# 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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### LP problem: Giving variables the same value or 0

If I have the following objective function: $$\min X_1 + X_2 + X_3 + X_4$$ How could I ensure that the variables $X_1, X_2, X_3$ and $X_4$ either have the value of 0 or they could have a random ...
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### Given a number N, how to construct a set of different numbers that has a maximal product, and the sum of these numbers equal N?

Note that: N is positive integer. The set also consists of positive integers. The set consists of different integers. (The thread suggested by @hardmath doesn't have this constraint.) For example: ...
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### How to optimize $\min_w\frac{1}{2}\|w-w_t\|^2 : w^\top(y-\hat y)\ge 1$?

I have some trouble minimizing the following problem: $$\min_w\frac{1}{2}\|w-w_t\|^2 : w^\top(y-\hat y)\ge 1$$ Considering $w$, $w_t$ are vectors, so are $y$ and $\hat y$. The idea is to find a $w$ ...
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### What operation is this in maths

I need to develop an algorithm for a problem then translate it into code, but I am sure someone would had done it before, my question is that what is the name of operation in below sudo code ? ...
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### I'm walking towards my car - when should I try the remote, in an optimal sense?

I'm interested to learn about how discrete/'event' based elements are incorporated into optimisation problems. Hopefully this is an interesting problem in its own regard, it's inspired by a daily ...
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### How to solve this problem through bisection search or any other method?

I have an optimization problem in the form $$\text{Minimize}\hspace{1mm}D$$ $$\text{subject to}$$ $$\sigma_1+\sigma_2=\sigma$$ $$\rho_1+\rho_2=\rho$$ $$\epsilon\le\rho_i\le c_i\hspace{1mm},i=1,2$$...
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### Question on optimization algorithm to train peculiar regression

I've been in my operations research course, and we have been working on optimization in particular problems within regression. We hypothesize that for variables $h,s,d,t,$ there is this set ...
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### optimize pasting text

Someone asked me how can he paste a string 1000 times in Windows notepad. While this can be done easily using editors like Vi, I'm trying to answer his question using notepad only. So the problem goes ...
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### Variation of TSP - Revisit Nodes

I have a problem where I have an symmetric graph and I want to find that shortest path that visits every node at least once (not exactly once). In order to solve this problem, I have found that we ...
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### Maximizing this parametric expression with a certain range of integer inputs

Let $a,b$ be integers with $1 \le b < a \le n$ and $s,t$ be integers with $0 \le s < t \le m$ I would like to maximize the expression: $b^s (a^{t-s} - b^{t-s})$ My intuition says this should ...
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### Minimize linear function with $\ell_1$ norm regularization and positive semidefinite constraint

I am running into the problem like this: $\underset{\mathbf{X}\succ0}{\text{minimize }} vec(\mathbf{A})^{\top}vec(\mathbf{X}) + \lambda ||\mathbf{X}||_1$ I am think about maybe one can minimize a ...
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### The Hardest Sudoku Puzzle

I was playing a casual game of Sudoku today when a friend came by and asked "What's the hardest game of Sudoku possible?" My response: "A Sudoku puzzle with the minimal amount of starting numbers ...
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### Maximum of this parametric expression

Let $a,b$ be integers with $1 \le b < a \le n$ and $s,t$ be integers with $0 \le s < t \le m$ I would like to maximize the following expression: $b^s~(a^{t-s}-b^{t-s})$ My intuition says this ...
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### Alternative solution to a Lagrange Method Optimization Problem

Find extrema of $f(x,y,z)=\frac{1}{x^2}+\frac{1}{y^2}+\frac{1}{z^2}$ subject to $\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=1$ by reducing variables and then using the Single Variable Method or by using ...
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### Set of marginals is convex [closed]

Let $[n] = \{1,2,\cdot,n\}$ and $[m] = \{1,2,\cdot,m\}$. Let $Z_{1,2}$ denote the set of all probability distribution on the Cartesian product $[m]\times [n]$. Let $S_{1,2}$ denote a convex closed ...
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### How to minimize objective function involving convolution?

My objective function is \begin{align} \underset{\mathbf{p},\mathbf{q}}{\text{min}}\hspace{4mm} (\mathbf{p*q})^T \mathbf{A}(\mathbf{p*q}) \hspace{4mm} \\ s.t \hspace{4mm}\mathbf{p^Te_p}-1=0\\\mathbf{...
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### Easier way of finding out whether a given linear programming problem has optimal solution or not

I have the linear program $$\begin{array}{ll} \text{minimize} & -2x-5y\\ \text{subject to} & 3x + 4y \geq 5\\ & x, y \geq 0\end{array}$$ I can solve it using Simplex algorithm, but I ...
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### Weight Modification for Computationally-Efficient Nonlinear Least Squares Optimization

There was a time where I could figure this out for myself, but my math skills are rustier than I thought, so I have to humbly beg for help. Thank you in advance. I am solving a weighted nonlinear ...
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### Comparing the task complexity of installing three different offenses for American style football in three days

I want to identify the inherent difficulty of installing three separate American rules football offenses by their complexity of practice schedules in three days then relate those offenses back to one ...
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### Exponent to maximize the expression $log_b \left(a\frac{b-1}{b^k-1}\right)$

Given $a, b \in \mathbb N$, how to maximize the expression $$log_b \left(a\frac{b-1}{b^k-1}\right) \in \mathbb N$$ Put differently, what is the minimum $k \in \mathbb N$ verifying  a\frac{b-1}{...
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### How did the answer key get $h=40-2r$?

A cone has radius of $20\ \rm cm$ and a height of $40\ \rm cm$. A cylinder fits inside the cone, as shown below. What must the radius of the cylinder be to give the cylinder the maximum ...
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### Engineering/mathmatics question

I have an equation $M(x)= -15.328x^2+176.44x-352.88$ (a parabola) and also $V(x) = -30.657x + 176.44$. I want to know how to find $x$ where the values of $M$ and $V$ combined are the lowest, I'm ...
Find the absolute minimum and maximum values of $f (x, y) = xy e^{−2x^2 −2y^2}$ on the set $\Delta =$ {$(x,y)\in\mathbb{R^2} | x^2+y^2\le1$} i know i should take the partial derivatives and set ...