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.

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Problem with finding Karush-Kuhn-Tucker points and checking for global or local minima.

I need to solve the following optimization problem $$\begin{align*} & \mathrm{Min}:\quad f(x_1,x_2)=x_1-10x_2\\ & \mathrm{subject \ to}: \quad x_1^2 -x_2 \geq 0\\ & \qquad \qquad \qquad x_1^2x_2^2 \leq 1 \end{align*}$$ So first I tried to find all…
user313212
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Minimizing convex functions without compatible gradients

I've been working on a minimization problem for a while, involving "simple" conditions, but haven't been able to figure it out. I've tried using Lagrange Multipliers and KKT, but the presence of $\frac{1}{\gamma^2}$ and $\frac{1}{n^2}$ in $\nabla f$…
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Functional analysis with KKT conditions

I want to solve an optimization problem min $F(x_{ik}) $ subject to $x \in X$. $F$ here is a function or functional that I wish to determine. I want my optimal solution to satisfy certain properties. The goal is to find all such functions which in…
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Strong Duality for Euclidean distance

i have an optimization problem in the form: $min ||x - y||$ sbj to: $A.x = 0$ $A.y = 0$ $l_x \leq x \leq u_x, l_y \leq y \leq u_y$ I'm trying to find the dual form of this optimization problem, kkt can be used in this case, but i'm trying to avoid…
Sara Amr
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Minimize $\|\mathbf{x-y}\|^2 $ subject to $x \in $ set $S=\{\mathbf{x} \in \mathbb{R}^n \;\;\;\mid \;\;\; \|\mathbf{x-x_c}\|^2\leq r^2 \}$

We are given the set $S=\{\mathbf{x} \in \mathbb{R}^n \;\;\;\mid \;\;\; \|\mathbf{x-x_c}\|^2\leq r^2 \}$ and a point $\mathbf{y} \in \mathbb{R}^n$. Our goal is to find point $\mathbf{\hat{x}}$ which minimizes $\|\mathbf{x-y}\|_2^2$. If…
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Connection between method of Lagrange multipliers and KKT conditions?

I understand that in general, the KKT conditions are not sufficient for optimality. However, if the primal problem is a convex optimization problem, then the KKT conditions are sufficient for optimality (given smoothness of functions). I'm trying to…
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KKT conditions for nonlinear problem

I need to state the KKT conditions for the following problem: Minimise $x_1^2 + 2x_2^2$ subject to $(x_1-1)^2 + x_2^2 \le 1$ and $x_2 = 1$. I have that these conditions are: $f(x^*) \le 0$ $h(x^*) = 0$ $\lambda^* \ge 0$ $\lambda^*_if_i(x^*) = 0$…
Nique
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Can lagrange multiplier(Kuhn tucker multipliers?) change in corner solution?

If we want to maximize $f(x)$ subject to two constraints, one which says that $x< c$ $c>0$, and another that says that $x\geq 0 $. Assume there are no problems with either $x=0, x>0$ or $\mu =0 ,\mu >0$, where $\mu$ is the multiplier on the second…
majmun
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Kuhn-Tucker's Conditions for optimization problem with non linear inequalities constraints

My problem is to minimize the function \begin{align*} f(x,y,z,t)=& 3 t \left(2 x^2+4 x z\right) \left(2 t x y+t x z-2 t y^2-2 x z+4 y z\right) \\ &+\left(-t x^2+4 t x y+4 t y z+4 x z-8 y z\right)^2 \end{align*} under…
Zbigniew
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KKT conditions (equations) for Generalized Assignment Problem or Binary integer programming problem

I have this formulated Generalized Assignment Problem (GAP) or it can also be considered as Binary integer programming problem. Solving this problem can be achieved through Branch and Bound Technique. enter image description here $max \text{ }…
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General KKT problem

Consider the following problem, where $a_j,b$ and $c_j$ are positive constants: Minimize $\sum_{j=1}^n \frac{c_j}{x_j}$, subject to $\sum_{j=1}^n a_j x_j = b, x_j ≥ 0$ for $j= 1,...,n$. Write the KKT condition and solve the point $\bar{x}$…
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Linear programming with kernel

Can anyone please help me with solving the constrained minimization problem below? $$\mathbf{x}^* = \arg\min \sum_{i=1}^m q_i e^{-2x_i} $$ $$s.t.$$ $$\sum_{i=1}^m x_i = c$$ $$x_i\geq0, i = 1,\cdots,m$$ where $\mathbf{x}=[x_1,\cdots,x_m]^T$ is the…
Sohrab
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optimisation with inequality constraints

I'm struggling with this question: $ \max \{ \ln(y) - (x-1)^2 \} $ s.t. $x + y \leq t$ and $y > 0$ I'm trying to use the Lagrange/Kuhn-Tucker method but don't know how to progress after getting first order conditions $1/y = \lambda$ and $-2(x-1) =…
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Finding KKT conditions for nonlinear optimization problem.

I have an optimization like below: $\text{ minimize } \sum_k - \log_2 x_k $ $\text{subject to: } x_k \leq q , k =1,2, \cdots, N .$ I can form the Lagrange of the problem as below: $L(x, \lambda) = \sum_{k=1}^{N} - \log_2 x_k + \sum_{k=1}^{N}…
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KKT system with rank-deficient constraints

I have an optimization problem of the following form: $$ \begin{aligned} \operatorname*{minimize}_x & \quad \frac{1}{2}||x - a||^2 \\ \operatorname{subject~to} & \quad D^Tx = b \end{aligned} $$ I attempted to solve it, but I…
Alex Shtof
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