Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [karush-kuhn-tucker]

In mathematical optimization, the Karush–Kuhn–Tucker (KKT) conditions are first order necessary conditions for a solution in nonlinear programming to be optimal, provided that some regularity conditions are satisfied.

In mathematical optimization, the Karush–Kuhn–Tucker (KKT) conditions, also known as the Kuhn–Tucker conditions, are 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. 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.

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.

The KKT conditions include stationarity, primal feasibility, dual feasibility, and complementary slackness.

511 questions
4
votes
2 answers

Does KKT works for non-convex problems as well?

I want to make sure that the following claim is correct. Please let me know what you think. "Let us assume that we have a constrained non-convex and nonlinear minimization problem. The objective function and all the constraints are differentiable.…
Shahram Shahsavari
  • 103
2
votes
1 answer

KKT condition with equality and inequality constraints

find the KKT point of the following problem: $$\min\quad f(x_1,x_2)=(x_1-3)^{2}+(x_2-2)^{2}\\ subject\quad to\qquad \qquad \qquad \qquad \qquad\qquad\\ x_1^{2}+x_2^{2}\le5\\ x_1+2x_2=4\\ x_1\ge0,x_2\ge0$$ what I tried: for the general…
burg1ar
  • 73
2
votes
2 answers

Kuhn-Tucker conditions

I can not find out when to use positive or negative Lagrange multipliers. Does it depend on if I am looking for MAX or MIN ? or maybe it depends on inequality I mean if it is bigger than zero or lower than zero ? I have a feeling that I get right…
Plokijuh
  • 21
1
vote
0 answers

Complementarity in KKT conditions

What is the meaning of Complementarity in Karush–Kuhn–Tucker? "In mathematical optimization, the Karush–Kuhn–Tucker (KKT) conditions, also known as the Kuhn–Tucker conditions, are first-order necessary conditions for a solution in nonlinear…
kangarooo
  • 65
1
vote
0 answers

Binding constraint in Kuhn Tucker

I am struggling a bit with a Kuhn-Tucker maximization problem. The problem is as follows: $\max_{v} u(y_{h} + vy_{w}) - g(v)$ $\text{s. t. } u(y_{w}(1-v)) - h(v) \geq \overline{u}(y_{w})$ u() is increasing, concave, and h and g increasing convex.…
Charlotte Ringdal
  • 11
  • 1
0
votes
1 answer

Kuhn-Tucker conditions: why does $\lambda$ $\dfrac{\partial L}{\partial \lambda}=0$?

Satifying Kuhn-Tucker conditions. Given $\lambda$ is a row, and $\dfrac{\partial L}{\partial \lambda}$ is a column, why does $\lambda$ $\dfrac{\partial L}{\partial \lambda}=0$ ?
Eóin Éamon Ó Cnoic
  • 1
0
votes
1 answer

Are KKT regularity conditions always fulfilled for a single inequality?

When I was studying the KKT regularity conditions it appeared to me that they would always be fulfilled for a single inequality constraint since LICQ only requires the gradients to be independent at the extreme point. Is this understanding correct?
Jannis
  • 163
0
votes
0 answers

Kuhn - Tucker Conditions Optimization problem

I will be really grateful if someone can give me som advice. I have the following problem: Given the program: \begin{align*} & \mathrm{opt.}:\quad &&(x_{1} - 4)^2 + (x_{2} -3)^2\\ & \mathrm{subject \ to}: &&x^2_{1} +x^2_{2} \leq 25\\ & &&x_{1} +…
Jorge
  • 1
0
votes
1 answer

Can KKT conditions be used if some constraint is a strict inequality?

I want to know if the Kuhn Tucker conditions can be used to identify a global maximum if one or more of the constraints is a strict inequality. What information would I need to answer this question? Edit: The problem that motivated me to ask this is…
bitter-sweet
  • 377
-1
votes
2 answers

Langrarian multiplier

Consider the following function $$f(x, y)=x^4-y^2$$ And Set $A=\{(x,y)\in R^2: x^2+y^2=1\}$ is required. find the Lagrangian equation that determines the extreme point of $F$ on $A$ and calculates the solution for this equation. characterize the…
RudY the Heck
  • 1
  • 2