# 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.

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Any idea how to solve this using MATLAB? I know about quadprog. However, because of the presence of $x^*$ and $x_0$, I am not able to define single coordinate transformation that will include both the ...
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### Maximum volume with given surface area [closed]

How to prove that a given closed (smooth continuous two dimensional) surface area encloses a maximum volume for the sphere?
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I read an paper on Quadratic Programming: Paul A. Jensen, Jonathan F. Bard: Operations Research Models and Methods Nonlinear Programming Methods.S2 Quadratic Programming Available here: https://www.me....
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### Proper explanation of KKT equations and Basic constraint qualification works?

I am learning Non-Linear optimisation. Last time I read about KKT (Prof notes and some googling) and Basic constraint qualification and somehow tried to convince myself this and that but now when I am ...
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### Maximize $x^2 + y^2$ subject to $1 \leq 2 x^2 + 4 y^2 \leq 4$

$$\begin{array}{ll} \text{maximize} & x^2 + y^2\\ \text{subject to} & 1 \leq 2 x^2 + 4 y^2 \leq 4\end{array}$$ I don't know how to apply the KKT conditions here, maybe there is other method ...
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### Confusion in understanding a simple convex optimization problem's solution

I am learning optimization through a course on Youtube. I have one confusion in solving the following problem. As per my understanding, the objective function is not convex. The problem is a ...
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### NDCQ Explanation

Can someone explain the non degenerate constraint qualification intuitively? I don’t understand why it needs to be full rank or match number of variables in objective function
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### Constraint Qualification and Non Linear Optimization

Why is the objective function a linear combination of gradient of constraints at optimal point for non linear inequality optimizarion under KKT? Intuitive answers please as only in process of masters ...
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### KKT and Constraint Qualification

Have studied lagrangian and optimization primarily via Khan academy, as got bogged down with Simon and Blume Mathematics for Economists/Chiang and Wainwright Fundamental Methods of Mathematical ...
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### Frizt John and KKT with optimization problem

I am trying to find all candidates for a local minimizer in $min\{ -x_1| x_2-(1-x_1)^3\leq 0 , -x_2\leq 0\}$. Denote $g_1(x)= x_2-(1-x_1)^3$, $g_2(x)= -x_2$, and $I(\bar x)=\{ i|g_i(\bar x)=0\}$ I ...
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### Minimize function subject to inequality constraints using KKT conditions

Task: $$f(x)=x_1^2-2x_2+x_3^2\to \min$$ $$x_1+x_3=1$$ $$2x_1+x_2-x_3\le 2$$ $$x_1\ge0$$ Wolfram Mathematica result: $x_1=0, x_2=3, x_3=1, F(x)=-5$. GNU Octave result: $x_1=0, x_2=3, x_3=1, F(x)=-5$. ...
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### KKT-Conditions in a functional setting

Let $F:L^2([0,1])\rightarrow \mathbb{R}$ be a convex functional. Consider the minimization problem \begin{align} \underset{f(\cdot) \in L^2([0,1])}{\min} F(f)\,\,\text{ subject to } \|f(\cdot)...
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### KKT multipliers and “active” and “inactive” constraints on the generalized Lagrangian $L$

The textbook Deep Learning by Goodfellow, Bengio, and Courville, says the following in a section on constrained optimization: The inequality constraints are particularly interesting. We say that a ...
$\renewcommand{\vec}{\mathbf}$ Consider a non-linear minimization problem $\mathcal{P}$ and suppose to have found a point $\vec{x}$ satisfying the first order KKT conditions. Suppose further that ...