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 Homework

I need help with this math question: A farmer with 720 ft of fencing wants to enclose a rectangular area and then divide it into four pens with fencing parallel to one side of the rectangle. What is ...
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SVD and least squares proof

I've seen it claimed that the solution to the minimization problem $$\underset{\textbf{b}}{\text{argmin}} \ ||\textbf{A} \textbf{b}|$$ subject to a constraint $||\textbf{b}||=1$, is given by first ...
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Intuition about multiplicative gradient descent

Suppose we want to minimize a function $f(x)$ wrt $x$, i.e., we want to solve, $$x^* = \arg \min_x f(x)$$ One method to solve such problems is gradient descent. In gradient descent, one uses the ...
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Minimization of Sum of Squares Error Function

Given that $y(x,{\bf w}) = w_0 + w_1x + w_2x^2 + \ldots + w_mx^m = \sum_{j=0}^{m} w_jx^j$ and there exists an error function defined as $E({\bf w})=\frac{1}{2} \sum_{n=1}^{N} \{y(x_n, w)-t_n\}^2$ ...
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real-time linear programming

I'm going to implement in C a light-weight embedded lp-solver for a production system. I need to be able to sequentially solve a series of (possibly unrelated) linear programming problems with ~6-60 ...
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197 views

How to find the maxima/minima of a function using a software?

I want to find the maxima/minima of a function in an interval. Is there any software or online tool available for this? The particualr function I want to maximize is, $$f(x) = x(1 - \sqrt{2e} \cdot ...
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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: ...
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Selecting k vectors with maximum spread out of a set of n vectors

Given a set $\mathcal{V}$ of $n$ vectors, find a subset $\mathcal{V}_k = \mathcal{V} - \mathcal{V}_{n-k}$ containing $k$ maximally spread vectors. Intuitively, these $k$ vectors should be spread as ...
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514 views

Solving optimization problems using derivatives and critical points

I have a homework question which I have completed 2/3 of; however I am stuck on the last part of the question. The question is: A drug used to treat cancer is effective at low doses with an ...
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a basic doubt on a multivariable optimization problem

Suppose we have a production system where there is a known production function $f(x_1,x_2,\dots,x_n)$ that gives the amount of the commodity produced as a function of the amounts $x_i$ of the inputs, ...
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How to maximise functions of this shape $y=2\cdot3^{-x}$

How can I find the maximum of $2\cdot 3^{-x}$? I know its close to $1$ because I have seen its graph, but when I differentiate the function and set it equal to zero (to get a maximum) I get $-2\cdot ...
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Factory Optimization

First I feel like a disclaimer here is needed that this is NOT a homework problem but I am going to ask the question in something that looks like one. I don't know how to ask my question any other ...
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What are non-orthogonal eigenvectors?

Given a symmetric matrix $A$, the maximum of the trace, $Tr(Z^TAZ)$ under the assumption that $Z^TZ=I$ occurs when $Z$ has the eigenvectors of $A$, as $Tr(U^TAU)= \lambda_1 +\lambda_2+...\lambda_ d$ ...
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195 views

Multivariable local maximum proof

Suppose we have a twice differentiable function $f: \mathbb{R} ^n \to \mathbb{R}$, a point ${\bf x^0} = (x_1 ^0 , \ldots , x_n ^0)$ and we know that $\nabla f({\bf x}^0) = 0$ $({\bf x - x^0})H({\bf ...
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Linear equation question with maximisation

A person, with a certain given amount, can buy either 8 apples or 14 oranges or 3 water melons. The weight of either 10 apples is equal to the weight of 12 oranges which is in turn equal to the weight ...
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44 views

How can I find the minimum and maximum?

Lets have the following equation: $f(x,y,y) = cos(x)^2+\frac{1}{1+x^3}+y^3+z$ I would like to find the minimum and maximum where $-2<x<2$ $-1<y<1$ $-2<z<1$ How can I do that, I ...
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Confusion over a proof that gradient is perpendicular to the level set

To prove that the vector $\nabla{f}(x_0)$ is orthogonal to the tangent vector to "an arbitrary smooth curve" passing through $x_0$ on the level set determined by $f(x)=f(x_0)$ the following proof is ...
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Linear Programming- are my equations correct?

A dairy produces cheese, milk, sour cream, and yogurt. Suppose: Every 100 lbs of cheese requires 2 units of plant capacity, 3 workers, and 7 gallons of culturing additive, and gives $1,500 in ...
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Finding minimum $\frac{x+y}{z}+\frac{x+z}{y}+\frac{y+z}{x}$

I would appreciate if somebody could help me with the following problem Q. Finding maximum minimum $$\frac{x+y}{z}+\frac{x+z}{y}+\frac{y+z}{x}(\text{where} ~~x,y,z>0)$$
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When does the triangle have the smallest area?

The following triangle has an area $S$, and the sides $AO$ and $BO$ have the length $a$ and $b$, respectively. There is a fixed point $X$ at $(x,y)$. A point $C$ is put on the line segment $OA$, and ...
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How to minimize such an energy?

For the Energy, $$ Q(u)=\int_I(1+|u'(x)|^2)^{1/4} dx $$ where $u(0)=0$ and $u(1)=1$, $u$ is $C^1$ in $I$ and continuous up to the boundary, $I=(0,1)$. How to show the infimum of $Q$ is $1$?
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Reformulating objective function of canonical correlation analysis

Given two column vectors $X = (x_1, \dots, x_n)'$ and $Y = (y_1, \dots, y_m)'$ of random variables with finite second moments, canonical-correlation analysis seeks vectors $a$ and $b$ such that the ...
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How to find the point on a parabola where x and y are equal?

On a parabola how could i find the point at which the y and x points are equal and meet on a point of the graph, algebraically?
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the objective function $\|F\|_F^2$ is quasiconvex in the optimization?why?

I have read a paper, but I can not understand one optimization thoroughly.Generally, Frobenius norm of one matrix, $\|F\|_F^2$, as the objective function is convex, so we can resolve it not using the ...
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Sort objects into groups based on group size preference

I have a research question that involves human subjects being sorted into groups before playing a social game. Before sorting, each person decides on their preferred group size, from 1 to n; where n ...
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773 views

Are Legendre transforms of non-convex functions useful?

Do Legendre transforms have any applications that do not appeal to convexity? What is the intuitive interpretation of the Legendre transform of a non-convex function?
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How to get the closed form solution of a non-convex optimization problem?

I want to know if there is a closed form expression for the optimal objective function? How can I get it, if it does exist? Condition: $h,f\in \mathbb{C}^{N\times1}, \epsilon > 0 $. $\max \ \ ...
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95 views

Limit of a function at its maximum

Let $f_n(x,y)$ be a real function in the domain $0\leq x\leq 1$ $0\leq y\leq 1$ I would like to compute $A = \lim_{x \to 0}\left( \arg \max \limits_{y} f(x,y) \right)$, The problem is that it ...
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satisfy the Euler-Lagrange equation

Two circles of unit radius, each normal to the line through their centers are a distance d apart. A soap film is formed between themas shown below; energetic considerations require the filem to ...
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(1)Maximize the trace of a matrix. (2)Minimize the trace of its inverse matrix. Are (1) and (2) equal?

I need to solve an optimization problem which aims to $\text{maximize}$ $\text{Trace}[(U^HU)^{-1}]$. The diagonal elements of $U^HU$ are all positive. I want to know whether it is equal to solve the ...
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Minimisation of a distance sum

I have a list $L$ of $N$ numbers, and I want to choose $k$ numbers $\{x_1,x_2, \ldots,x_k\} \subseteq L$ in such a way value $S$ of the those K numbers is minimum. $$ S = \sum_{0< i < j <= k} ...
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finding range of function of three variables

Three real numbers $x$, $y$, $z$ satisfy the following conditions. $x^{2}+y^{2}+z^{2}=1~$, $~y+z=1$ Find the range of $~x^{3}+y^{3}+z^{3}~$ without calculus. I solved this problem only with ...
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Maximize two-variable linear function

How would you maximize the following function (with integer domain) $$f(x,y) = a * x + b * y$$ subject to $$c * x + d * y \leq N$$ $$x \geq 0, y \geq 0$$ the constants $a, b, c, d, N$ are known ...
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Optimization of wave drag formula ( gradient calculation of double summation)

i want to ask a question about gradient calculation of double summation term wave drag formula The formula (objective function to minimize !) shown above calculates the wave drag of an aircraft, S is ...
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Maximize $W(x) - (\ln(x) - \ln{\ln{x}})$

How can you maximize $f(x) = W(x) - (\ln(x) - \ln{\ln{x}})$ for $x\geq 2$? Numerically the answer seems to be at around $x \approx 41$ where you get $f(41) \approx 0.31$. Mathematica suggests the ...
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How to solve linear system of equations in 1 and inf-norm?

I have the problem to find a linear program that is equivalent to solving the problem that finds a minimum for $||Ax-b||_1$ and $||Ax-b||_{\infty}$. We defined a linear program as follows: $min_{x} ...
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Simplifying a difficult expression to input it in R

The function is $\Large f(X,Y|\mu_1,\mu_1,\theta)=\frac{\phi (X-\mu_1)\phi (Y-\mu_2)\theta(1-e^{-\theta})e^{-\theta (\Phi(X-\mu_1)+\Phi(Y-\mu_2))}}{[1-e^{-\theta}-(1-e^{-\theta \Phi ...
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How $(P_{r+1}-P_r)$ is maximized?

This is from DeGroot's "PROBABILITY and STATISTICS"(Second edition)(Cf. pages 87 to 92).I am rewriting the relevant stuff. Let $r$ be a positive integer and $r\leq n$. ...
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Minimum moves to transform a list to another?

Given two list of n positive elements. We are allowed to perform only one transformation which is to increment each element of the list except one. What are the minimum number of transformation ...
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What is “anti-optimization” about?

I was wondering what "anti-optimization" is about? Is it related to optimization? What topics does it cover? All I can find out from Google is this paper. It looks like having some relation with ...
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Maximizing a ratio of convex matrix functions by minimizing a difference?

Given that $g(.),h(.)$ are twice-differentiable convex quadratic real functions whose domain is the set of all real matrices while the range is the set of positive real numbers, then: Is maximizing ...
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How to determine wether critical points (of the lagrangian function) are minima or maxima? [duplicate]

$f(x,y) = 2x+y$ subject to $g(x,y)=x^2+y^2-1=0$. The Lagrangian function is given by $$ \mathcal{L}(x,y,\lambda)=2x+y+\lambda(x^2+y^2-1), $$ with corresponding $$ \nabla \mathcal{L}(x,y,\lambda)= ...
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Determine all the extrema of a function subject to a non-linear constraint.

QUESTION Determine all extrema of the function $$f(x,y) = x+ 2y $$ subject to $$x^2 + y^2 - 80 = 0$$ ATTEMPT I don't think I understand what I'm supposed to do. This was in a test and I ended up ...
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Find the minimum,maximum, infimum and supremum of sets?

If $X$ is the intersection of all the intervals $(1-\frac{1}{n^2},1+\frac{5}{n^3}]$ for $n=1$ to infinity, what is the minimum, maximum, supremum and infimum of $X$? If $Y$ is the intersection of all ...
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For each of the following functions, what do the first- and the second-order optimality conditions say about wether 0 is a minimum on $\mathbb{R}$.

For each of the following functions, what do the first- and the second-order optimality conditions say about wether 0 is a minimum on $\mathbb{R}$. $f_1(x)=x^2$ $f_2(x)=x^3$ $f_3(x)=x^4$ ...
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how to minimize this convex function?

$x_i$ and $y_j$ are variables. I intend to minimize this function and obtain the optimal value of $x$ and $y$: $\begin{align} ...
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Applying the Lagrangian function to find critical points

So I have the following function $$ f(x,y) = x^2+y^2 $$ subject to $$ g(x,y) = x+y-1 = 0. $$ And I have to use the Lagrangian to find the critical points, and determine wether they are ...
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What's a really good book for a course titled “Optimization and Control Theory”?

I can't seem to find one that shows a lot of examples with the theory. Could I get some help? Also, it would be a bonus if the book/material is readily available online so I can download it onto my ...
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This question is a basic optimization problem, also a linear algebra question:

Let $p$ be a direction of unboundedness for the constraints $$Ax = b, x ≥ 0.$$ Prove that $−p$ cannot be a direction of unboundedness for these constraints.
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Derivation of soft thresholding operator

I was going through the derivation of soft threholding at http://dl.dropboxusercontent.com/u/22893361/papers/Soft%20Threshold%20Proof.pdf. It says the three unique solutions for $\operatorname{arg ...