# Tagged Questions

Optimization is the process of choosing the "best" value among possible values. They are often formulated as questions on the minimization/maximization of functions, with or without constraints.

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### Find point on a line that is nearest to the origin

Can you help me with this exercise? Find the nearest point to the origin $(0,0,0)$ in the line given by the intersection of planes $x+y+z=2$ and $12x+3y+3z=12$. The intersection of the planes is ...
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### Upper bound of $x_1x_2x_3+x_2x_3x_4+\dots+x_{n}x_1x_2$

Let $n\geq 3$ be a positive integer and let $x_i$'s be non-negative real numbers with $x_1+x_2+\dots+x_n=1$. What is the maximum of $x_1x_2x_3+x_2x_3x_4+\dots+x_{n}x_1x_2$? If the sum were symmetric ...
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### Relationship between Newton's method in root finding and optimization

In both root finding and optimization, there are Newton's method. Wikipedia has 2 links here and here. Root finding is using first order derivative and optimization is using Hessian. What's the ...
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### Find maximum value of a function [closed]

$a$, $b$, and $c$ are real numbers, and $a+b+c=0$ and $a^2+b^2+c^2=2$. I need help finding the maximum value of: $$\big|a^2b^2(a-b)+b^2c^2(b-c)+c^2a^2(c-a)\big|$$ To be honest, I don't know where ...
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### For which point of $x+y+3z+k=10$, the expression $x^2+y^2+9z^2+4k^2$ is minimal? [closed]

For which $x,y,z,k \in \mathbb R$ of $x+y+3z+k=10$ is the expression $x^2+y^2+9z^2+4k^2$ minimal?
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### how to minimize $tr (X^T W - Y^T W_p)(X^T W -Y^TW_p)^T$ in closed form

Assume we are dealing with matrices. Then how to minimize $$E(W,W_p) = tr (X^T W - Y^T W_p)(X^T W -Y^TW_p)^T$$ w.r.t both $W, W_p$ simultaneously? I can calculate the derivatives of $W$ and $W_p$ ...
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### Scheduling grid optimization

I am trying to optimize the programming of multiple TV channels for a given week. For each show (a day, a time and a TV show) it is possible to forecast in advance the number of people that will watch ...
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### Working with Lagrange multipliers, reducing gradients is okay, right?

I am employing the method of Lagrange multipliers to determine a maximum. As part of this, I arrive at the following equation involving two gradients and the parameter $\lambda$, as is common for ...
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### Finding dual of certain problem

Could anyone help me finding lagrangian function and lagrangian dual of the following problem: \begin{split} \max_{X}\quad & \operatorname{trace}(H X H^T)\\ \text{s.t} \quad &...
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### Economics maximization problem linear activity

Consider the vectors: $a_1 = \begin{pmatrix} 0 \\ -1 \\ 1 \\0 \end{pmatrix}, a_2 = \begin{pmatrix} 0 \\ 0 \\ -1 \\1 \end{pmatrix}, a_3 = \begin{pmatrix} 2 \\ 0 \\ 0 \\ 1\end{pmatrix}$ Find a single ...
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### Convex set equals convex functions within optimization?

Can optimizing a convex function subject to convex constraints be written as optimizing the function subject to a convex set? Does the intersection of convex nonlinear ineualities necessarily describe ...
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### What is the easiest way to optimize the weighted sum of L2 norms?

I have the following cost function (solving for $M$ - the $x_i$s are known): minimize $\sum_i\sum_j(w_{ij} \cdot (x_i-x_j)^T\cdot M\cdot(x_i-x_j))$ ($w_{ij} \in [-1,1]$) subject to: $M \succeq 0$ (...
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### Optimization with box constraints - via nonlinear function

I have the following convex optimization problem over $\mathbb{R}^n$ with box constraints: \begin{align}\text{minimize }&\;f(x)\\ \text{subject to }&\;x \in [-1,1]^n\end{align} I can see ...
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### how to find closely related values from a set?

I have a set of values, for eg. {20, 1, 1, 21, 8, 22, 11, 40, 5, 21} and will need to find n closely related values. If n is 4 in the given example, the result should be {20, 21, 21, 22} because these ...
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### Is there a way to measure how (non)convex a function is, maybe analogous to condition number?

Consider the functions $f(x) = \sin x$ and $g(x) = (x+1)^2 (x-1)^2$. We know that $f$ has an infinite number of local minimizers and is nonconvex on a non-compact subset of $R$. We know that $g$ has ...