Karush–Kuhn–Tucker conditions

Consider the following nonlinear minimization or maximization problem:

Optimize ${\displaystyle f(\mathbf {x} )}$

subject to

${\displaystyle g_{i}(\mathbf {x} )\leq 0,}$

${\displaystyle h_{j}(\mathbf {x} )=0.}$

where ${\displaystyle \mathbf {x} \in \mathbf {X} }$ is the optimization variable chosen from a convex subset of ${\displaystyle \mathbb {R} ^{n}}$, ${\displaystyle f}$ is the objective or utility function, ${\displaystyle g_{i}\ (i=1,\ldots ,m)}$ are the inequality constraint functions and ${\displaystyle h_{j}\ (j=1,\ldots ,\ell )}$ are the equality constraint functions. The numbers of inequalities and equalities are denoted by ${\displaystyle m}$ and ${\displaystyle \ell }$ respectively. Corresponding to the constraint optimization problem one can form the Lagrangian function

${\displaystyle L(\mathbf {x} ,\mathbf {\mu } ,\mathbf {\lambda } )=f(\mathbf {x} )+\mathbf {\mu } ^{\top }\mathbf {g} (\mathbf {x} )+\mathbf {\lambda } ^{\top }\mathbf {h} (\mathbf {x} )}$

where ${\displaystyle \mathbf {g} (\mathbf {x} )=\left(g_{1}(\mathbf {x} ),\ldots ,g_{m}(\mathbf {x} )\right)^{\top }}$, ${\displaystyle \mathbf {h} (\mathbf {x} )=\left(h_{1}(\mathbf {x} ),\ldots ,h_{\ell }(\mathbf {x} )\right)^{\top }}$. The Karush–Kuhn–Tucker theorem then states the following.

Theorem. If ${\displaystyle (\mathbf {x} ^{\ast },\mathbf {\mu } ^{\ast })}$ is a saddle point of ${\displaystyle L(\mathbf {x} ,\mathbf {\mu } )}$ in ${\displaystyle \mathbf {x} \in \mathbf {X} }$, ${\displaystyle \mathbf {\mu } \geq \mathbf {0} }$, then ${\displaystyle \mathbf {x} ^{\ast }}$ is an optimal vector for the above optimization problem. Suppose that ${\displaystyle f(\mathbf {x} )}$ and ${\displaystyle g_{i}(\mathbf {x} )}$, ${\displaystyle i=1,\ldots ,m}$, are convex in ${\displaystyle \mathbf {x} }$ and that there exists ${\displaystyle \mathbf {x} _{0}\in \mathbf {X} }$ such that ${\displaystyle \mathbf {g} (\mathbf {x} _{0})<0}$. Then with an optimal vector ${\displaystyle \mathbf {x} ^{\ast }}$ for the above optimization problem there is associated a non-negative vector ${\displaystyle \mathbf {\mu } ^{\ast }}$ such that ${\displaystyle L(\mathbf {x} ^{\ast },\mathbf {\mu } ^{\ast })}$ is a saddle point of ${\displaystyle L(\mathbf {x} ,\mathbf {\mu } )}$.[5]

Since the idea of this approach is to find a supporting hyperplane on the feasible set ${\displaystyle \mathbf {\Gamma } =\left\{\mathbf {x} \in \mathbf {X} :g_{i}(\mathbf {x} )\leq 0,i=1,\ldots ,m\right\}}$, 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]

Suppose that the objective function ${\displaystyle f:\mathbb {R} ^{n}\rightarrow \mathbb {R} }$ and the constraint functions ${\displaystyle g_{i}:\,\!\mathbb {R} ^{n}\rightarrow \mathbb {R} }$ and ${\displaystyle h_{j}:\,\!\mathbb {R} ^{n}\rightarrow \mathbb {R} }$ are continuously differentiable at a point ${\displaystyle x^{*}\in \mathbb {R} ^{n}}$. If ${\displaystyle x^{*}}$ is a local optimum and the optimization problem satisfies some regularity conditions (see below), then there exist constants ${\displaystyle \mu _{i}\ (i=1,\ldots ,m)}$ and ${\displaystyle \lambda _{j}\ (j=1,\ldots ,\ell )}$, called KKT multipliers, such that the following four groups of conditions hold:

Inequality constraint diagram for optimization problems

Stationarity

For minimizing ${\displaystyle f(x)}$: ${\displaystyle \nabla f(x^{*})+\sum _{i=1}^{m}\mu _{i}\nabla g_{i}(x^{*})+\sum _{j=1}^{\ell }\lambda _{j}\nabla h_{j}(x^{*})=\mathbf {0} }$

For maximizing ${\displaystyle f(x)}$: ${\displaystyle -\nabla f(x^{*})+\sum _{i=1}^{m}\mu _{i}\nabla g_{i}(x^{*})+\sum _{j=1}^{\ell }\lambda _{j}\nabla h_{j}(x^{*})=\mathbf {0} }$

Primal feasibility

${\displaystyle g_{i}(x^{*})\leq 0,{\text{ for }}i=1,\ldots ,m}$

${\displaystyle h_{j}(x^{*})=0,{\text{ for }}j=1,\ldots ,\ell \,\!}$

Dual feasibility

${\displaystyle \mu _{i}\geq 0,{\text{ for }}i=1,\ldots ,m}$

Complementary slackness

${\displaystyle \sum _{i=1}^{m}\mu _{i}g_{i}(x^{*})=0.}$

The last condition is sometimes written in the equivalent form: ${\displaystyle \mu _{i}g_{i}(x^{*})=0,{\text{ for }}i=1,\ldots ,m.}$

In the particular case ${\displaystyle m=0}$, i.e., when there are no inequality constraints, the KKT conditions turn into the Lagrange conditions, and the KKT multipliers are called Lagrange multipliers.

If some of the functions are non-differentiable, subdifferential versions of Karush–Kuhn–Tucker (KKT) conditions are available.[8]

One can ask whether a minimizer point ${\displaystyle x^{*}}$ 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 ${\displaystyle x^{*}}$ of a function ${\displaystyle f(x)}$ in an unconstrained problem has to satisfy the condition ${\displaystyle \nabla f(x^{*})=0}$. 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 ${\displaystyle g_{i}}$ and ${\displaystyle h_{j}}$ 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 ${\displaystyle x^{*}}$.
Mangasarian-Fromovitz constraint qualification	MFCQ	The gradients of the equality constraints are linearly independent at ${\displaystyle x^{*}}$ and there exists a vector ${\displaystyle d\in \mathbb {R} ^{n}}$ such that ${\displaystyle \nabla g_{i}(x^{*})^{\top }d<0}$ for all active inequality constraints and ${\displaystyle \nabla h_{j}(x^{*})^{\top }d=0}$ for all equality constraints.[9]
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 ${\displaystyle x^{*}}$ 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 ${\displaystyle x^{*}}$ with non-negative scalars associated with the inequality constraints, then it remains linearly dependent in a neighborhood of ${\displaystyle x^{*}}$.
Quasi-normality constraint qualification	QNCQ	If the gradients of the active inequality constraints and the gradients of the equality constraints are linearly dependent at ${\displaystyle x^{*}}$ with associated multipliers ${\displaystyle \lambda _{j}}$ for equalities and ${\displaystyle \mu _{i}\geq 0}$ for inequalities, then there is no sequence ${\displaystyle x_{k}\to x^{*}}$ such that ${\displaystyle \lambda _{j}\neq 0\Rightarrow \lambda _{j}h_{j}(x_{k})>0}$ and ${\displaystyle \mu _{i}\neq 0\Rightarrow \mu _{i}g_{i}(x_{k})>0.}$
Slater's condition	SC	For a convex problem (i.e., assuming minimization, ${\displaystyle f,g_{i}}$ are convex and ${\displaystyle h_{j}}$ is affine), there exists a point ${\displaystyle x}$ such that ${\displaystyle h(x)=0}$ and ${\displaystyle g_{i}(x)<0.}$

It can be shown that

LICQ ⇒ MFCQ ⇒ CPLD ⇒ QNCQ

and

LICQ ⇒ CRCQ ⇒ CPLD ⇒ QNCQ

(and the converses are not true), although MFCQ is not equivalent to CRCQ.[10] 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 ${\displaystyle f}$ of a maximization problem is a concave function, the inequality constraints ${\displaystyle g_{j}}$ are continuously differentiable convex functions and the equality constraints ${\displaystyle h_{i}}$ are affine functions. Similarly, if the objective function ${\displaystyle f}$ of a minimization problem is a 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.[11][12]

Often in mathematical economics the KKT approach is used in theoretical models in order to obtain qualitative results. For example,[13] consider a firm that maximizes its sales revenue subject to a minimum profit constraint. Letting ${\displaystyle Q}$ be the quantity of output produced (to be chosen), ${\displaystyle R(Q)}$ be sales revenue with a positive first derivative and with a zero value at zero output, ${\displaystyle C(Q)}$ be production costs with a positive first derivative and with a non-negative value at zero output, and ${\displaystyle G_{\min }}$ 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 ${\displaystyle -R(Q)}$

subject to

${\displaystyle G_{\min }\leq R(Q)-C(Q)}$

${\displaystyle Q\geq 0,}$

and the KKT conditions are

${\displaystyle {\begin{aligned}&\left({\frac {{\text{d}}R}{{\text{d}}Q}}\right)(1+\mu )-\mu \left({\frac {{\text{d}}C}{{\text{d}}Q}}\right)\leq 0,\\[5pt]&Q\geq 0,\\[5pt]&Q\left[\left({\frac {{\text{d}}R}{{\text{d}}Q}}\right)(1+\mu )-\mu \left({\frac {{\text{d}}C}{{\text{d}}Q}}\right)\right]=0,\\[5pt]&R(Q)-C(Q)-G_{\min }\geq 0,\\[5pt]&\mu \geq 0,\\[5pt]&\mu [R(Q)-C(Q)-G_{\min }]=0.\end{aligned}}}$

Since ${\displaystyle Q=0}$ would violate the minimum profit constraint, we have ${\displaystyle Q>0}$ and hence the third condition implies that the first condition holds with equality. Solving that equality gives

${\displaystyle {\frac {{\text{d}}R}{{\text{d}}Q}}={\frac {\mu }{1+\mu }}\left({\frac {{\text{d}}C}{{\text{d}}Q}}\right).}$

Because it was given that ${\displaystyle {\text{d}}R/{\text{d}}Q}$ and ${\displaystyle {\text{d}}C/{\text{d}}Q}$ are strictly positive, this inequality along with the non-negativity condition on ${\displaystyle \mu }$ guarantees that ${\displaystyle \mu }$ is positive and so the revenue-maximizing firm operates at a level of output at which marginal revenue ${\displaystyle {\text{d}}R/{\text{d}}Q}$ is less than marginal cost ${\displaystyle {\text{d}}C/{\text{d}}Q}$ — 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:

${\displaystyle {\text{Maximize }}\;f(x)}$

${\displaystyle {\text{subject to }}\ }$

${\displaystyle g_{i}(x)\leq a_{i},h_{j}(x)=0.}$

The value function is defined as

${\displaystyle V(a_{1},\ldots ,a_{n})=\sup \limits _{x}f(x)}$

${\displaystyle {\text{subject to }}\ }$

${\displaystyle g_{i}(x)\leq a_{i},h_{j}(x)=0}$

${\displaystyle j\in \{1,\ldots ,\ell \},i\in \{1,\ldots ,m\},}$

so the domain of ${\displaystyle V}$ is ${\displaystyle \{a\in \mathbb {R} ^{m}\mid {\text{for some }}x\in X,g_{i}(x)\leq a_{i},i\in \{1,\ldots ,m\}\}.}$

Given this definition, each coefficient ${\displaystyle \mu _{i}}$ is the rate at which the value function increases as ${\displaystyle a_{i}}$ increases. Thus if each ${\displaystyle a_{i}}$ is interpreted as a resource constraint, the coefficients tell you how much increasing a resource will increase the optimum value of our function ${\displaystyle f}$. This interpretation is especially important in economics and is used, for instance, in utility maximization problems.

With an extra multiplier ${\displaystyle \mu _{0}\geq 0}$, which may be zero (as long as ${\displaystyle (\mu _{0},\mu ,\lambda )\neq 0}$), in front of ${\displaystyle \nabla f(x^{*})}$ the KKT stationarity conditions turn into

${\displaystyle {\begin{aligned}&\mu _{0}\,\nabla f(x^{*})+\sum _{i=1}^{m}\mu _{i}\,\nabla g_{i}(x^{*})+\sum _{j=1}^{\ell }\lambda _{j}\,\nabla h_{j}(x^{*})=0,\\[4pt]&\mu _{j}g_{i}(x^{*})=0,\quad i=1,\dots ,m,\end{aligned}}}$

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.

• Farkas' lemma
• Lagrange multiplier
• The Big M method, for linear problems, which extends the simplex algorithm to problems that contain "greater-than" constraints.
• Slack variable

• Andreani, R.; Martínez, J. M.; Schuverdt, M. L. (2005). "On the relation between constant positive linear dependence condition and quasinormality constraint qualification". Journal of Optimization Theory and Applications. 125 (2): 473–485. doi:10.1007/s10957-004-1861-9. S2CID 122212394.
• Avriel, Mordecai (2003). Nonlinear Programming: Analysis and Methods. Dover. ISBN 0-486-43227-0.
• Boltyanski, V.; Martini, H.; Soltan, V. (1998). "The Kuhn–Tucker Theorem". Geometric Methods and Optimization Problems. New York: Springer. pp. 78–92. ISBN 0-7923-5454-0.
• Boyd, S.; Vandenberghe, L. (2004). "Optimality Conditions" (PDF). Convex Optimization. Cambridge University Press. pp. 241–249. ISBN 0-521-83378-7.
• Kemp, Murray C.; Kimura, Yoshio (1978). Introduction to Mathematical Economics. New York: Springer. pp. 38–73. ISBN 0-387-90304-6.
• Rau, Nicholas (1981). "Lagrange Multipliers". Matrices and Mathematical Programming. London: Macmillan. pp. 156–174. ISBN 0-333-27768-6.
• Nocedal, J.; Wright, S. J. (2006). Numerical Optimization. New York: Springer. ISBN 978-0-387-30303-1.
• Sundaram, Rangarajan K. (1996). "Inequality Constraints and the Theorem of Kuhn and Tucker". A First Course in Optimization Theory. New York: Cambridge University Press. pp. 145–171. ISBN 0-521-49770-1.

• Karush–Kuhn–Tucker conditions with derivation and examples
• Examples and Tutorials on the KKT Conditions