Mathematical optimization (alternatively spelled optimisation) or mathematical programming is the selection of a best element, with regard to some criterion, from some set of available alternatives. Optimization problems of sorts arise in all quantitative disciplines from computer science and engineering to operations research and economics, and the development of solution methods has been of interest in mathematics for centuries.In the simplest case, an optimization problem consists of maximizing or minimizing a real function by systematically choosing input values from within an allowed set and computing the value of the function. The generalization of optimization theory and techniques to other formulations constitutes a large area of applied mathematics. More generally, optimization includes finding "best available" values of some objective function given a defined domain (or input), including a variety of different types of objective functions and different types of domains.
This is the code that i wrote
Clear["Global`*"]
Z = 500;
W = 100000;
G = 250;
H = 100;
K = 0.5;
T = 30;
L = 4000;
P = 5;
S = 2.5;
Y = 1;
A = 0.1;
V = 2.5;
J = 8000;
f[x_] := 1/
x {(J*Z*x*(2*Y - x))/(
2*Y) - ((W + T*G) + ((L + T*P)*2*Z*Y*(1 - ((Y - x)/Y)^1.5))/
3 + (H + T*S +...
Hello,
I am using the Lagrange multipliers method to find the extremums of ##f(x,y)## subjected to the constraint ##g(x,y)##, an ellipse.
So far, I have successfully identified several triplets ##(x^∗,y^∗,λ^∗)## such that each triplet is a stationary point for the Lagrangian: ##\nabla...
Hi,
I was working through the following problem and I am getting confused with the solution's definition of the dual.
Problem:
Given the optimization problem:
minimize ## x^2 + 1 ##
s.t. ## (x - 2) (x - 4) \leq 0 ##
Attempt:
I can define the Lagrangian as:
L(x, \lambda) = (x^2 + 1) + \lambda...
Gradient descent is numerical optimization method for finding local/global minimum of function. It is given by following formula: $$ x_{n+1} = x_n - \alpha \nabla f(x_n) $$ There is countless content on internet about this method use in machine learning. However, there is one thing I don't...
Consider the following theorem:
Theorem: Let ##f## be a concave differentiable function and let ##g## be a concave function. Then: ##y \in argmax_{x} {f(x)+g(x)}## if and only if ##y \in argmax_{x} {f(y)+f'(y)(x-y)+g(x)}.##
The intuition is that local maxima and global maxima coincide for...
I have a set of data (representing the strength distribution of samples), and I would like to fit a normal-Weibull grafted distribution. To the left of a specified graft point, the distribution is Weibull, and to the right it's normal. At the graft point, the value and the first derivative are...
A figure is made from a semi circle and square. With the following dimensions, width = w, and length = l.
Find the maximum area when the combined perimiter is 8 meter.
I first try to construct the a function for the perimeter.
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Next I insert this...
I have a data set of samples, and I made up some variables that filter out unwanted samples from the data set.
Say each sample has values ##\vec{x}=x_1, x_2,..., x_n##, and I know the values of ##\vec{x}## for each sample.
Also, if I sum up all of the samples, I get a value, ##Z##, which tells...
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Hi,
I have the following optimization problem. I have a list of tasks that I should be able to perform with my tools. Each tool costs a certain amount of money, and may be used to carry out a finite number of tasks. The goal is to choose an optimal set of tools in such a way that the toolset can...
Homework Statement
Homework Equations
I have yet to figure out any relevant equations, but I do believe that the constraint equation for the optimization problem is the y=64-x^6 listed above.
The Attempt at a Solution
I am currently trying to figure out methods to begin my optimization...
Summary:
Some help would be greatly appreciated for finding the ideal taper and x/y dimensions of a trebuchet throwing arm to optimize strength and minimize the moment of inertia.
Long Version:
I am a high school junior currently in the process of working on an ambitious project to build a 21...
Hi, I am trying to solve a control problem where I have to minimize the fuel consumption of a vehicle:
$$J=\int_{0}^{T} L(x(t), u(t),t) + g(x(T),T)dt$$
##L(u(t),v(t))=\sum\limits_{i,j=0}^{2} K_{i,j} u(t)^i v(t)^j ## is convex (quadratic) and the term ##g(x(T),T)## is to have a constraint in the...
I have this word problem, and was wondering how I would go about creating a system of equations.
Here is the question:
Problem: You are a small forest landowner, and decide you want to sustainably harvest some of timber on your property. There are costs related to the infrastructure needed to...
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The problem: I have a complex histogram and a...
Homework Statement
Given a list of integers and a single sum value, return the first two values (parse from the left please) in order of appearance that add up to form the sum.
sum_pairs([11, 3, 7, 5], 10)
# ^--^ 3 + 7 = 10
== [3, 7]
sum_pairs([4, 3, 2, 3, 4]...
I'm trying to solve a problem that amounts to:
Given b0, ..., bn-1 where1 <= bi, find the max of |a0 - a1| + |a1 - a2| + ... + |an-2 - an-1| where 1 <= ai <= bi.
I'm 100% confident that each ai is either 1 or bi.
I'm 90% confident that the elements a0, ..., an-1 are either
1, b0, 1, b1...
Dear Physics Forum friends,
I am currently stuck with the following question about the integer optimization:
"Produce a family of 0,1 knapsack sets (having an increasing number n of variables) whose associated family of minimal covers grows exponentially with n."
My thought is that I need to...
http://tinypic.com/r/1570ojk/9
Please see the image. I don't understand how to solve such a problem. It has three criteria. Any hints or guide would be appreciated or even a solution. This is actually just an example from the book, but the book doesn't even solve the question, so I don't even...
Im trying to solve the following problem from the book 'Learning with kernels', and would really appreciate a little help.
Background information
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I try to find out how to minimize functions i R by using nlm function:
> f<-function(x,y){x^2+y^2+10-5*x-y}
> nlm(f,0.1,0.1)
That only gives me an estimate for x. How would write the code to get x and y?
I have seen the implementation of L-BFGS-B by authors in Fortran and ports in several languages. I am trying to implement the algorithm on my own.
I am having difficulty grasping a few steps. Is there a worked out example using L-BFGS or L-BFGS-B ? Something similar to...
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##I_0 = [\frac{-R_2\omega^2C_1C_2 + [C_1+C_2]R_1R_2\omega^3C_1C_2 - R_1\omega^2[C_1+C_2]^2...
Hi,
I am trying to understand the functional form of B3LYP from the Gaussian output file. I have tried to relate the details in the output file with the functional form of B3LYP. But I am not sure what certain terms correspond to. I have mentioned below the details. Can you pleaese help me.
In...
Hello, I am working on an optimization problem but I am not sure if the problem can be formulated and solved with conventional solvers.
Assume the minimization problem for a set of elements ##\mathcal{N} =\{ 1,\dots, h, \dots, i,\dots, N \}##
$$
\mathrm{minimize}\quad C = \sum_{i=1}^{N}...
Hi everyone,
I am new here. I am working in geophysics and I would like to invert for a simple layered velocity model using CMA-ES optimization method. I downloaded the purecmaes.m code in Matlab here: https://www.lri.fr/~hansen/cmaes_inmatlab.html, and also implemented one in Fortran 90. I...
First time posting here so excuse me if I don't know the rules so well. I figured this would be the best place to post this question.
I'm trying to optimize the force produced by a solenoid that is no bigger than 15mm in diameter (D). My goal is to get just the right balance of number of wire...
Homework Statement
The cylinder x^2 + y^2 = 1 intersects the plane x + z = 1 in an ellipse. Find the point on the ellipse furthest from the origin.
Homework Equations
$f(x) = x^2 + y^2 + z^2$
$h(x) = x^2 + y^2 = 1$
$g(x) = x + z = 1$
The Attempt at a Solution
$\langle 2x, 2y, 2z \rangle...
Homework Statement
A piece of wire, 100 cm long, needs to be bent to form a rectangle. Determine the dimensions of a rectangle with the maximum area.
Homework Equations
P = 2(l+w)
A = lw
The Attempt at a Solution
This is what I don't understand, the solutions that I saw from looking around...
Max: 3x + 5y
s.t. x + 2y ≤ 5
x ≤ 3
y ≤ 2
x,y ≥0
By the simplex method, the profit is $14. Using sensitivity analysis I changed the RHS of the 1st constraint and keeping everything else constant, I get the best profit value of $19 at RHS of 7.
What other methods can I use such as the...
C \in \mathbb{R}^{m \times n}, X \in \mathbb{R}^{m \times n}, W \in \mathbb{R}^{m \times k}, H \in \mathbb{R}^{n \times k}, S \in \mathbb{R}^{m \times m}, P \in \mathbb{R}^{n \times n}
##{S}## and ##{P}## are similarity matrices (symmetric).
##\lambda##, ##\alpha## and ##\beta## are...
So, I'm trying to plot a 3D "dipole" (an arrow with a small torus around it basically) in mathematica, and want to rotate it according to Euler angles...
I use this code for the rotation matrix:
rot[a, b, g] := RotationMatrix[g, {1, 0, 0}].RotationMatrix[b, {0, 1, 0}].RotationMatrix[a, {0, 0...
Hi everyone,
I am working on a mathematical optimization model for a fuel cell.
Currently I am facing a problem with the ramp-up of the cell.
I have a modulation ramp of 4% of the nominal power (58.3 kW) per minute.
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A common problem with optimization of machines is that occasionally, some of the variable combinations will not be geometrically feasible. For example, if magnet angle is being parameterized in an IPMSM, it might be that at some point, the magnet skews so much that it juts out of the rotor...
Homework Statement
A ski jumper leaves the ski track moving in the horizontal direction with a speed of 25.0 m/s as shown in Figure 4.14. The landing incline below her falls off with a slope of 35.0°. Where does she land on the incline? I've attached an image of the problem, my work is below...
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Hi. My colleagues and I are doing a research on transformers (single-phase) and we stumbled across the following equations involving hysteresis and eddy current losses:
Wh = ηBmaxxfV
where Wh = hysteresis losses
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Let F:V\rightarrow{}\mathbb{R}^{+}_{0} be a differentiable function. V is the set of all positive real-valued 2\times{}2 matrices, so
V=\left\{\left[
\begin{array}{cc}
a & b \\
c & d \\
\end{array}\right]\mbox{ with }a,b,c,d\in\mathbb{R}^{+}\right\}
Here are the two constraints for F...
Homework Statement
Find the extremizing (maximum) value of the function f(x) = sin x / x using Newton's 1D method.
Homework Equations
[/B]
The Attempt at a Solution
I know the maximum point in this equation is (0, 1). When I differentiated the equation twice and used the formula above, I...
I am at the moment working on a project in which I try to minimize the annual running costs of a chemical manufacturing plant. To predict annual running costs I created a model with over 50 inputs, including things such as the type of chemicals and equipment used at different points in the...
ƒ(ß)=.5sec(ß) + √[1+(sec2(ß)/4)+tan(ß)/√(2)]
Without graphing it or using calculus find the minimum. I already know the answer but want to know how to do it. It s at π/12 and is something like 1.5.
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Homework Statement
There is a typo in the problem, ”R > Σ n i=1 σi − n max 1≤i≤n σi” which should be R > n max (1≤i≤n) σi − (Σ n i=1 σi )
Homework Equations
The Attempt at a Solution
Not sure where to go with part B or where to start...
Hi, I have an mathematics assignment to do, and I wonder if the topic I have chosen is doable for me. I want to minimize the surface area of a cobbler cocktail shaker, and until now my plan was to get the curve equation for the side of it, and get the area equation from surface of revolution...
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First post on these forums, thanks all for your help!
Homework Statement
It is estimated that the growth rate of the fin whale population (per year) is rx(1 - x/K), where r = 0.08 is the intrinsic growth rate, K = 400,000 is the maximum sustainable population, and x is the current population...