# 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.

245 questions
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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 ...
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### Can KKT be used in minimization s.t to constant param

Can KKT be used : min g(x) s.t x>=constant where constant > 0 I have read this The Kuhn-Tucker method: here says that This is an alternative, and ...
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### Local optimality of non-convex quadratic minimization with linear constraints

We are interested in minimizing a quadratic function, which is not convex. The feasible set is a polyhedron. We know that the global minimum is at a vertex when the function is concave. Also, there ...
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### How to solve this Non linear optimization problem?

I need to minimize the below-mentioned expression. $L = min (a_0-b_0*(p_1+p_2))^2 + (p_1*y1+p_2*y)$ ,with s.t p_1 >=0 ,p_2 >=0 Here ...
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### Explain what happens if you use KKT to solve this problem

Explain what happens if you use KKT to solve this problem: \begin{equation*} \begin{aligned} & \underset{(x,y)\epsilon \mathbb{R}^2}{\text{minimize}} & & x+y \\ & \text{subject to} &...
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### Strong Duality in Hilbert Spaces for convex problems?

Setting I'm looking for a proof of the following fact, if $f,g$ are lower-semi-continuous and convex function from a Hilbert space $\mathscr{H}$ to $\mathbb{R}$, with $f$ strictly convex continuous. ...
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### Silly Quadratic programming

Suppose that I want to minimize the function $x^2$ subject to the contraint $$ax\leq b,$$ for some $b> 0$. I solved the problem if the contraint is an equality but I'm not sure how to go about ...