Mathematics Stack Exchange
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Questions tagged [optimization]

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.

In mathematics, computer science, economics, or management science, mathematical optimization (alternatively, optimization or mathematical programming) is the selection of a best element (with regard to some criteria) from some set of available alternatives.

An optimization problem can be represented in the following way: given a function $f:A\to\mathbb{R}$ from some set $A$ to the real numbers, we want to find an element $x_0\in A$ such that $f(x_0)\le f(x)$ for all $x \in A$ ("minimization") or such that $f(x_0)\ge f(x)$ for all $x \in A$ ("maximization").

22512 questions
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Homogeneous function has its derivative homogeneous of one less degree

This is from Simon and Blume's Mathematics for Economists: But, for LHS, applying Chain rule goes:$$\dfrac{\partial f}{\partial(tx_1)}(tx_1,\dots,tx_n)\cdot\dfrac{\partial(tx_1)}{\partial x_1}=\dfrac{\partial…
Silent
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Mathematical derivation of successive parabolic interpolation

I'm studying for an exam, and in some past papers they ask to provide 'a mathematical derivation of the iteration scheme'. Now, I understand successive parabolic interpolation, I can do it (or I certainly hope I can!) but I can't find any…
Squimmy
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The conjugate of conjugate of the function $f=yx^2$

Suppose $f=yx^2$ and the arguments are $x\in\mathbb{R},y>0$. According to the definition of conjugate, $f^*(x,y)=\max\limits_{y',x'}xx'+yy'-y'x'^2$ Because $f$ is not a convex function, I cannot use the first optimal condition. Furthermore, I prefer…
Heather
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How do I find a minimum of a function satisfying conditions

Given the following function $y=(x_1-2)^2+(x_2-3)^2+(x_3-5)^2$ and the following conditions $x_3-x_2\geq2$ $x_2-x_1\geq2$ How do I find $x_1$, $x_2$ and $x_3$ such that the $y$ is minimum. I need a general idea how to solve such equations not just…
vrepsys
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Proof that this matrix is TUM

Suppose that A is a mxn TU Matrix. (Totally unimodular). Proof that [A I]^T (so I mean the column vector with A the first element and I the second, where I is the identity matrix) is also TUM. Can somebody help me?
Roos Jansen
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Finding minimum value of multi-variable expression without partial derivatives

Minimize where $a$ and $b$ are positive real numbers $\sqrt{a^{2}\; +\; 4}\; +\; \sqrt{\left( 3-a \right)^{2}\; +\; \left( b-2 \right)^{2}}\; +\; \sqrt{25\; +\; \left( 6-b \right)^{2}}$ I could take the partial derivatives, equate them to 0, and…
1110101001
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What does it mean for a value to be minimised?

I am trying to solve this problem, but I do not understand what is meant bythe following expression being minimised $$ D = P_{k} + P_{j} $$
user87870
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How to maximize number of appointments in a day?

I am trying to figure out how to maximize the number of appointment blocks in a day given the current appointment blocks with their specific times: Appointment Type A 8-10:30am 10:30-1pm 1-3:30pm 3:30-6pm 6-8:30pm Appointment Type…
T.S.
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Optimisation: Minimise series

Let $a_i\geqslant 0$ for $i=1,\ldots,n$. Show how to minimize $$\sum_{i=1}^n\frac 1 {a_i+x_i}$$ subject to $$\sum_{i=1}^n x_i = b$$ where $x_i\geqslant 0$ for $i=1,\ldots,n$ and $b>0$. I'm stuck on how to do this problem.
Raul
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minimization of log function

I am trying to minimize a log function of 3 variables $f(x,y,z) = a\log(x)+b\log(y)+c\log(z)$ such that $x+y+z=k$ for some constant $k$. for $a,b,c > 0$ and $x,y,z > 1$ Now $f(x,y,z)$ is concave so Lagrangian method won't work to minimize. Any…
aniket
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Why this optimization problem is solved like that? any explanations, help? Thanks!

Ok.so I do understand up to step 9. But then it gets all confusing to me...Normally what I would do for these sort of problems is to isolate the lambda symbols in equations 8 and 9 and then equal them, so I can isolate lets say x1. I worked it…
Maximilian1988
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A semicircle has a radius of 2 m. Determine the dimensions of a rectangle with the greatest area that is inscribed in it.

$y^2 + x^2 = 4$ $A(x) = 2xy$ (make base of semicircle = $2x$) plug it in: $A(x) = (\space 2\sqrt{(4-y^2)}\space)\cdot y$ Final derivative: $$\begin{align} A'(x) & = \frac{-2y^2}{\sqrt{4-y^2} + 2\sqrt{(4-y)}} \\ & \\ & \\ 0 & =…
Jessica
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Prove that $\text{argmin}_x f(x) \leq \text{argmin}_x (f(x) - mx)$ for any $f(x)$ and $m\geq0$

How to prove that $\text{argmin}_x f(x) \leq \text{argmin}_x (f(x) - mx)$ for any function $f(x)$ and $m\geq0$?
Uwe
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Non linear constraints

I have to minimize $f(x_1,x_2,...,x_k)$ where $f$ is an affine function, subject to constraints where products between these variables are involved. The number of variables can vary from 10 to 100. What kind of optimization problem this one above…
Dingo13
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Minimization problem with parameter

$a>0,\; b>0,\; S=$parameter $>0$. $$a+b+\dfrac{S-2}{2(a+b)} \longrightarrow min$$ With condition that $a\cdot b =1$ Using inequality of arithmetic and geometric means we get: $$a+b+\dfrac{S-2}{2(a+b)} \geq 2+ \dfrac{S-2}{2(a+b)}$$ With equality when…
Karlis Olte
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