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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Optimization of sum of logs

I have an optimization problem of the form $$\operatorname*{argmax}_{\mathbf{w}} \sum_i \log(1 + \mathbf{w} \cdot \mathbf{k_i})$$ given some set of vectors, $\mathbf{ \{k_i\} }$. I have tried both ...
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Monotonicity of $f(x)= \frac{ e^{\alpha x}-1}{\alpha x}$ where $\alpha \geq1 $ and $x >0$.

There is this function I encountered when I was solving a problem and I am trying to study its monotonicity. So the function is $f(x)= \frac{ e^{\alpha x}-1}{\alpha x}$ where $\alpha \geq1 $ and $x &...
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243 views

Dealing with free variables in Linear Programming

I have a free variable in my formulation. In the objective function, this free variable has a cost, and another cost coefficient which is only incurred when the free variable is negative. I used the ...
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173 views

proving that the shortest line conntecting a point and a line will be perpendicular to that line

So I have a problem for my final math project that I've been fiddling with for hours without success. I have to use calculus to prove that the shortest line connecting a point to a line will always be ...
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31 views

How to solve this kind of Lagrangian function?

Suppose $\mathbf{a} = (a_{0}, \dots, a_{N-1})$ and $\mathbf{b} = (b_{0}, \dots, b_{N-1})$ with $a_{i}\geq0$, $b_{i}\geq 0$. I would like to minimize $$-\sum_{i=0}^{N-1}a_{i}b_{i}$$ subject to $$\...
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30 views

Find a critical point satisfied the Lagrange condition is not local extremum

We know that Lagrange Multiplier gives necessary conditions for an extremum.It locates all possible condidates.But not all such points need be extrma. I want to find an example of the point is ...
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97 views

Line of Best Fit Optimization Problem (Stewart's Early Transcendentals, 14.7, #55)

I know posting pictures is kind of frowned upon here, but I didn't want to type the whole problem out, diagram and all. I'm feeling pretty lost on this one. We've been learning about absolute minima ...
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Problem on EU commission

Consider the following problem. A collection of $n$ countries $C_1, \dots, C_n$ sit on an EU commission. Each country $C_i$ is assigned a voting weight $c_i$. A resolution passes if it has the ...
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39 views

How can I mathematically model the combinatory problem?

I have the following problem, and I would like to model it using a mathematical formula, for a purpose of optimization problem: let's say that I have two tasks $[T_1, T_2]$, and $3$ resources $[R_1,...
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41 views

Find local and global extrema for $f(x,y) = y^4 -3xy^2 +x^3$

above you find a function and some questions I have to answer. I'll give you a more or less detailed input of what I did. I'll be glad if you could help me with the questions I inserted with "->". ...
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48 views

Simple Lagrange Multiplyers Problem

Can anyone please help me with the following: Find the stationary values of $u=x^2+y^2$ subject to the constraint $t(x,y) = 4x^2 + 5xy + 3y^2 = 9$. The answer is given as $u = 9$ and $x = \pm 3/\...
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number of maximizers

Suppose f(x) is continuous in x∈X, where X is compact. Let T(x):=argmaxf(x) be the set of maximum of f(x), where the maxf(x) is bounded. Then under what condition the set T(x) cannot contain ...
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Homework help on eigenvalues function minimization

so I actually have two separate questions which are homework bonuses for my numerical methods course. Unfortunately, because of the time of the semester, our TAs are not available so I don't have many ...
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26 views

The functional take its maximal value for $y(t)=-t$

I want to show that the functional $J(y)=\int_0^1 [y'(t) \sin{(\pi y(t))-(t+y(t))^2}]dt$ ,where $y$ is a continuously differentiable function on $[0,1]$, takes its maximal value $\frac{2}{\pi}$ for ...
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84 views

Bivariate probability distribution(s) over unit square, uniform marginals, midpoint is saddlepoint

Construct a bivariate probability distribution--or family of such distributions--over the unit square (corners $(0,0), (0,1), (1,1), (1,0)$) with uniform marginals and having a saddlepoint at $(1/2,1/...
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How to explain lagrange multipliers to a lay audience?

So I will be giving a seminar to a scientifically mature lay audience (think bio/social science undergrad level). I have been told that I should count on less than half the audience to have experience ...
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485 views

How to maximize (baking) surface area?

I like eating crust, so I am trying different baking molds to try to get the most crust per dough. More generally, I'm interested in the reverse of this more specific question — how to maximize the ...
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384 views

Traveling salesman problem: why visit each city only once?

According to wikipedia, the Traveling Salesman Problem (TSP) is: Given a list of cities and the distances between each pair of cities, what is the shortest possible route that visits each city ...
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Smallest bound for convex combination of columns of non-negative matrix

The problem can be formulated as following linear program: $\min_{\mathbf{x},y}\;\;y$ subject to: $\mathbf{Ax}\le y\mathbf{1}$ $\mathbf{x}^T\mathbf{1}=1$ and $x_i \ge 0,\;\forall i$ Here, ...
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Find $\min_{y \in \mathcal{A}} J(y)$, if it exists.

Let $\mathcal{A}$ be the set of continuously differentiable functions at the interval $[a,b]$. Let $J$ be the functional $$J(y)=\int_a^b \sqrt{1+y'(x)^2}dx$$ Find $\min_{y \in \mathcal{A}} J(y)$, if ...
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How to find parameters from logistic equation

I have an function and assume that that is convex function. I want to use gradient decent to find parameters in that equation. Could you suggest to me the way to do it. Thanks. This is my function $$J=...
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34 views

Maximization of a function I came up while studying

So in a problem I am trying to solve, after calculations I came up with the following function: \begin{equation*} f(\overline{y},\theta)=\frac{e^{n\,min\{\overline{y},\theta)}-1}{n\theta} \end{...
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69 views

Wouldn't this Greedy Algorithm achieve the highest possible of money in this situation?

I am doing a practice question from Midterm Dynamic Programming The Problem : Consider a row of n numbers a1, ..., an. The numbers are all positive, and n is even. We play a game against an ...
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45 views

A maximization problem within the simplex

Let $\lambda_i$ be an ordered list of $N$ positive numbers, $\lambda_1<\lambda_2<\dots<\lambda_N$. I'm looking for the extrema of the function $$ f=\left(\sum_{i=1}^N p_i \lambda_i \right)\...
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Generalization of minimisation problem

First I would like indtroduce my problem ! There is an easy way to solve this one : Find the value of $$ \inf_{(a,b)\in \mathbb{R}^2} \int_0^1 (t^2-at-b)^2 dt $$ and precise for which values $a$ ...
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Converting a max-min problem to a max problem with a constraint

The objective is to find the greatest lower bound of the variable $\mu$. The lower bound is resulting from the positive-semidefinite (PSD) constraint $$\tilde{\mathbf{T}}:=\left( \begin{array}{ccc} ...
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Optimization of $e^{x^2 + y}$ on $x+y \leq 2$

Let $f(x,y) = e^{x^2 + y}$ and $M = {(x,y): x+y \leq 2}$. A. $f(x,y)$ on M is bounded above and not bounded below B. $f(x,y)$ on M achieves global minimum(a). C. (0,0) is point of local ...
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How can I solve the following exercise

Find the critical curves for the following functional : $$J[y(x),z(x)]=\int_{0}^{1}(y'^2+z'^2-xyz'-yz)dx$$ With the conditions : $$K[y(x),z(x)]=\int_{0}^{1}(y'^2-xy'-z'^2)dx=2$$ $$y(1)=z(1)=1$$ $$y(0)=...
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How does the value of a functional change when you perturb the extremizing function?

In deriving the Euler equation for etremizing a functional \begin{equation*} J[y] = \int_a^b F(x,y,y')\,dx, \end{equation*} we look at: \begin{equation*} J[y+h]-J[y] = \int_a^b(F_yh+F_{y'}h')\,dx + ...
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198 views

Which shape optimize the perimeter for a given area?

I was wondering what is shape that maximize the perimeter for a given area? In fact I would like to know what would be the most optimized perimeter that I can include in a rectangle. See image below....
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Binary depending on the sign of another variable

I'm writing a mixed integer linear problem, where I have an indicator function in the objective function counting the instances of negative values of a decision variable. I thought of defining a ...
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87 views

How to determine the optimal step size in a quadratic function optimization

I have the following optimization problem: $$\underset{\alpha\in\mathbb{R}}{\text{min}}:\;\;f(\textbf{x}+\alpha\textbf{d})$$ $$\text{subject to}:\;\;0\leq\alpha\leq \alpha_{max},$$ where $f(\...
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56 views

Quadratic programming with constrained number of free variables

I started with a (positive-definite) quadratic programming problem subject only to a single equality constraint. i.e. $$ f(x)=x^{T}Qx+c^{T}x $$ $$ s.t. x_1+x_2+x_3+...+x_n=y $$ I now have to find ...
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31 views

How to do optimization

My teacher gave me a very complicated explanation on how to solve an optimization problem so I just wanted clarification. To do so I have laid out what I think is the simplest way to solve it. Take ...
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382 views

Solving an optimization problem with KKT-conditions

I've been studying about KKT-conditions and now I would like to test them in a generated example. My task is to solve the following problem: $$\text{minimize}:\;\;f(x,y)=z=x^2+y^2$$ $$\text{...
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38 views

Find the edges of a polyhedron P.

Given the polyhedron $P = \{v \in \mathbb R^2 \mid Av \le b\}$ with $A = \begin{bmatrix} -1 & -1 \\ 2 & -1 \\ -1 & 2 \\ 1 & 2 \end{bmatrix}$ and $b = \begin{bmatrix} 0 \\ 1 \\ 1 \\ 2 ...
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Trace minimization when some matrix is unknown

The problem is as follows: $\displaystyle\min_{V}$ trace($V^TH^T\Phi HV$)$\\$ s.t. $V^TV=I_d$ in the case when $H$ is not known. When $H$ is known, the solution is given by the eigenvectors ...
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Minimizing Norm

I have below problem: Find $\bf C$ to minimize $\|\mathbf A-\mathbf B\mathbf C\|_F$. Given ${\bf B} \in \mathbb R^{m \times n}$, ${\bf B}$ has lin. ind. col. A satisfies: ${\bf DA} = {\bf E}$ , ${\...
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Maximum of norm

Given a matrix $A$ with $N$ rows and $d$ columns, I would like to prove (or disprove) the following. Let $q(f)=\|(\begin{pmatrix} f_1&0&0&\cdots&0\\0&f_2&0&\cdots&0\\0&...
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36 views

$v^TAv\ge \left\|A\right\|\cdot\left\|v\right\|_2$ for all positive definite $A\in\mathbb{R}^{n\times n}$

Let $A\in\mathbb{R}^{n\times n}$ be positive definite and $v\in\mathbb{R}^n$. Let $\left\|\cdot\right\|_2$ be the Euclidean norm. Can we prove $$v^TAv\ge \left\|A\right\|\cdot\left\|v\right\|_2$$ for ...
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77 views

Definition functions, integrals on $\mathbb R^{|N|}, \mathbb R^{\mathbb R}$

Is there a standard/reasonable way of defining functions on the sets $\mathbb R^{|\mathbb N|}, \mathbb R^{\mathbb R} $. How about defining integrals over these sets? I guess a function on $\mathbb R^{|...
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92 views

How to simplify the summation of log

I have a summation that involve log. I don't know how to solve this summation. I want to find an expression (even a good approximation is enough) for this summation. $\sum_{k=0}^{n}{log(a_k)}$ ...
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An easy question about NP-hard

Consider an optimization problem includes two variables. If we fix the value of one variable, then the optimization problem over the other variable is NP-hard. Can it be concluded that the original ...
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Effect on Minimizer of Tightening Constraints

The Statement of the Problem: Consider the minimization problem $f(x,y)=14x+20y$ under the constraints $x+2y \ge 4 $, $7x+6y \ge 20$, and $x,y \ge 0$. Don't use the simplex method! (i) Draw the ...
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492 views

Minimal circle containing set of points

Suppose that there are $n$ points in the plane $x_1, x_2, \dots x_n$, and $C$ is the minimal circle (the circle with the minimal radius) that contains all of them. If there is another point $p$ ...
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23 views

Grouping constrained optimization

I am looking for an efficient solution to solve the following problem. Can anybody help? S is a finite set of elements $k_i$ V is a subset of S, e.g. $v_4$={$k_1$,$k_3$} E is a finite ensemble of V,...
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Shortest smooth paper Möbius Strip

I want to make a familiar Möbius strip of width 1 unit satisfying the physical properties of paper. Assume paper is a ruled surface, and the strip has to be smooth and non-self-intersecting. What is ...
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28 views

Derivatives - optimization (minimum of a function)

For which points of $x^2 + y^2 = 25$ the sum of the distances to $(2, 0)$ and $(-2, 0)$ is minimum? Initially, I did $d = \sqrt{(x-2)^2 + y^2} + \sqrt{(x+2)^2 + y^2}$, and, by replacing $y^2 = 25 - x^...
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341 views

Let $x$ and $y$ be real numbers such that $x^2 + y^2 = 1$. Find the maximum value of $2x - 5y$.

Let $x$ and $y$ be real numbers such that $x^2 + y^2 = 1$. Find the maximum value of $2x - 5y$. I do know how to solve this problem using trigonometry, however I need to solve it by using vectors. ...
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optimization for the area of a garden

I have been working this problem for awhile and cannot seem to solve it even though its probably easier than I think... There is a rectangular garden that needs fencing. For one side of the fencing ...