesis in 1939.[3][4]

Nonlinear optimization problem

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Consider the following nonlinear optimization problem in standard form:

minimize  
subject to
 
 

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 — (sufficiency) If   is a saddle point of   in  ,  , then   is an optimal vector for the above optimization problem.

(necessity) Suppose that   and  ,  , are convex in   and that there exists   such that   (i.e., Slater's condition holds). Then with an optimal vector   for the above optimization problem there is associated a vector   satisfying   such that   is a saddle point of  .[5]

Since the idea of this approach is to find a supporting hyperplane on the feasible set  , the proof of the Karush–Kuhn–Tucker theorem makes use of the hyperplane separation theorem.[6]

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.[7]

Necessary conditions

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Suppose that the objective function   and the constraint functions   and   have subderivatives 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:[8]

 
Inequality constraint diagram for optimization problems
Stationarity
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.

Proof

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Theorem — (sufficiency) If there exists a solution   to the primal problem, a solution   to the dual problem, such that together they satisfy the KKT conditions, then the problem pair has strong duality, and   is a solution pair to the primal and dual problems.

(necessity) If the problem pair has strong duality, then for any solution   to the primal problem and any solution   to the dual problem, the pair   must satisfy the KKT conditions.[9]

Proof

First, for the   to satisfy the KKT conditions is equivalent to them being a Nash equilibrium.

Fix  , and vary  : equilibrium is equivalent to primal stationarity.

Fix  , and vary  : equilibrium is equivalent to primal feasibility and complementary slackness.

Sufficiency: the solution pair   satisfies the KKT conditions, thus is a Nash equilibrium, and therefore closes the duality gap.

Necessity: any solution pair   must close the duality gap, thus they must constitute a Nash equilibrium (since neither side could do any better), thus they satisfy the KKT conditions.

Interpretation: KKT conditions as balancing constraint-forces in state space

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The primal problem can be interpreted as moving a particle in the space of  , and subjecting it to three kinds of force fields:

  •   is a potential field that the particle is minimizing. The force generated by   is  .
  •   are one-sided constraint surfaces. The particle is allowed to move inside  , but whenever it touches  , it is pushed inwards.
  •   are two-sided constraint surfaces. The particle is allowed to move only on the surface  .

Primal stationarity states that the "force" of   is exactly balanced by a linear sum of forces   and  .

Dual feasibility additionally states that all the   forces must be one-sided, pointing inwards into the feasible set for  .

Complementary slackness states that if  , then the force coming from   must be zero i.e.,  , since the particle is not on the boundary, the one-sided constraint force cannot activate.

Matrix representation

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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:

Stationarity
For maximizing  :  
For minimizing  :  
Primal feasibility
 
 
Dual feasibility
 
Complementary slackness
 

Regularity conditions (or constraint qualifications)

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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:

Constraint Acronym Statement
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.[10]
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  

The strict implications can be shown

LICQ ⇒ MFCQ ⇒ CPLD ⇒ QNCQ

and

LICQ ⇒ CRCQ ⇒ CPLD ⇒ QNCQ

In practice weaker constraint qualifications are preferred since they apply to a broader selection of problems.

Sufficient conditions

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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 differentiable concave function, the inequality constraints   are differentiable convex functions, the equality constraints   are affine functions, and Slater's condition holds.[11] Similarly, if the objective function   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.[12][13]

Second-order sufficient conditions

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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,

 

then,

 

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.

Economics

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Often in mathematical economics the KKT approach is used in theoretical models in order to obtain qualitative results. For example,[14] 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

Minimize  
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.

Value function

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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.

Generalizations

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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.

See also

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References

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  1. ^ Tabak, Daniel; Kuo, Benjamin C. (1971). Optimal Control by Mathematical Programming. Englewood Cliffs, NJ: Prentice-Hall. pp. 19–20. ISBN 0-13-638106-5.
  2. ^ Kuhn, H. W.; Tucker, A. W. (1951). "Nonlinear programming". Proceedings of 2nd Berkeley Symposium. Berkeley: University of California Press. pp. 481–492. MR 0047303.
  3. ^ W. Karush (1939). Minima of Functions of Several Variables with Inequalities as Side Constraints (M.Sc. thesis). Dept. of Mathematics, Univ. of Chicago, Chicago, Illinois.
  4. ^ Kjeldsen, Tinne Hoff (2000). "A contextualized historical analysis of the Kuhn-Tucker theorem in nonlinear programming: the impact of World War II". Historia Math. 27 (4): 331–361. doi:10.1006/hmat.2000.2289. MR 1800317.
  5. ^ Walsh, G. R. (1975). "Saddle-point Property of Lagrangian Function". Methods of Optimization. New York: John Wiley & Sons. pp. 39–44. ISBN 0-471-91922-5.
  6. ^ Kemp, Murray C.; Kimura, Yoshio (1978). Introduction to Mathematical Economics. New York: Springer. pp. 38–44. ISBN 0-387-90304-6.
  7. ^ Boyd, Stephen; Vandenberghe, Lieven (2004). Convex Optimization. Cambridge: Cambridge University Press. p. 244. ISBN 0-521-83378-7. MR 2061575.
  8. ^ Ruszczyński, Andrzej (2006). Nonlinear Optimization. Princeton, NJ: Princeton University Press. ISBN 978-0691119151. MR 2199043.
  9. ^ Geoff Gordon & Ryan Tibshirani. "Karush-Kuhn-Tucker conditions, Optimization 10-725 / 36-725" (PDF). Archived from the original (PDF) on 2022-06-17.
  10. ^ Dimitri Bertsekas (1999). Nonlinear Programming (2 ed.). Athena Scientific. pp. 329–330. ISBN 9781886529007.
  11. ^ Boyd, Stephen; Vandenberghe, Lieven (2004). Convex Optimization. Cambridge: Cambridge University Press. p. 244. ISBN 0-521-83378-7. MR 2061575.
  12. ^ Martin, D. H. (1985). "The Essence of Invexity". J. Optim. Theory Appl. 47 (1): 65–76. doi:10.1007/BF00941316. S2CID 122906371.
  13. ^ Hanson, M. A. (1999). "Invexity and the Kuhn-Tucker Theorem". J. Math. Anal. Appl. 236 (2): 594–604. doi:10.1006/jmaa.1999.6484.
  14. ^ Chiang, Alpha C. Fundamental Methods of Mathematical Economics, 3rd edition, 1984, pp. 750–752.

Further reading

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