# Tagged Questions

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.

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### KKT condition of linearly inseparable Support Vector Machine (SVM)

In the paper Sequential Minimal Optimization:A Fast Algorithm for Training Support Vector Machines, the optimization problem for linearly inseparable SVM is \begin{align} \min\limits_{\boldsymbol{w},...
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### Reducing KKT system

I was using CVXOPT library to solve one of my quadratic programming problem. I found that, CVXOPT library solves KKT system efficiently by reducing a 3x3 matrox into 2x2 blocks which has the following ...
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### KKT conditions for different inputs.

So I have the following problem: I'm trying to get a demand function for a nonlinear 2 variable optimisation problem. There are 3 inequality constraints. Doing the usual thing I get the following ...
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### What does it mean if the KKT conditions do not result in a real solution for a convex problem?

Given a convex optimization problem with equality and inequality constraints, the KKT conditions are sufficient and necessary conditions for optimality. What does it mean if the KKT conditions do not ...
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### Optimize nonlinear model over time period (w/ Kuhn-Tucker Conditions)

I'm working on a nonlinear model that includes a time interval of 12 months. The goal is to maximize the total net benefit (NB) over the entire time period given the constraints listed below. I've ...
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### Kuhn Tucker conditions with strict inequality constraints?

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? ...
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### Solving Binary Linear Programming Problem Using KKT

Execuse me, I know that if I searched a lot I could find the answer, However I have already did my research and I am running out of time. I need the detailed solution of the following linear problem (...
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### Numerical methods and KKT in NLP

I am studying numerical methods and NLP. I started with gradient based methods, newton methods and KKT conditions. I found the following sentence: A local minimum is found by solving KKT conditions, ...
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### Proving equivalent optimization problems

Consider the problems $\min f(x) , x \in X$ and $\min g(x), x \in X$. two optimization problems are said to be equivalent if an optimal solution to one, is also optimal to another. I would like to ...
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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 ...
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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, ...
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### Theoretically understanding the algorithmic solution to this Quadratically Constrained Quadratic Optimization Problem

I have a simple QCQP problem to solve: $\min_{t} x(t)^{T}Ax(t)$ subject to constraints $x(t)^{T}Ax(t) > 1$ where A is a positive definite matrix and $x(t) \in \mathbb{R}^2$ is some time ...
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### Quadratic programing problem and MATLAB

I have a little problem with quadratic programing problem: ${\bf v}^T \Sigma {\bf v} \rightarrow min$, and constrains are ${\bf v}^T \mu = \mu_*, {\bf v}^T {\bf 1 }= 1, 0 \leq {\bf v}.$ Where ...
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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 $\mathbf{\hat{x}}$ ...
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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 ...
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### KKT conditions for a convex optimization (optimal crowdsourcing with budget constraint)

I am having some troubles deriving the optimal solution of the following convex optimization problem, $w_j$, $c_{ij}$, and $B$ are fixed and non negative. \begin{align} & \underset{n_{ij}}{\text{...
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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\...
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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. ...
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### Optimization by KKT-method

I need to solve the following problem by KKT method. $$\text{min} \ \ 2xy + 2yz + 2zx \\ \text{subject to} \ x^2 + y^2 ≤ 2, \ 2x + 2y + z = 0$$ I have gotten as far as setting up the system of ...
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### How to use the Karush–Kuhn–Tucker conditions?

From what I read, the Karush-Kuhn-Tucker conditions are a generalization of the Lagrange Multiplier Method. For the Lagrange Multiplier Method I have been able to find a serie of steps I must do to ...
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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 ...
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### Minimize $x^2+y^2$, subject to… (optimal points, KKT conditions, dual theories)

I am new to this. I am self learning to get ahead of my next years course and came across this question. I thought it would be a good question to look at due to it touching an many different aspects ...
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### dual feasibility of Kuhn-Tucker condition?

minimize $f(x)$ subject to \begin{align} f_i(x) & \le 0, \quad i \in \left\{ 1,\ldots,m \right\} \\ h_i(x) & = 0, \quad i \in \left\{ 1,\ldots,p \right\} \end{align} Then the Lagrange ...
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### Some True or False questions on Nonlinear Optimisation (Exam Preparation)

I am currently preparing for a Nonlinear Optimisation exam and am working through some old question papers and came across these True or False questions: When minimizing a convex function over ...
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I need to maximize a function with equality constraint, so I made the Lagrangian function and I found the partial derivatives, which are these: \frac{|h_1|^2}{\sigma^2(\frac{x b}{\sigma^2+wd}+\... 0answers 35 views ### KKT condition and linear program Since all linear program all convex, and the Slater's condition always hold for linear programs. Is it always possible to solve the linear programs with KKT conditions? because it will convert the ... 0answers 17 views ### Maximize using KKT conditions. Let T\ge1 be some finite integer. Maximize \sum_{t=1}^{T} (\frac{1}{2})^t \sqrt(x_t) Subject to \sum_{t=1}^{T} x_t \le 1  and  x_t \ge 0, t = 1,...,T. I believe that constraints  x_t \... 1answer 41 views ### Karush-Kuhn-Tucker NLP Consider the nonlinear program Minimize: \begin{align}f(x,y) = \frac{1}{2}x^2 - 10xy + 10y^2\end{align} Subject to: \begin{align}2x +y^2 &\le 5 \implies g_1(x,y)=2x + y^2 -5 \le0 \\ x^2 -... 1answer 68 views ### KKT condition - minimization problem y^2-8 \ln(x+4)\rightarrow min, such that -x^2 -y^2+9 \geq 0, y \geq 0 *I have to find all possible optimal points.* Lagragian function is: L(x,y,γ_1,γ_2) = y^2 - 8\ln(x+4)+γ_1(x^2+y^2-... 1answer 50 views ### 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 ... 1answer 64 views ### applying KKT to non-convex optimization I am trying to find a solution for following \begin{eqnarray} \text{minimize }~~ -\sum_{i=1}^K \frac{1}{a_i+b_i2^{(-2x_i)}} \\ \text{s.t.} ~~~ \sum_{i=1}^Kx_i=C \end{eqnarray} where a_i,b_i,x_i\... 0answers 39 views ### Solving the Lagrange dual problem If we have a convex constraint problem \text{min}\quad f_0(x)\\ f_i(x)\le0\\ h_i(x)=0,  where $f_1,\ldots,f_n$ are convex and $h_1,\ldots,h_r$ are affine. Assuming Slater's conditions we know ...
I have a continuous, twice differentiable function $f:\mathbb{R}^2\rightarrow \mathbb{R}$. My objective is to maximise the function $f$ subject to quasiconcave function \$g_i(x,y)\le 0,\, i\in\{1,2\}\, ...