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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Connected graph where edge costs depend on a parameter $t$. Find the $t^*$ which gives the minimum cost minimum spanning tree.

The set-up: Let $G=(\,V,\,E\,)$ be a connected graph. Associated with every edge $e\in E$ is a cost/weight function $f_e(t) = a_e t^2 + b_e t + c_e $, where $a_e>0$. For a fixed $t$ we can define ...
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1answer
27 views

Need help with the graph of a function

In the optimization problem max: $$6x+2xy-2x^2-2y^2$$ subject to $x+2y\le2$ and $-x+y^2\le1$ I need to draw the graph of the feasible region in order to determine if the problem has global solutions, ...
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39 views

Local minimum and gradient [duplicate]

But the proof here below is specially elegant. Is there any function $f$ such that $f$ has a local minimum at $x$ but $\nabla f(x) \neq 0$? Only assumption on $f$ is that it has to be differentiable ...
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18 views

Sum of squares of series of boolean variables

I am going to simplify the following series: $$\sum^4_{v=1} \left(1 - \sum^4_{i=1} x_{v,i}\right)^2 + \sum^4_{i=1} \left(1 - \sum^4_{v=1} x_{v,i}\right)^2$$ Since $x_{i,j}$ is a boolean variable, ...
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23 views

How do I know if I have imaginary numbers when using Newton Raphson Method?

I am studying Newton-Raphson Method but I am facing questions in my head. As far as I know Newton Raphson Method works on real values, but what if Newton Raphson Method faces an imaginary number when ...
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10 views

Imaginary roots and Real values when using Newton-Raphson Values

I am studying Newton-Raphson Method but I am facing questions in my head. How do I know if I have an imaginary number or imaginary numbers? and What to do when I have them when using Newton Raphson ...
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16 views

Shortest path problem: dual formulation and proof of total unimodularity

The IP formulation of the shortest path problem looks as follows: \begin{align*} \min & \sum_{u,v \in A} c_{uv} x_{uv}\\ \text{s.t } & \sum_{v \in V^{+}(s)} x_{sv} - \sum_{v \in ...
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1answer
19 views

Sums of positive and negative distances to the least squares plane

Let $A_{1}, A_{2}, \ldots, A_{n}$ be points in $\mathbb{R}^{3}$ and $\pi_{*}$ be the least squares plane, i. e. $$ \sum \limits_{i = 1}^{n}\rho^{2}(A_{i}, \pi_{*}) = \min_{\pi}\sum \limits_{i = ...
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26 views

Optimize $x^2 + y^2 +2z^2 +z(x^2-y^2)$ subject to $x+y=2$

$$x^{2}+y^{2}+2z^{2}+zx^{2}-zy^{2}\overset{\left(x=2-y\right)}{\longrightarrow}4-4y+2y^{2}+2z^{2}+4z-4yz\rightarrow FOC: \; \begin{cases} -4+4y-4z=0\\ 4z+4-4y=0 \end{cases}\rightarrow y=1+z\rightarrow ...
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1answer
40 views

Find max and min subject to constraint ||x|| = 4

$Q(x,y)=7x^{2}+12xy+12y^{2}$ I only know how to do this is $\|(x,y)\|=1$ If $\|(x,y)\|=1$, the eigenvalues are $16$ and $3$. So obviously $\min=3,\max=16$. I don't know what to do if ...
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2answers
41 views

How do I know if a function has x roots on x-axis?

I am currently studying Newton Raphson Method. Now I am kind of having a question that how I know if the function ever has a x-root or roots on x-axis? Please let me hear your advice. I am sorry if I ...
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26 views

An error in least square optimization problem in Matlab

I am new to MATLAB and I want to formulate the following lease square expression in Matlab. I have some codes that I am typing here. But the optimization problem solution seems not to be correct. Does ...
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16 views

Minimization problem with amplitude constraint

I have the following minimization problem: $$\left\| \bf{A}x - y\right\|^2 \to min $$ $$s.t. \left|x_i\right| < 1, \forall i,$$ where $\bf{A}$ is the complex matrix with size of $(n\times m)$, ...
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22 views

optimization problem with integrals

There is a maximization problem of the following form \begin{equation} \max_{l(a)} \sum \int \bigg(U(c, 1-l(a)) \bigg) x(a,e) da \end{equation} where $$ c = a(1+ f(L)) + e G(L)l(a) - h $$ $$ L = ...
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24 views

How to find the global minimum or maximum of a data set

From some experiment, I am getting noisy data. I am interested in highest maximum value from data. Somehow data is periodic and I want to get the highest maximum value from first period. I am quite ...
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1answer
45 views

Help me out with this optimization problem

This excercise has been taken from an exam. In the following problem: opt:x+y^2-2 subject to y^2<=x and x<=2-y and y>=0 I've found the green area to be the feasible region. (Sorry for the ...
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77 views

Example of a math problem that is not an optimization problem [on hold]

Is there a problem in mathematics that cannot be formulated as an optimization problem? If so, any example? Thanks Edit: To clarify the point and spirit of the question. The question is meant to ask ...
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1answer
17 views

Norman Window Optimization

A Norman window has the shape of a rectangle surmounted by a semicircle. Find the dimensions of a Norman window of perimeter 24 ft that will admit the greatest possible amount of light. I know that I ...
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2answers
58 views

Two-way matrix optimization

I have run into a problem like this. Looks a bit unusual, but I think should be doable. Find $U$ achieving $$\min_U \left( \| A - UW \|_2^2 + \| RU - H \|_2^2 \right)$$ $A,U,W,R,H$ are all ...
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31 views

explicit function between transformation matrix and vertex in polyhedron

recently I am stuck in solving a geometric problem. I hope someone could give me some tips, thanks for all in advance!!! Question 1: given a constant polygon $M1$ with 4 vertices: ...
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1answer
38 views

Need help with second derivative test

In an optimization problem with restrictions, when I have already found the critical points of a function and I have to classify those points (they can either be maxima or minima or saddle points), do ...
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First and Second Order Necessary Conditions [on hold]

1)Let Ω = {x belong to R : Ax=b} and min cT x .Find all points satisfying in the First and Second Order Necessary Conditions. 2)Let f(x) = cT x. Show that if c ≠ 0, then we cannot have an optimal ...
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16 views

Quadratic Optimization Problem with Box Constraints

I want to solve a problem of form $$\min_x x'Ax + b'x \;\;\mbox{ s.t. } l\leq x \leq u$$ where $A$ is a positive semidefinite matrix, thus the function I'm optimizing should be convex. However the ...
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18 views

Differentiation of cost function in adaptive CFO estimator

I'me trying to simulate the steepest descent algorithm for CFO estimation using null subcarriers (OFDM wireless). And some mathematic difficulties have arised. In the core of algorithm lies cost ...
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3answers
35 views

Needing help with this problem

can anybody help me out with this? opt: $x^2+y^2$ subject to $(x-1)^2-y^2=0$ I couldn't even find the critical points.
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1answer
34 views

Optimization Calculus Problem- Flight

If exactly 230 people sign up for a charter flight, the operators of a charter airline charge Dollars 330 for a round-trip ticket. However, if more than 230 people sign up for the flight, then fare is ...
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2answers
31 views

Rectangular Box Optimization Problem

A rectangular box is to have a square base and a volume of 40 ft3. If the material for the base costs \$0.31 per square foot, the material for the sides costs $0.05 per square foot, and the material ...
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20 views

Optimizing the distance [duplicate]

A painting is mounted on a wall. The bottom of the painting is 5 feet above eye level, and the top of the painting is 14 feet above eye level. If you stand directly underneath the painting, you cannot ...
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1answer
20 views

Convex hulls for a finite amount of points

I'm trying to understand what a convex hull intuitively is, and say given for a set of points $(x,y)\in\mathbb{R}^2$ how is it generated from these points? I tried reading the wikipedia article and ...
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55 views

Minimizing Question

A closed box constructed from a tin sheet has a square base and a volume of $343 \text{in}^3$. Find the dimensions of the box, assuming the minimum amount of material was used in its construction. ...
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22 views

Select machines to minimise latencies between them

I am working in an optimisation problem. I am still trying to model it and solve it. The problem is: There is a number of different types of virtual machines. Each type has different hourly cost ...
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2answers
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Optimize volume of an open cardboard box made from flat square of cardboard…

Consider the following problem: A box with an open top is to be constructed from a square piece of cardboard, 3 ft wide, by cutting out a square from each of the four corners and bending up the sides. ...
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1answer
107 views

Find maximum of $P$

Let $$P = \frac{{{x^2}}}{{{x^2} + yz + x + 1}} + \frac{{y + z}}{{x + y + z + 1}} - \frac{{1 + yz}}{9}.$$ Find maximum of $P$ where $x, y,z$ are nonnegative real numbers such that ${x^2} + {y^2} + ...
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1answer
47 views

Finding the maximum and minimum

Can't understand how to find the maximum and minimum with the given definitions (with both x and y).. can someone explain step by step?
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22 views

A simple optimization problem of reciprocal function

Can someone tell me the answer to this question? I cannot seem to figure it out The function $y=\frac{2}{x}$ is decreasing in?? a.$(0,\infty)$ b.$(-\infty,0)$ c.$(0,2)$ d,$(-\infty,\infty)$ I ...
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Determining the coordinate of C to minimize the area of a triangle ABC

Given $A=(0,-10)$ and $B=(2,0)$. Determine the coordinate of $C$ in the curve $y=x^2$ which minimalize the area of triangle $ABC$.
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To minimize $x^TAx$ where $A$ is not necessarily positive semi-definite with constrains?

Let $A\in \mathbb{n\times n}$ be a symmetric matrix. Let $x\in \mathbb{R}^{n\times 1}$ be an unknown vector. The problem is $$\min \limits_x \{E(x)=x^TAx\}$$ where $x\in C$, $C$ is a convex set. ...
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1answer
33 views

To minimize $x^TAx$ where $A$ is not necessarily positive semi-definite.

Let $A\in \mathbb{n\times n}$ be a symmetric matrix. Let $x\in \mathbb{R}^{n\times 1}$ be an unknown vector. The problem is $$\min \limits_x x^TAx.$$ Since $A$ is an input, I am not sure 1 it ...
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1answer
42 views

Logistic function approximation of the real valued Riemann $\zeta(x)$ function

Given the function: $$f(x)=\dfrac{a}{1-b\exp(-cx)}+d$$ where: $a = 0.7071$, $b = 2.21$, $c = 0.7672$, $d = 0.2942$, I found the following inequality: $$|\zeta(x) - f(x)|\lt \epsilon$$ for ...
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1answer
36 views

Solve: $\sum_{i=1}^n \max\left\{x-a_i,0 \right\}=1.$

Given $a_1,a_2,\ldots,a_n \in\mathbb{R}$. Solve the following equation on $\mathbb{R}$: $$\sum_{i=1}^n \max\left\{x-a_i,0 \right\}=1.$$ I am not sure that a closed-form solution exists, so iterative ...
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Nonlinear Optimization problem

Function $f(x) \in \mathbb{R}^n$, $(n\geq 1)$, depend on one parameter $x \in \mathbb{R}$. Performing a nonlinear transformation of $f(x)$, we obtain function $g(y) \in \mathbb{R}^n$. This ...
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Is there any available method to solve $A^TAA^TA+A^TAPA^TA-Q=0$

Let $P, Q\in \mathbb{R}^{m\times m}$ are symmetric matrixes. $A$ is an unknown matrix $\mathbb{R}^{m\times m}$ which satisfies the following equality and $A$ is not sure to be unique, ...
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1answer
14 views

geometric significance of the largest possible dimensions of a rectangle

Find the largest possible rectangular area you can enclose, assuming you have 128 meters of fencing. what is the (geometric) significance of the dimensions of this largest possible enclosure? My ...
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1answer
16 views

How to maximize this function

We are in an euclidian space, and we have to maximize the quadratic form : $x\in B\rightarrow (x|u) (x|v) $where $u$ and $v$ are two given vectors, and $B=\{x:||x||\leq1\}$ I don't find where i have ...
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optimization word problem in calculus

You are asked to build an open cylindrical can (i.e. no top) that will hold $665.5$ cubic inches. To do this, you will cut its bottom from a square of metal and form its curved side by bending a ...
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1answer
31 views

Optimization question for calculus

Could anyone tell me where is my mistake? I took the derivative and I solved for r and ended up with this answer
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3answers
30 views

Optimization in Calculus

As you can see I found the equation but I don't know how to find the points. As far as I tried was $(7, 49)$ but it was wrong.
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Random Rotation of Points using Householder matrices

I have $N$ points in $D$ dimensions, were $D$ is big, for sure more than $100$. $N$ is also big. The goal is to produce an algorithm in my code, that will take as input this dataset and will give ...
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29 views

How to find all stationary points of $ \alpha\|v\|^2-\|x^Tv\|^2+\|g^Tv\|^2$

Let $v,x,g$ be three vectors and $\alpha$ be a constant. The problem is $$\min\limits_v \{\alpha\|v\|^2-\|x^Tv\|^2+\|g^Tv\|^2\}$$ where $\|v\|^2=\sum\limits_{i=1}^{|v|}v_i^2$ and $|v|$ is the ...
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7 views

Proper name for the problem (finding optimal discrete function)

Given a set $D = \{d_1, d_2, ..., d_N\}$, a set of some subsets of $D$, $D^\ast$ and a set of classes, $C = \{c_1, c_2, ..., c_M\}$, I want to find function, that maps a sequence $({d_i}_1^\ast, ...