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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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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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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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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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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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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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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126 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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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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242 views

Derivation of Euler-Lagrange equation

Here is a simple (probably trivial) step in the derivation of the Euler-Lagrange equation. If we denote $Y(x) = y(x) + \epsilon \eta(x) $, I want to know why is $\dfrac{\partial ...
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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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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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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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226 views

Gradient descent with inequality constraints

Suppose we are given a convex function $f(\cdot)$ on $[0,1]$. One wants to solve the following optimization problem: \begin{equation} \begin{aligned} & \text{minimize} && \sum_{i=1}^n ...
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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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1answer
17 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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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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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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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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Extrema homework — maximizing the viewing angle of a picture on a wall

I have hit a problem in my homework and don't know how to solve it. Here it is: "A picture with height of 1.4 meters hangs on the wall, so that the bottom edge of the picture is 1.8 meters from the ...
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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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22 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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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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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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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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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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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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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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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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177 views

Characterization of sphere.

I'm editing the question because I think the previous formulation was leaving a key element of the problem out and that was making it impossible to answer the question. I tried to update/improve the ...
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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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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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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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674 views

How do I construct the Jacobian for use in a Levenberg-Marquardt algorithm.

I am working on a 3D reconstruction system and I am looking to use a Levenberg-marquardt algorithm to do bundle adjustment. I am not too sure about how LM works and what it requires. The model I am ...
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1answer
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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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288 views

How to calculate the hessian of a matrix function

I am estimating a model minimizing the following objective function, $ M(\theta) = (Z'G(\theta))'W(Z'G(\theta))$ $Z$ is an $N \times L$ matrix of data, and $W$ is an $L\times L$ weight matrix, ...
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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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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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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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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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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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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, ...
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Dependency of the Lagrange multipliers

Let $F, g$ be a polynomias in $n$ variables and consider the optimization task $\min F(x) s.t. g(x)=0$ . In Order to solve this with the Lagrange method, one has to find a multipliers. Can one say ...
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Objective function reaches a plateau

I have an objective function and I am trying to minimize it. I noticed that if the objective function is far away from the "solution" it decreases (convergence). Once I start close to the solution the ...
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Primal-dual subgradient method

In these notes, an extension of the subgradient method is presented in Section 8 (page 30). The method is described so quickly and neither convergence analysis (compared to classical subgradient for ...
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Optimization problem: Maximize the sum of minimum.

Given positive integers $L$ and a set of non-negative integers $N$. Find maximum of: $$\large \sum_{i = 1}^{4L}\ N_i\cdot(\min(\vert i - c\vert, 4L - \vert i - c\vert))$$ with $c \in \{1, 2,\dots ...
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Division of plane into equal area regions

We divide a plane ($\mathbb{R}^2$) into infinite number of regions each of area equal $1$. We can use only (one-dimensional) curves which may meet at points. Fix a point $p$ on a plane and consider ...
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Rank degenerate non negative least squares

I'm following an algorithm in the book "Solving Least Squares Problems" by Lawson and Hanson (#15 in Siam's Classics in Applied Mathematics) for solving non negative least squares. That is, minimize ...
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Minimizing a multivariable function in several variables

I would like to show that a certain function is negative, to help establish asymptotic stability via a Lyapunov function for a system of differential equations. This is exactly what I need help on: ...
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Estimating objective function weights from Pareto front

Let's say I have 2 functions $f_1(x)$ and $f_2(x)$. I ran a multi-objective optimization method to obtain the Pareto Front. Now, take any point P on the Pareto front. Assuming the Pareto Front to be ...