# 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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### Interpretation of this Lagrange Multiplier

I have the following utility maximization problem with inequality constraints: Objective function given by $U(x_1,x_2)=\ln(x_1)+\beta \ln(x_2)$ where $0<\beta<1$, and the constraints are given ...
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### KKT condition for the proximal algorithm

This slide shows that the KKT condition for the proximal gradient descent is this inequality. I don't know where this comes from. Using KKT , we can only get equality for the stationary condition, ...
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### Dual of linear function with convex and non-convex constraints

I would like to compute the dual of the following problem by using the KKT conditions. However, due to form of the first constraint I am not able to obtain the dual. The problem is the following \...
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### Optimizing a function under strictly positive constraint

Find x and y that optimise \begin{align} f(x,y) &= (-a-y)(\Psi(y)-\Psi(x+y)) + (b-x)(\Psi(x)-\Psi(x+y)) \\ &-\log \Gamma(x+y) + \log\Gamma(x) + \log\Gamma(y) \end{align} where a, b are ...
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### How to handle optimization problems when optimization variable is matrix?

Suppose we have the following optimization problem $$\min_{0\preceq M \preceq I} y^TMy$$ where $y \in \mathbb{R}^n$ and $M \in \mathbb{R}^{n \times n}$ is a positive semi-definite matrix. Notice ...
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### Minimize $\left\|BA\right\|_2$ under these constrain

Minimize $\left\|BA\right\|_2$ while B is a given $m*n$ matrix with rank n and A is an $n*t$ matrix which is not given. Such that $B'*u_{1}*v_{1}' = a*u*v$; $\left\|A\right\|_2 = b$. While ...
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### Lagrange duality compared with Lagrange multiplier method

As we all know, Lagrange multiplier method says: in order to find the extremum of $f(x)$ over $x$, s.t. $g(x)=0$, one instead finds the extremum of $f(x)+\lambda g(x)$ over $x$ and $\lambda$. Note ...
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### Find KKT point of $\min_{x \in \mathbb{R}^4} x^Tx$ subject to $x^TAx \geq 1$.

onsider the following problem: $$\min_{x \in \mathbb{R}^4} x^Tx$$ over $C=\{x \in \mathbb{R}^4 \mid x^TAx \geq 1\}$ where $A \in \mathbb{R}^{4 \times 4}$ is a symmetric matrix with two distinct ...
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### When is LICQ useful in KKT conditions?

KKT establishes a set of criteria for differentiable optimisation problems related to strong duality (i.e. when primal optimal equals dual optimal). In particular, KKT conditions are necessary for ...
In Boyd's Convex Optimisation, the following optimisation problem is considered  \begin{align} \min\quad & f_0(x)\\ \text{s.t.}\quad & f_i(x)\le 0,\quad i=1:m,\quad m\in\Bbb Z_{\ge 0}\\ &...