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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Absolute value of eigenvalues

It is well known that if $A$ is a real symmetric square matrix of size $n$ with eigenvalues $\lambda_1 \leq \cdots \leq \lambda_n$, then $$ \lambda_k = \min \{ \max \{ \langle Ax, x \rangle : \|x\| = ...
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Maximize sum of logs subject to constraints

I have the optimization problem $$\begin{array}{ll} \text{maximize} & \displaystyle\sum_{i=1}^n \log(c_i + x_i)\\ \text{subject to} & \displaystyle\sum_{i=1}^n x_i = 1\\ & x_i\ge0\end{...
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normalization of constraints $ 0 \leq x \leq 1 $ in Lagrangian KKT

With Lagrangian we have an objective function and a set of equality constraints of form $ g_{i}(x_{j}) = 0 $ . With KKT we can have another set of inequality constraints of the form $ h_{i}(x_{j}) \...
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Directional derivative and lagrange multipliers

Find the points $(x,y)\in \mathbb R^2$ and unit vectors $\vec u$ such that the directional derivative of $f(x,y)=3x^2+y$ has the maximum value if $(x,y)$ is in the circle $x^2+y^2=1$ My attempt: ...
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discrete convexity arising in a simple discrete optimization problem

Let $S$ be a fixed integer satisfying $S \ge 1$, let $a$ range over the integers between $1$ and $S$ inclusive, and for $i = 1, \dotsc, a$, let each $x_i$ range over the nonnegative integers, such ...
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Index of a stationary point of constrained optimization

For an unconstrained optimization problem with objective function $F(x,y,z)$ the index of a stationary point is well-defined: If $(x^*, y^*, z^*)$ is a point where the gradient of $F(x,y,z)$ vanishes, ...
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Minimize $-\sum\limits_{i=1}^n \ln(\alpha_i +x_i)$

While solving PhD entrance exams I have faced the following problem: Minimize the function $f(x)=- \sum_{i=1}^n \ln(\alpha_i +x_i)$ for fixed $\alpha_i >0$ under the conditions: $\sum_{i=1}^n ...
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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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Using the Lagrange method to find max/min of $f(x,y) = \frac{x^3}3 + y$

Problem Use the Lagrange method to find max/min of $f(x,y) = \frac{x^3}3 + y$ Subject to the constraint $x^2 + y^2 = 1$ My attempt The constraint gives us $g(x,y) = x^2 + y^2 - 1$ $\displaystyle\...
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finding curve along which a function extremizes via theory of calculus of variations [closed]

Consider $$ I(y)= \int \limits _0 ^1 [y'(x)]^2dx \ +y(1)^2$$ with $y$ subsjected to the initial condition $y(0)=1$. Find the equation of curve along which $y$ extremizes.
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Which way should you run from the lions?

This is a fun problem that I saw somewhere on the internet a long time ago: Suppose you are at the center of an equilateral triangle with side length $s$. At each of its vertices, there is a lion ...
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Find the maximum and minimum values of $\sin^2\theta+\sin^2\phi$ when $\theta+\phi=\alpha$

Find the maximum and minimum values of $\sin^2\theta+\sin^2\phi$ when $\theta+\phi=\alpha$(a constant). $\theta+\phi=\alpha\implies\phi=\alpha-\theta$ $\sin^2\theta+\sin^2\phi=\sin^2\theta+\sin^2(\...
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33 views

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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LP: add extra costs in the objective function for every variable which is not equal to $0$

I am trying to optimise an LP problem but extra costs should be added if a variable is larger than $0$. For example, if we have the following objective function: $$\text{minimize} \qquad 2X_1 + 3X_2 ...
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If $f(x)=\frac{1}{\pi}\left(\arcsin x+\arccos x+\arctan x\right)+\frac{x+1}{x^2+2x+10}\;,$ Then $\max$ value of $f(x)$

If $\displaystyle f(x)=\frac{1}{\pi}\left(\arcsin x+\arccos x+\arctan x\right)+\frac{x+1}{x^2+2x+10}\;,$ Then $\max$ value of $f(x)$ $\bf{My\; Try::}$ Here Domain of $\arcsin x\;,\arccos x$ is $\...
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Kullback-Leibner divergence true distribution

I have an image with an object which I treat as 2-dimensional Gaussian random vector with mean equal to the center of the object surrounded by, roughly, 3-sigma ellipsoid. On the other hand I feed the ...
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Geometric median (or Fertmat-Webber problem), including continuous case

For a finite set $X\subset \mathbb R^n$ the geometric median is defined as the point in $\mathbb R^n$ for which the sum of distances to all points of $X$ attains its minimum. Here is a wiki article: ...
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Intuitive understanding of Maximin Principle

From the the book page $324$, does someone could explain to me the Theorem $2$. Maximin principle? I have a bit of difficulties to well understand how works this theorem. A simple example would be ...
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Properties of unit vector scaling

What properties are kept when we scale a vector to unit length, i.e. $\frac{\mathbf{v}}{||\mathbf{v}||_1}$? Imagine that we have an unconstrained optimization problem, and we obtain as solution $x_i ...
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What is the maximum of $x^3y^3 + x^3z^3 + y^3z^3$ subject to $x+y+z=1$?

All variables are positive reals. This is a math competition problem. I've tried solving it using boundary value optimization, but it's not elegant at all. Thanks for any ideas.
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Find the absolute max and min of a multivariable function on a bounded by a circle?

So i do understand everything up the square rectangle, in the photo here i mean, how did he come up with $(±2,0), (0,±1)$ is it because of $g(2cos x, sin x)$ and if that is the case why would he ...
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Shortest distance as measured in norm $||\cdot ||$ from point to a sphere in norm $||*||$

I recently found this theorem, which is used in some clustering algorithms: Let $x,v \in \mathbb{R}^p$, $r>0$, $||\cdot ||_{\ast}$ be a given norm on $\mathbb{R}^p$ and $\partial B_{||\cdot||_{\...
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Minimizing an integral — Hilbert space

Find the real values of $a, b$ which minimize $$\int_1^{\infty} \left| \frac{1}{x^2} - a \frac{1}{x^3} - b\frac{1}{x^4}\right|^2 \; dx.$$ Hint : Work in an appropriate Hilbert space. Here is why I ...
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How to solve this optimization question with the Extreme Value Theorem?

Consider the region in the x-y plane that is bounded by the x-axis and the function $f(x)=b-ax^2$. Construct a rectangle whose base lies on the x-axis and is centered at the origin, and whose sides ...
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How can I show the corresponding dual solution is unique when the given primal solution is nondegenerate, basic feasible?

the given problem is to show that if $x_1,...,x_n$ is a nondegenerate basic feasible solution of the primal LP max $\sum_{j=1}^{n}c_jx_j$ s.t. $\sum_{j=1}^na_{ij}x_j\leq b_i, \forall i\in\{1,...,...
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Rectangular prism optimization using extreme values

A box with a rectangular base, whose length is twice its width, is to have a closed top. The area of the material in the box is to be $192in^2$. What should the dimensions of the box be in order to ...
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Binary optimization problem

I am facing the following problem: Let P be a fixed m x n finite matrix and D be a matrix of ones and zeros with the same dimensionality as P plus the following constraints: sum of row entries <=...
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Nonlinear optimization with constraints; is changing variables an reliable approach?

I have a optimization problem as follows, $$ \begin{array}{cll} [\hat{x_1},\hat{x_2},\hat{x_3}] = & \text{argmin}_{x_1,x_2,x_3} \sum_{i = 1}^N \sum_{t = 1}^T \left[ \ln(f_{i,t}(x_1,x_2,x_3)) + \...
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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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60 views

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 ...
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Find the absolute max and min values of a multivariable function bounded by a circular boundary

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 ...
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Can Extragradient method be expressed with proximal steps?

As we know, for solving saddle point problems, forward-backward algorithm is generally not guaranteed to converge. But extragradient method converge Structured Prediction via the Extragradient Method ...