# 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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### Maxima of $f(x)/e^x$ where $f(x)$ is an approximation of $e^x$ using Stirling's

Let $$f(x)=1+\sum_{n=1}^\infty\frac{x^n}{\sqrt{2\pi n}(n/e)^n}\tag1$$ and let $$g(x)=\frac{f(x)}{e^x}\tag2$$ If we plot $g(x)$ we get a graph that looks like this: Clearly there is a maximum at ...
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### Find min/max $\|x\|_{1}$ subject to $Ax = b$, using the simplex method

Let $Ax = b$ be a linear system with $a_{i,j} \in \{0,1\}$ and $b_i \in \{0,1,2,3,4,5,6,7,8\}$. The constraints on $x$ are $x_i \in \{0,1\}$. We suppose that the system admits at least one solution....
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### Shortest possible distance to locate an unknown road

You are stranded in the middle of a large desert and the only way home is a through a straight road, which unfortunately you do not know the location of. If the perpendicular distance from you to ...
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### Projected gradient descent with momentum

Can we apply momentum to projected gradient descent? If so, how should we do that? In the domain I'm working on, momentum greatly speeds up gradient descent. However, I want to do projected ...
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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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### Generate a class of matrices via optimization

I want to generate a matrix (using Matlab) with the following properties: (1) $A = (a_{ij}) \in \mathbb{R}^{n \times n}$; (2) $a_{ij} \in \{0,1\}$ and $a_{ii} = 0$ for all $i\in\{1,2,\cdots, n\}$; (...
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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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### primal to dual solution conversion ??

i have an optimization problem $$\text{ maximize } z=3x+4y$$ $$\text{ such that: } x+y ≤ 450 \text{ and } 2x+y ≤ 600$$ the optimal solution to this problems comes to be $x=0$; $y=450$; $p=150$ (...
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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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### How to maximize the volume of a cylinder with no top

A cylindrical can without a top is made using $A \text{ cm}^2$ of material. Find the dimensions that will maximize the volume of the can. What I have done was similar to the question: Optimization ...
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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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### related to biconcave optimization

I have a bivariate function $f(x,y)$ both $x,y$ can assume values within closed interval i.e. $x_1\leq x\leq x_2$ and similarly $y_1 \leq y \leq y_2$. I know that for a fix value of $x$ the function ...
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### Is 0-1 integer programming always NP-hard?

I have the following problem. Maximize $\sum\limits_{m=1}^M\sum\limits_{n=1}^N x_{mn}$ subject to: \$\sum\limits_{\substack{m^\prime=1\\ m^\prime \neq m}}^M\sum\limits_{\substack{n^\prime=1\\ n^\...