Karush–Kuhn–Tucker conditions#Regularity conditions .28or constraint qualifications.29

Consider the following nonlinear optimization problem in standard form:

:minimize $f(\backslash mathbf\{x\})$

:subject to

::$g\_i(\backslash mathbf\{x\})\; \backslash leq\; 0,$

::$h\_j(\backslash mathbf\{x\})\; =0.$

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

$$\backslash mathcal\{L\}(\backslash mathbf\{x\},\backslash mathbf\{\backslash mu\},\backslash mathbf\{\backslash lambda\})\; =\; f(\backslash mathbf\{x\})\; +\; \backslash mathbf\{\backslash mu\}^\backslash top\; \backslash mathbf\{g\}(\backslash mathbf\{x\})\; +\; \backslash mathbf\{\backslash lambda\}^\backslash top\; \backslash mathbf\{h\}(\backslash mathbf\{x\})=L(\backslash mathbf\{x\},\; \backslash mathbf\{\backslash alpha\})=f(\backslash mathbf\{x\})+\backslash mathbf\{\backslash alpha\}^\backslash top\; \backslash begin\{pmatrix\}$$

\mathbf{g}(\mathbf{x}) \\

\mathbf{h}(\mathbf{x})

\end{pmatrix}

where

\mathbf{g}\left(\mathbf{x}\right) = \begin{bmatrix}

g_{1}\left(\mathbf{x}\right) \\

\vdots \\

g_{i}\left(\mathbf{x}\right) \\

\vdots \\

g_{m}\left(\mathbf{x}\right)

\end{bmatrix},

\quad

\mathbf{h}\left(\mathbf{x}\right) = \begin{bmatrix}

h_{1}\left(\mathbf{x}\right) \\

\vdots \\

h_{j}\left(\mathbf{x}\right) \\

\vdots \\

h_{\ell}\left(\mathbf{x}\right)

\end{bmatrix},

\quad

\mathbf{\mu} = \begin{bmatrix}

\mu_{1} \\

\vdots \\

\mu_{i} \\

\vdots \\

\mu_{m} \\

\end{bmatrix},

\quad

\mathbf{\lambda} = \begin{bmatrix}

\lambda_{1} \\

\vdots \\

\lambda_{j} \\

\vdots \\

\lambda_{\ell}

\end{bmatrix}

\quad \text{and} \quad

\mathbf{\alpha} = \begin{bmatrix}

\mu \\

\lambda

\end{bmatrix}. The Karush–Kuhn–Tucker theorem then states the following.

{{Math theorem

| math_statement =

(sufficiency) If $(\backslash mathbf\{x\}^\{\backslash ast\},\backslash mathbf\{\backslash alpha\}^\backslash ast)$ is a saddle point of $L(\backslash mathbf\{x\},\backslash mathbf\{\backslash alpha\})$ in $\backslash mathbf\{x\}\; \backslash in\; \backslash mathbf\{X\}$, $\backslash mathbf\{\backslash mu\}\; \backslash geq\; \backslash mathbf\{0\}$, then $\backslash mathbf\{x\}^\{\backslash ast\}$ is an optimal vector for the above optimization problem.

(necessity) Suppose that $f(\backslash mathbf\{x\})$ and $g\_i(\backslash mathbf\{x\})$, $i\; =\; 1,\; \backslash ldots,\; m$, are convex in $\backslash mathbf\{X\}$ and that there exists $\backslash mathbf\{x\}\_\{0\}\; \backslash in\; \backslash operatorname\{relint\}(\backslash mathbf\{X\})$ such that (i.e., Slater's condition holds). Then with an optimal vector $\backslash mathbf\{x\}^\{\backslash ast\}$ for the above optimization problem there is associated a vector $\backslash mathbf\{\backslash alpha\}^\backslash ast=\backslash begin\{bmatrix\}\backslash mu^*\; \backslash \backslash \; \backslash lambda^*\backslash end\{bmatrix\}$ satisfying $\backslash mathbf\; \backslash mu^*\backslash ge\; \backslash mathbf\; 0$ such that $(\backslash mathbf\{x\}^\{\backslash ast\},\backslash mathbf\{\backslash alpha\}^\backslash ast)$ is a saddle point of $L(\backslash mathbf\{x\},\backslash mathbf\{\backslash alpha\})$.{{cite book |first=G. R. |last=Walsh |title=Methods of Optimization |location=New York |publisher=John Wiley & Sons |year=1975 |isbn=0-471-91922-5 |chapter=Saddle-point Property of Lagrangian Function |pages=39–44 |chapter-url=https://books.google.com/books?id=K0EZAQAAIAAJ&pg=PA39 }}

}}

Since the idea of this approach is to find a supporting hyperplane on the feasible set $\backslash mathbf\{\backslash Gamma\}\; =\; \backslash left\backslash \{\; \backslash mathbf\{x\}\; \backslash in\; \backslash mathbf\{X\}\; :\; g\_i(\backslash mathbf\{x\})\; \backslash leq\; 0,\; i\; =\; 1,\; \backslash ldots,\; m\; \backslash right\backslash \}$, the proof of the Karush–Kuhn–Tucker theorem makes use of the hyperplane separation theorem.{{cite book |first1=Murray C. |last1=Kemp |first2=Yoshio |last2=Kimura |title=Introduction to Mathematical Economics |location=New York |publisher=Springer |year=1978 |isbn=0-387-90304-6 |pages=[https://archive.org/details/introductiontoma0000kemp/page/38 38–44] |url=https://archive.org/details/introductiontoma0000kemp/page/38 }}

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.{{cite book|last1=Boyd|first1=Stephen|last2=Vandenberghe|first2=Lieven|title=Convex Optimization|publisher=Cambridge University Press|location=Cambridge |year=2004|pages=244|isbn=0-521-83378-7|mr=2061575}}

Suppose that the objective function $f\backslash colon\backslash mathbb\{R\}^n\; \backslash rightarrow\; \backslash mathbb\{R\}$ and the constraint functions $g\_i\backslash colon\backslash mathbb\{R\}^n\; \backslash rightarrow\; \backslash mathbb\{R\}$ and $h\_j\backslash colon\backslash mathbb\{R\}^n\; \backslash rightarrow\; \backslash mathbb\{R\}$ have subderivatives at a point $x^*\; \backslash in\; \backslash mathbb\{R\}^n$. If $x^*$ is a local optimum and the optimization problem satisfies some regularity conditions (see below), then there exist constants $\backslash mu\_i\backslash \; (i\; =\; 1,\backslash ldots,m)$ and $\backslash lambda\_j\backslash \; (j\; =\; 1,\backslash ldots,\backslash ell)$, called KKT multipliers, such that the following four groups of conditions hold:

{{cite book |last=Ruszczyński |first=Andrzej |title=Nonlinear Optimization |publisher=Princeton University Press |year=2006 |isbn=978-0691119151 |location=Princeton, NJ |mr=2199043 |author-link=Andrzej Piotr Ruszczyński}}

File:Inequality constraint diagram.svg

;Stationarity

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

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

;Primal feasibility

:$h\_j(x^*)\; =\; 0,\; \backslash text\{\; for\; \}\; j\; =\; 1,\; \backslash ldots,\; \backslash ell\; \backslash ,\backslash !$

:$g\_i(x^*)\; \backslash le\; 0,\; \backslash text\{\; for\; \}\; i\; =\; 1,\; \backslash ldots,\; m$

;Dual feasibility

:$\backslash mu\_i\; \backslash ge\; 0,\; \backslash text\{\; for\; \}\; i\; =\; 1,\; \backslash ldots,\; m$

;Complementary slackness

:$\backslash sum\_\{i=1\}^m\; \backslash mu\_i\; g\_i\; (x^*)\; =\; 0.$

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

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

{{Math theorem

| math_statement =

(sufficiency) If there exists a solution $x^*$ to the primal problem, a solution $(\backslash mu^*,\; \backslash lambda^*)$ to the dual problem, such that together they satisfy the KKT conditions, then the problem pair has strong duality, and $x^*,\; (\backslash mu^*,\; \backslash lambda^*)$ is a solution pair to the primal and dual problems.

(necessity) If the problem pair has strong duality, then for any solution $x^*$ to the primal problem and any solution $(\backslash mu^*,\; \backslash lambda^*)$ to the dual problem, the pair $x^*,\; (\backslash mu^*,\; \backslash lambda^*)$ must satisfy the KKT conditions.{{Cite web |last=Geoff Gordon & Ryan Tibshirani |title=Karush-Kuhn-Tucker conditions, Optimization 10-725 / 36-725 |url=http://www.cs.cmu.edu/~ggordon/10725-F12/slides/16-kkt.pdf |archive-url=https://web.archive.org/web/20220617163414/http://www.cs.cmu.edu/~ggordon/10725-F12/slides/16-kkt.pdf |archive-date=2022-06-17}}

}}

{{Math proof|proof=

First, for the $x^*,\; (\backslash mu^*,\; \backslash lambda^*)$ to satisfy the KKT conditions is equivalent to them being a Nash equilibrium.

Fix $(\backslash mu^*,\; \backslash lambda^*)$, and vary $x$: equilibrium is equivalent to primal stationarity.

Fix $x^*$, and vary $(\backslash mu,\; \backslash lambda)$: equilibrium is equivalent to primal feasibility and complementary slackness.

Sufficiency: the solution pair $x^*,\; (\backslash mu^*,\; \backslash lambda^*)$ satisfies the KKT conditions, thus is a Nash equilibrium, and therefore closes the duality gap.

Necessity: any solution pair $x^*,\; (\backslash mu^*,\; \backslash lambda^*)$ 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.

}}

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

• $f$ is a potential field that the particle is minimizing. The force generated by $f$ is $-\backslash partial\; f$.
• $g\_i$ are one-sided constraint surfaces. The particle is allowed to move inside $g\_i\; \backslash leq\; 0$, but whenever it touches $g\_i=0$, it is pushed inwards.
• $h\_j$ are two-sided constraint surfaces. The particle is allowed to move only on the surface $h\_j$.

Primal stationarity states that the "force" of $\backslash partial\; f(x^*)$ is exactly balanced by a linear sum of forces $\backslash partial\; h\_j(x^*)$ and $\backslash partial\; g\_i(x^*)$.

Dual feasibility additionally states that all the $\backslash partial\; g\_i(x^*)$ forces must be one-sided, pointing inwards into the feasible set for $x$.

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

The necessary conditions can be written with Jacobian matrices of the constraint functions. Let $\backslash mathbf\; g(x)\; :\; \backslash ,\backslash !\backslash mathbb\{R\}^n\; \backslash rightarrow\; \backslash mathbb\{R\}^m$ be defined as $\backslash mathbf\{g\}(x)\; =\; \backslash left(\; g\_\{1\}(x),\; \backslash ldots,\; g\_\{m\}(x)\; \backslash right)^\backslash top$ and let $\backslash mathbf\; h(x)\; :\; \backslash ,\backslash !\backslash mathbb\{R\}^n\; \backslash rightarrow\; \backslash mathbb\{R\}^\{\backslash ell\}$ be defined as $\backslash mathbf\{h\}(x)\; =\; \backslash left(\; h\_\{1\}(x),\; \backslash ldots,\; h\_\{\backslash ell\}(x)\; \backslash right)^\backslash top$. Let $\backslash boldsymbol\{\backslash mu\}\; =\; \backslash left(\; \backslash mu\_1,\; \backslash ldots,\; \backslash mu\_m\; \backslash right)^\backslash top$ and $\backslash boldsymbol\{\backslash lambda\}\; =\; \backslash left(\; \backslash lambda\_1,\; \backslash ldots,\; \backslash lambda\_\{\backslash ell\}\; \backslash right)^\backslash top$. Then the necessary conditions can be written as:

;Stationarity

:For maximizing $f(x)$: $\backslash partial\; f(x^*)\; -\; D\; \backslash mathbf\; g(x^*)^\{\backslash top\}\; \backslash boldsymbol\{\backslash mu\}\; -\; D\; \backslash mathbf\; h(x^*)^\{\backslash top\}\; \backslash boldsymbol\{\backslash lambda\}\; =\; \backslash mathbf\; 0$

:For minimizing $f(x)$: $\backslash partial\; f(x^*)\; +\; D\; \backslash mathbf\; g(x^*)^\{\backslash top\}\; \backslash boldsymbol\{\backslash mu\}\; +\; D\; \backslash mathbf\; h(x^*)^\{\backslash top\}\; \backslash boldsymbol\{\backslash lambda\}\; =\; \backslash mathbf\; 0$

;Primal feasibility

:$\backslash mathbf\; g(x^*)\; \backslash le\; \backslash mathbf\; 0$

:$\backslash mathbf\; h(x^*)\; =\; \backslash mathbf\; 0$

;Dual feasibility

:$\backslash boldsymbol\; \backslash mu\; \backslash ge\; \backslash mathbf\; 0$

;Complementary slackness

:$\backslash boldsymbol\; \backslash mu^\{\backslash top\}\; \backslash mathbf\; g\; (x^*)\; =\; 0.$

One can ask whether a minimizer point $x^*$ 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 $x^*$ of a function $f(x)$ in an unconstrained problem has to satisfy the condition $\backslash nabla\; f(x^*)=0$. 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:

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 $f$ of a maximization problem is a differentiable concave function, the inequality constraints $g\_j$ are differentiable convex functions, the equality constraints $h\_i$ are affine functions, and Slater's condition holds.{{cite book|last1=Boyd|first1=Stephen|last2=Vandenberghe|first2=Lieven|title=Convex Optimization|publisher=Cambridge University Press|location=Cambridge |year=2004|pages=244|isbn=0-521-83378-7|mr=2061575}} Similarly, if the objective function $f$ 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.{{cite journal |first=D. H. |last=Martin |journal=J. Optim. Theory Appl. |volume=47 |issue=1 |pages=65–76 |title=The Essence of Invexity |year= 1985 |doi=10.1007/BF00941316 |s2cid=122906371 }}{{cite journal |first=M. A. |last=Hanson |title=Invexity and the Kuhn-Tucker Theorem |journal=J. Math. Anal. Appl. |volume=236 |issue=2 |pages=594–604 |year=1999 |doi=10.1006/jmaa.1999.6484 |doi-access=free }}

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

The solution $x^*,\; \backslash lambda^*,\; \backslash mu^*$ found in the above section is a constrained local minimum if for the Lagrangian,

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

then,

:$s^T\backslash nabla\; ^2\_\{xx\}L(x^*,\backslash lambda^*,\backslash mu^*)s\; \backslash ge\; 0$

where $s\; \backslash ne\; 0$ is a vector satisfying the following,

:$\backslash left[\; \backslash nabla\_x\; g\_i(x^*),\; \backslash nabla\_x\; h\_j(x^*)\; \backslash right]^T\; s\; =\; 0\_\{\backslash mathbb\{R\}^\{2\}\}$

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

If $s^T\backslash nabla\; ^2\_\{xx\}L(x^*,\backslash lambda^*,\backslash mu^*)s\; =\; 0$, the third order Taylor expansion of the Lagrangian should be used to verify if $x^*$ is a local minimum. The minimization of $f(x\_1,x\_2)=(x\_2-x\_1^2)(x\_2-3x\_1^2)$ is a good counter-example, see also Peano surface.

{{see also|Profit maximization}}

Often in mathematical economics the KKT approach is used in theoretical models in order to obtain qualitative results. For example,Chiang, Alpha C. Fundamental Methods of Mathematical Economics, 3rd edition, 1984, pp. 750–752. consider a firm that maximizes its sales revenue subject to a minimum profit constraint. Letting $Q$ be the quantity of output produced (to be chosen), $R(Q)$ be sales revenue with a positive first derivative and with a zero value at zero output, $C(Q)$ be production costs with a positive first derivative and with a non-negative value at zero output, and $G\_\{\backslash min\}$ 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 $-R(Q)$

:subject to

:$G\_\{\backslash min\}\; \backslash le\; R(Q)\; -\; C(Q)$

:$Q\; \backslash ge\; 0,$

and the KKT conditions are

: $$

\begin{align}

& \left(\frac{\text{d} R}{\text{d} Q} \right) (1+\mu ) - \mu \left( \frac{\text{d} C}{\text{d} Q} \right) \le 0, \\[5pt]

& Q \ge 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} \ge 0, \\[5pt]

& \mu \ge 0, \\[5pt]

& \mu [R(Q) - C(Q) - G_{\min}] = 0.

\end{align}

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

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

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

:$\backslash text\{Maximize\; \}\backslash ;\; f(x)$

:$\backslash text\{subject\; to\; \}\backslash$

::$g\_i(x)\; \backslash le\; a\_i\; ,\; h\_j(x)\; =\; 0.$

The value function is defined as

:$V(a\_1,\; \backslash ldots,\; a\_n)\; =\; \backslash sup\backslash limits\_x\; f(x)$

:$\backslash text\{subject\; to\; \}\backslash$

::$g\_i(x)\; \backslash le\; a\_i\; ,\; h\_j(x)\; =\; 0$

::$j\; \backslash in\; \backslash \{1,\backslash ldots,\; \backslash ell\backslash \},\; i\backslash in\backslash \{1,\backslash ldots,m\backslash \},$

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

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

With an extra multiplier $\backslash mu\_0\backslash geq0$, which may be zero (as long as $(\backslash mu\_0,\backslash mu,\backslash lambda)\backslash neq0$), in front of $\backslash nabla\; f(x^*)$ the KKT stationarity conditions turn into

: $$

\begin{align}

& \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_jg_i(x^*)=0, \quad i=1,\dots,m,

\end{align}

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.
• Interior-point method a method to solve the KKT conditions.
• Slack variable
• Slater's condition

{{Reflist}}

• {{cite journal |first1=R. |last1=Andreani |first2=J. M. |last2=Martínez |first3=M. L. |last3=Schuverdt |title=On the relation between constant positive linear dependence condition and quasinormality constraint qualification |journal=Journal of Optimization Theory and Applications |volume=125 |issue=2 |pages=473–485 |year=2005 |doi=10.1007/s10957-004-1861-9 |s2cid=122212394 }}
• {{cite book |last=Avriel |first=Mordecai |year=2003 |title=Nonlinear Programming: Analysis and Methods |publisher=Dover |isbn=0-486-43227-0 }}
• {{cite book |first1=V. |last1=Boltyanski |first2=H. |last2=Martini |first3=V. |last3=Soltan |title=Geometric Methods and Optimization Problems |location=New York |publisher=Springer |year=1998 |isbn=0-7923-5454-0 |pages=78–92 |chapter=The Kuhn–Tucker Theorem |chapter-url=https://books.google.com/books?id=YD7UBwAAQBAJ&pg=PA78 }}
• {{cite book |first1=S. |last1=Boyd |first2=L. |last2=Vandenberghe |year=2004 |chapter=Optimality Conditions |chapter-url=https://web.stanford.edu/~boyd/cvxbook/bv_cvxbook.pdf#page=255 |title=Convex Optimization |publisher=Cambridge University Press |isbn=0-521-83378-7 |pages=241–249 }}
• {{cite book |first1=Murray C. |last1=Kemp |first2=Yoshio |last2=Kimura |title=Introduction to Mathematical Economics |location=New York |publisher=Springer |year=1978 |isbn=0-387-90304-6 |pages=[https://archive.org/details/introductiontoma0000kemp/page/38 38–73] |url=https://archive.org/details/introductiontoma0000kemp/page/38 }}
• {{cite book |first=Nicholas |last=Rau |chapter=Lagrange Multipliers |title=Matrices and Mathematical Programming |location=London |publisher=Macmillan |year=1981 |isbn=0-333-27768-6 |pages=156–174 }}
• {{cite book |first1=J. |last1=Nocedal |first2=S. J. |last2=Wright |year=2006 |title=Numerical Optimization |publisher=Springer |location=New York |isbn=978-0-387-30303-1 }}
• {{cite book |first=Rangarajan K. |last=Sundaram |chapter=Inequality Constraints and the Theorem of Kuhn and Tucker |title=A First Course in Optimization Theory |location=New York |publisher=Cambridge University Press |year=1996 |isbn=0-521-49770-1 |pages=145–171 |chapter-url=https://books.google.com/books?id=yAfug81P-8YC&pg=PA145 }}

• [http://www.onmyphd.com/?p=kkt.karush.kuhn.tucker Karush–Kuhn–Tucker conditions with derivation and examples]
• [http://apmonitor.com/me575/index.php/Main/KuhnTucker Examples and Tutorials on the KKT Conditions]

{{DEFAULTSORT:Karush-Kuhn-Tucker conditions}}

Category:Mathematical optimization

Category:Mathematical economics