Inmathematical optimization, theKarush–Kuhn–Tucker (KKT)conditions, also known as theKuhn–Tucker conditions, arefirst derivative tests (sometimes called first-ordernecessary conditions) for a solution innonlinear programming to beoptimal, provided that someregularity conditions are satisfied.

Allowing inequality constraints, the KKT approach to nonlinear programming generalizes the method ofLagrange 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 afterHarold W. Kuhn andAlbert W. Tucker, who first published the conditions in 1951.^[2] Later scholars discovered that the necessary conditions for this problem had been stated byWilliam Karush in his master's thesis in 1939.^[3][4]

Consider the following nonlinear optimization problem instandard form:

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

subject to

${\displaystyle g_i(\mathbf{x})\le 0,}$

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

where${\displaystyle \mathbf{x}\in \mathbf{X}}$ is the optimization variable chosen from aconvex subset of${\displaystyle {\mathbb{R}}^n}$,${\displaystyle f}$ is theobjective orutility function,${\displaystyle g_i(i=1,\dots ,m)}$ are the inequalityconstraint functions and${\displaystyle h_j(j=1,\dots ,\ell )}$ are the equalityconstraint functions. The numbers of inequalities and equalities are denoted by${\displaystyle m}$ and${\displaystyle \ell }$ respectively. Corresponding to the constrained optimization problem one can form the Lagrangian function

$${\displaystyle \mathcal{L}(\mathbf{x},\mu ,\lambda )=f(\mathbf{x})+{\mu }^{\mathrm{\top }}\mathbf{g}(\mathbf{x})+{\lambda }^{\mathrm{\top }}\mathbf{h}(\mathbf{x})=L(\mathbf{x},\alpha )=f(\mathbf{x})+{\alpha }^{\mathrm{\top }}\left(\begin{array}{c}\mathbf{g}(\mathbf{x})\\ \mathbf{h}(\mathbf{x})\end{array}\right)}$$

where

$${\displaystyle \mathbf{g}\left(\mathbf{x}\right)=\left[\begin{array}{c}{g}_1\left(\mathbf{x}\right)\\ ⋮\\ g_i\left(\mathbf{x}\right)\\ ⋮\\ g_m\left(\mathbf{x}\right)\end{array}\right],\mathbf{h}\left(\mathbf{x}\right)=\left[\begin{array}{c}{h}_1\left(\mathbf{x}\right)\\ ⋮\\ h_j\left(\mathbf{x}\right)\\ ⋮\\ h_{\ell }\left(\mathbf{x}\right)\end{array}\right],\mu =\left[\begin{array}{c}{\mu }_1\\ ⋮\\ {\mu }_i\\ ⋮\\ {\mu }_m\end{array}\right],\lambda =\left[\begin{array}{c}{\lambda }_1\\ ⋮\\ {\lambda }_j\\ ⋮\\ {\lambda }_{\ell }\end{array}\right]\text{and}\alpha =\left[\begin{array}{c}\mu \\ \lambda \end{array}\right].}$$ TheKarush–Kuhn–Tucker theorem then states the following.

Since the idea of this approach is to find asupporting hyperplane on the feasible set${\displaystyle \mathbf{\Gamma }=\left\{\mathbf{x}\in \mathbf{X}:g_i(\mathbf{x})\le 0,i=1,\dots ,m\right\}}$, the proof of the Karush–Kuhn–Tucker theorem makes use of thehyperplane 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 aclosed-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 theobjective function${\displaystyle f:{\mathbb{R}}^n\to \mathbb{R}}$ and the constraint functions${\displaystyle g_i:{\mathbb{R}}^n\to \mathbb{R}}$ and${\displaystyle h_j:{\mathbb{R}}^n\to \mathbb{R}}$ havesubderivatives at a point${\displaystyle x^{\ast }\in {\mathbb{R}}^n}$. If${\displaystyle x^{\ast }}$ is alocal optimum and the optimization problem satisfies some regularity conditions (see below), then there exist constants${\displaystyle {\mu }_i(i=1,\dots ,m)}$ and${\displaystyle {\lambda }_j(j=1,\dots ,\ell )}$, called KKT multipliers, such that the following four groups of conditions hold:^[8]

Stationarity

For minimizing${\displaystyle f(x)}$:${\displaystyle \mathrm{\partial }f(x^{\ast })+\sum _{j=1}^{\ell }{\lambda }_j\mathrm{\partial }{h}_j(x^{\ast })+\sum _{i=1}^m{\mu }_i\mathrm{\partial }{g}_i(x^{\ast })\ni \mathbf{0}}$

For maximizing${\displaystyle f(x)}$:${\displaystyle -\mathrm{\partial }f(x^{\ast })+\sum _{j=1}^{\ell }{\lambda }_j\mathrm{\partial }{h}_j(x^{\ast })+\sum _{i=1}^m{\mu }_i\mathrm{\partial }{g}_i(x^{\ast })\ni \mathbf{0}}$

Primal feasibility

${\displaystyle h_j(x^{\ast })=0,\text{ for }j=1,\dots ,\ell }$

${\displaystyle g_i(x^{\ast })\le 0,\text{ for }i=1,\dots ,m}$

Dual feasibility

${\displaystyle {\mu }_i\ge 0,\text{ for }i=1,\dots ,m}$

Complementary slackness

${\displaystyle \sum _{i=1}^m{\mu }_i{g}_i(x^{\ast })=0.}$

The last condition is sometimes written in the equivalent form:${\displaystyle {\mu }_i{g}_i(x^{\ast })=0,\text{ for }i=1,\dots ,m.}$

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 calledLagrange multipliers.

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

• ${\displaystyle f}$ is a potential field that the particle is minimizing. The force generated by${\displaystyle f}$ is${\displaystyle -\mathrm{\partial }f}$.
• ${\displaystyle g_i}$ are one-sided constraint surfaces. The particle is allowed to move inside${\displaystyle g_i\le 0}$, but whenever it touches${\displaystyle g_i=0}$, it is pushed inwards.
• ${\displaystyle h_j}$ are two-sided constraint surfaces. The particle is allowed to move only on the surface${\displaystyle h_j}$.

Primal stationarity states that the "force" of${\displaystyle \mathrm{\partial }f(x^{\ast })}$ is exactly balanced by a linear sum of forces${\displaystyle \mathrm{\partial }{h}_j(x^{\ast })}$ and${\displaystyle \mathrm{\partial }{g}_i(x^{\ast })}$.

Dual feasibility additionally states that all the${\displaystyle \mathrm{\partial }{g}_i(x^{\ast })}$ forces must be one-sided, pointing inwards into the feasible set for${\displaystyle x}$.

Complementary slackness states that if, then the force coming from${\displaystyle \mathrm{\partial }{g}_i(x^{\ast })}$ must be zero i.e.,${\displaystyle {\mu }_i(x^{\ast })=0}$, since the particle is not on the boundary, the one-sided constraint force cannot activate.

The necessary conditions can be written withJacobian matrices of the constraint functions. Let${\displaystyle \mathbf{g}(x):{\mathbb{R}}^n\to {\mathbb{R}}^m}$ be defined as${\displaystyle \mathbf{g}(x)={\left(g_1(x),\dots ,g_m(x)\right)}^{\mathrm{\top }}}$ and let${\displaystyle \mathbf{h}(x):{\mathbb{R}}^n\to {\mathbb{R}}^{\ell }}$ be defined as${\displaystyle \mathbf{h}(x)={\left(h_1(x),\dots ,h_{\ell }(x)\right)}^{\mathrm{\top }}}$. Let${\displaystyle \mathit{\mu }={\left({\mu }_1,\dots ,{\mu }_m\right)}^{\mathrm{\top }}}$ and${\displaystyle \mathit{\lambda }={\left({\lambda }_1,\dots ,{\lambda }_{\ell }\right)}^{\mathrm{\top }}}$. Then the necessary conditions can be written as:

Stationarity

For maximizing${\displaystyle f(x)}$:${\displaystyle \mathrm{\partial }f(x^{\ast })-D\mathbf{g}(x^{\ast }{)}^{\mathrm{\top }}\mathit{\mu }-D\mathbf{h}(x^{\ast }{)}^{\mathrm{\top }}\mathit{\lambda }=\mathbf{0}}$

For minimizing${\displaystyle f(x)}$:${\displaystyle \mathrm{\partial }f(x^{\ast })+D\mathbf{g}(x^{\ast }{)}^{\mathrm{\top }}\mathit{\mu }+D\mathbf{h}(x^{\ast }{)}^{\mathrm{\top }}\mathit{\lambda }=\mathbf{0}}$

Primal feasibility

${\displaystyle \mathbf{g}(x^{\ast })\le \mathbf{0}}$

${\displaystyle \mathbf{h}(x^{\ast })=\mathbf{0}}$

Dual feasibility

${\displaystyle \mathit{\mu }\ge \mathbf{0}}$

Complementary slackness

${\displaystyle {\mathit{\mu }}^{\mathrm{\top }}\mathbf{g}(x^{\ast })=0.}$

One can ask whether a minimizer point${\displaystyle x^{\ast }}$ 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${\displaystyle x^{\ast }}$ of a function${\displaystyle f(x)}$ in an unconstrained problem has to satisfy the condition${\displaystyle \mathrm{\nabla }f(x^{\ast })=0}$. 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${\displaystyle h_j}$ areaffine 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 arelinearly independent at${\displaystyle x^{\ast }}$.
Mangasarian-Fromovitz constraint qualification	MFCQ	The gradients of the equality constraints are linearly independent at${\displaystyle x^{\ast }}$ and there exists a vector${\displaystyle d\in {\mathbb{R}}^n}$ such that for all active inequality constraints and${\displaystyle \mathrm{\nabla }{h}_j(x^{\ast }{)}^{\mathrm{\top }}d=0}$ 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${\displaystyle x^{\ast }}$ 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${\displaystyle x^{\ast }}$ with non-negative scalars associated with the inequality constraints, then it remains linearly dependent in a neighborhood of${\displaystyle x^{\ast }}$.
Quasi-normality constraint qualification	QNCQ	If the gradients of the active inequality constraints and the gradients of the equality constraints are linearly dependent at${\displaystyle x^{\ast }}$ with associated multipliers${\displaystyle {\lambda }_j}$ for equalities and${\displaystyle {\mu }_i\ge 0}$ for inequalities, then there is no sequence${\displaystyle x_k\to x^{\ast }}$ such that${\displaystyle {\lambda }_j\ne 0\Rightarrow {\lambda }_j{h}_j(x_k)>0}$ and${\displaystyle {\mu }_i\ne 0\Rightarrow {\mu }_i{g}_i(x_k)>0.}$
Slater's condition	SC	For aconvex 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${\displaystyle h_j(x)=0}$ 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.

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 differentiableconcave function, the inequality constraints${\displaystyle g_j}$ are differentiableconvex functions, the equality constraints${\displaystyle h_i}$ areaffine functions, andSlater's condition holds.^[10] Similarly, if the objective function${\displaystyle f}$ of a minimization problem is a differentiableconvex 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 1invex functions.^[11][12]

For smooth,non-linear optimization problems, a second order sufficient condition is given as follows.

The solution${\displaystyle x^{\ast },{\lambda }^{\ast },{\mu }^{\ast }}$ found in the above section is a constrained local minimum if for the Lagrangian,

${\displaystyle L(x,\lambda ,\mu )=f(x)+\sum _{i=1}^m{\mu }_i{g}_i(x)+\sum _{j=1}^{\ell }{\lambda }_j{h}_j(x)}$

then,

${\displaystyle s^T{\mathrm{\nabla }}_{xx}^{2}L(x^{\ast },{\lambda }^{\ast },{\mu }^{\ast })s\ge 0}$

where${\displaystyle s\ne 0}$ is a vector satisfying the following,

${\displaystyle {\left[{\mathrm{\nabla }}_x{g}_i(x^{\ast }),{\mathrm{\nabla }}_x{h}_j(x^{\ast })\right]}^{T}s=0_{{\mathbb{R}}^2}}$

where only those active inequality constraints${\displaystyle g_i(x)}$ corresponding to strict complementarity (i.e. where${\displaystyle {\mu }_i>0}$) are applied. The solution is a strict constrained local minimum in the case the inequality is also strict.

If${\displaystyle s^T{\mathrm{\nabla }}_{xx}^{2}L(x^{\ast },{\lambda }^{\ast },{\mu }^{\ast })s=0}$, the third order Taylor expansion of the Lagrangian should be used to verify if${\displaystyle x^{\ast }}$ is a local minimum. The minimization of${\displaystyle f(x_1,x_2)=(x_2-x_1^2)(x_2-3x_1^2)}$ is a good counter-example, see alsoPeano surface.

Often inmathematical 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 ofprofit, 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

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

${\displaystyle Q\ge 0,}$

and the KKT conditions are

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 whichmarginal revenue${\displaystyle \text{d}R/\text{d}Q}$ is less thanmarginal cost${\displaystyle \text{d}C/\text{d}Q}$ — a result that is of interest because it contrasts with the behavior of aprofit 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:

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

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

${\displaystyle g_i(x)\le a_i,h_j(x)=0.}$

The value function is defined as

${\displaystyle V(a_1,\dots ,a_n)=\underset{x}{sup}f(x)}$

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

${\displaystyle g_i(x)\le a_i,h_j(x)=0}$

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

so the domain of${\displaystyle V}$ is${\displaystyle \{a\in {\mathbb{R}}^m\mid \text{for some }x\in X,g_i(x)\le a_i,i\in \{1,\dots ,m\}\}.}$

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, inutility maximization problems.

With an extra multiplier${\displaystyle {\mu }_0\ge 0}$, which may be zero (as long as${\displaystyle ({\mu }_0,\mu ,\lambda )\ne 0}$), in front of${\displaystyle \mathrm{\nabla }f(x^{\ast })}$ the KKT stationarity conditions turn into

which are called theFritz John conditions. This optimality conditions holds without constraint qualifications and it is equivalent to the optimality conditionKKT or (not-MFCQ).

The KKT conditions belong to a wider class of the first-order necessary conditions (FONC), which allow for non-smooth functions usingsubderivatives.

• Farkas' lemma
• Lagrange multiplier
• TheBig M method, for linear problems, which extends thesimplex algorithm to problems that contain "greater-than" constraints.
• Interior-point method a method to solve the KKT conditions.
• Slack variable
• Slater's condition

• 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.

• Examples and Tutorials on the KKT Conditions

• Mathematical optimization
• Mathematical economics

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