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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Why does finding the $x$ that maximizes $\ln(f(x))$ is the same as finding the $x$ that maximizes $f(x)$?

I'm reading about maximum likelihood here. In the last paragraph of the first page, it says: Why does the value of $p$ that maximizes $\log L(p;3)$ is the same $p$ that maximizes $L(p;3)$. The ...
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1answer
139 views

Maximum of the sum of cube

(1) $-2\leq a_{i} \leq 2$ $~(i=1,2,3,4,5)$ (2) $\displaystyle\sum_{cyclic}a_{i}=0$ then, find the maximum value of $\displaystyle\sum_{cyclic}a_{i}^{3}$ also, can it be generalized as for ...
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207 views

Real approximation to the maximum using Laplace's method integral

The Laplace's Method states that under some conditions, it holds that: $ \sqrt{\frac{2\pi}{M(-g''(x_0))}} h(x_0) e^{M g(x_0)} \approx \int_a^b\! h(x) e^{M g(x)}\, dx \text { as } M\to\infty$ Where ...
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4answers
166 views

Three Variables-Inequality with $a+b+c=abc$

$a$,$b$,$c$ are positive numbers such that $~a+b+c=abc$ Find the maximum value of $~\dfrac{1}{\sqrt{1+a^{2}}}+\dfrac{1}{\sqrt{1+b^{2}}}+\dfrac{1}{\sqrt{1+c^{2}}}$
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How to interpret Hessian of a function

I know that gradient of a function gives the direction in which the directional derivative of the function is maximum. Is there any similar interpretation of Hessian ?
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2answers
136 views

Maxima and minima of multivariable function $f(x,y)=6x^3y^2-x^4y^2-x^3y^3$

$$f(x,y)=6x^3y^2-x^4y^2-x^3y^3$$ $$\frac{\delta f}{\delta x}=18x^2y^2-4x^3y^2-3x^2y^3$$ $$\frac{\delta f}{\delta y}=12x^3y-2x^4y-3x^3y^2$$ Points, in which partial derivatives ar equal to 0 are: ...
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1answer
88 views

Cubes, squares and minimal sums

I have trouble solving the following task: i need to find positive integers a and b such that 1) $a \neq b$ 2) $ \exists c \in \mathbb{N} : ~ a^2 + b^2 = c^3$ 3) $\exists d \in \mathbb{N}: ~ a^3 + ...
2
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2answers
749 views

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 ...
2
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1answer
256 views

Multiple solutions for both primal and dual

If matrix $A$ in an LP (or $A^T$ in its dual) has full row (column- in dual) rank, is it possible that both primal and dual have multiple solutions?
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2answers
102 views

Minimizing $f(x,y,z)=\dfrac{|x|+|y|+|z|}{xyz}$ on a sphere

I need to find the minimum of the function: $$f(x,y,z)=\dfrac{|x|+|y|+|z|}{xyz}$$ with the condition: $$x^2+y^2+z^2=r^2$$ Using numerical methods it's quite easy to solve the problem. How can I ...
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How to plot a phase portrait for this system of differential equations?

I beg your help.. I'd like the phase portrait for this system. I don't know how to use Mathematica/Matlab ... :( If anyone can make this portrait and post a print screen here, I would thank you ...
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0answers
187 views

Duality for Support Vector Machines

SVM classifier for two linearly separable classes is based on the following convex optimization problem: \begin{equation*} \frac{1}{2}\sum_{k=1}^{n}w_k^2 \rightarrow \min \end{equation*} ...
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Solving a set of equations with Newton-Raphson

I want to solve this set of equations with Newton-Raphson. Can anybody help me? $$ \cos(x_1)+\cos(x_2)+\cos(x_3)= \frac{3}{5} $$ $$ \cos(3x_1)+\cos(3x_2)+\cos(3x_3)=0 $$ $$ ...
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2answers
57 views

minximum and maximum of $P=x+y+z$

Let $x,y,z \in R;x \ge 1,y \ge 2,z \ge 3$ and $$\sum\limits_{\large{\text{cyc}}} {\frac{{{x^2} - x + 1}}{{x + \sqrt {x - 1} }} = 12} $$ Search minximum and maximum of $P=x+y+z$
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90 views

Explain this statement $\bar 0 \in \partial f(x^*)$ where $\partial f(x^*)$ is subgradient

I haven't understood this theorem "$x^*$ is global minimum iff $\bar 0\in \partial f(x^*)$". What does it mean? Visually? P.s. Studying Nonlinear-optimization -course, 2.3139.
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5answers
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Are derivatives defined at boundaries?

Given a differentiable function $f : [-5,5] \rightarrow \mathbb{R},$ I was under the impression that the derivative $f'$ has domain $(-5,5).$ However, according to Wikipedia ...a differentiable ...
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1answer
358 views

Simple question: the double supremum

Let $f:A\times B\to \mathbb R$. Is it always true that $$ f^* = \sup\limits_{a\in A,b\in B}f(a,b) = \sup\limits_{a\in A}\sup\limits_{b\in B}f(a,b). $$ I proved it by the $\varepsilon$-$\delta$ ...
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1answer
810 views

Why is Dantzig's solution to the knapsack problem only approximate

For a bunch of items with values $v_i$ and weights $w_i$, and with a total weight $W$ that our bag can carry, how do we achieve maximum total value without breaking the bag? Dantzig proposed that we ...
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1answer
2k views

The composition of two convex functions is convex

Let $f$ be a convex function on a convex domain $\Omega$ and $g$ a convex non-decreasing function on $\mathbb{R}$. prove that the composition of $g(f)$ is convex on $\Omega$. Under what conditions is ...
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8answers
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Maximizing the sum of two numbers, the sum of whose squares is constant

How could we prove that if the sum of the squares of two numbers is a constant, then the sum of the numbers would have its maximum value when the numbers are equal? This result is also true for ...
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2answers
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Math notation for location of the maximum

My question is about notation. I have maximum of the function $f(x)$. This can be expressed as $\max(f)$ How can I express in compact form that $x_0$ is the location of that maximum.
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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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1answer
96 views

Simple optimization trick

Let $f,g:X\to\Bbb R$ be two functions where $X$ is any set. Then $$ \left|\sup_x f(x) - \sup_x g(x)\right|\leq \sup_x|f(x) - g(x)|. $$ This fact is fairly easy to prove, but it seems to be a ...
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1answer
1k views

$f$ is convex function iff Hessian matrix is nonnegative-definite.

Let $f: \mathbb{R}^2 \rightarrow \mathbb{R}$, $f \in C^2$. Show that $f$ is convex function iff Hessian matrix is nonnegative-definite. $f(x,y)$ is convex if $f( \lambda x + (1-\lambda )y) \le ...
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1answer
177 views

Optimisation Problem on Cone

The problem I've got here is to prove that semi vertical angle of a cone with maximum volume with total surface area constant is equal to $arcsin(\frac{1}{3})$ I am trying to do that by making some ...
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1answer
273 views

Simple resource for Lagrangian constrained optimization?

Just had an optimization lecture. I understand unconstrained methods like Newton and Gradient descent just fine, as well as the ideas that give rise to them. I don't really understand the ideas that ...
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43 views

Computing $\sup_{\left \| u \right \|=1} d(u,F)$ in a closed subspace of a normed space.

I have come across with the following problem: Let $E$ be a normed space and $F$ a closed subspace of $E$. It's asking to compute $\sup\limits_{\left \| u \right \|=1} d(u,F)$. What I know it's ...
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1answer
66 views

Binomial Coefficient: monotonically decreasing in this range?

relating to this question, I'd like to ask a further one. Again we have $$f(x)={k-1 \choose x-1} p^x (1-p)^{k-x}$$ We know that this term is maximal for $x=kp$, before increasing, afterwards ...
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675 views

How to find the minimum/maximum distance of a point from elipse

I have the point $(1,-1)$ and the ellipse $$x^2/9 + y^2/5 = 1 $$ How to find the minimum and maximum distance of the point from the ellipse ? from exploring the ellipse I know that $$a = 3$$ , $$b ...
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Ladder Optimization Problem

A fence 4 feet tall runs parallel to a tall building at a distance of 4 feet from the building. What is the length of the shortest ladder that will reach from the ground over the fence to the wall of ...
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Maximizing a given function?

I have a function as below, $f(\alpha) = \frac{{1 - \alpha }}{2}\ln \left( {1 + \frac{{AB}}{{B + \frac{{1 - \alpha }}{{C\alpha }}}}} \right)$, where $A$, $B$, $C$ are constant, and $0 < \alpha ...
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1answer
50 views

A non-linear optimization problem

I have the following optimization problem on the variables $a_1, ..., a_n$: $$ minimize \frac{\sum_{k=1}^{n}\max(k\cdot a_{k},1)}{\sum_{k=1}^{n}a_{k}} $$ $$ such\ that\ \ 0\leq a_k\leq 1\ \ \ (k=1, ...
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1answer
143 views

Find the minimum of this expression

This is a problem in my exam and I can't find the solution using elementary inequality knowledge. Can anyone here help me solve this. Thanks $a,b,c $ are positive real numbers which satisfy ...
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28 views

How to set up matrix to compute best coefficients

Suppose we're given a non-linear spring with the following relationship between the applied weight ($x$) and displacement ($y$): $y = ax + bx^3$. I've done a sequence of $m$ tests measuring the ...
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How to perform the optimization when gradient is a matrix $\mathbf{R}^{n\times n}$

I am trying to optimize this cost function by using Gauss-Newton method. $$f = \sum_{i = 1}^n Tr{(Z^TZ)}$$ where $Z$ is a $4\times4$ matrix and it is a function of real vector ...
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1answer
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How to find the speed that minimizes the total cost of a trip?

Here are some facts about semi-trucks and a trip between Chicago and New Orleans. (a) The trip is 750 miles. (b) Running at 50 mph, the truck gets around 4 miles per gallon. (c) For each mph ...
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Linear Programming - Tableau Condition

The following tableau corresponds to an iteration of the simplex method: ...
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1answer
123 views

Finding Shortest distance between a Sphere and Ellipsoid?

Suppose that ,I have a Sphere and an ellipsoid as Sphere: $(x-x_1)^2 + (y-y_1)^2 + (z-z_1)^2 = R_1^2$ Ellipsoid: $\large\frac{(x-x_2)^2}{a^2} + \frac{(y-y_2)^2}{b^2} + \frac{(z-z_2)^2}{c^2} = 1$ ...
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2answers
56 views

To prove the existing and uniqueness of a solution

Let function $f$ be differentiable and convex in $R^{n}$. How can it be proved that $\forall \lambda > 0$ solution of system equations $f'(x) = -\lambda x$ exists exclusively ($\exists \hspace{3mm} ...
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Distance metrics with kmeans

Context: I'm trying to derive some formulas for computing the "mean" in the K-means algorithm. So given an assignment of $m$ data points to $k$ clusters, find a formula to recompute the mean of the ...
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1answer
75 views

How to solve this optimization problem?

Suppose I have the following problem: Maximize: $\quad\quad x_1+x_2+x_3+x_4$ Subject to: $\quad\quad \dfrac{\gamma\;a_1\;x_1}{\gamma\;a_2\;x_4+1}\geq1$, $\quad\quad\quad\quad\;\;\quad\quad ...
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Lagrange multipliers - finding maximum/minimum

I have solved the question, and obtained the critical points, but don't know how to show its a maximum or minimum of a function. I don't understand other answers because symbols confuse me so much and ...
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1answer
103 views

How to minimize cost of group of items given that weights of item sums up to fixed value and atmost 'n' number of items are allowed?

Given that we have a set of items :- { (c1, w1) , (c2, w2), (c3, w3) , ... } where (ci, wi) are the respective cost and weight of the ith item. Its required to minimize total cost of items C such ...
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1answer
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What's the largest possible volume of a taco, and how do I make one that big?

Let $f$ be a continuous, even, positive function over some interval $I=[-a,a]$ such that the total arc length of $f$ over $I$ is at least $2$, $f(0)=0$, and $f$ is increasing on $(0,a)$. View the ...
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Can Three Equilateral Triangles with Sidelength $s$ Cover A Unit Square?

A previous question on the site asked for a short proof of the fact that three equilateral triangles with unit side length cannot be arranged to cover a square with unit side lengths. Given the truth ...
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Operations research book to start with

for somebody having a quite strong background in Mathematics, which are some good books for the domain of Operations research? I guess there are textbooks covering topics like linear and nonlinear ...
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Do dynamic programming and greedy algorithms solve the same type of problems?

I wonder if dynamic programming and greedy algorithms solve the same type of problems, either accurately or approximately? Specifically, As far as I know, the type of problems that dynamic ...
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815 views

Math Wizardry - Formula for selecting the best spell

Imagine we have a wizard that knows a few spells. Each spell has 3 attributes: Damage, cooldown time, and a cast time. Cooldown time: the amount of time (t) it takes before being able to cast that ...
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354 views

If $A+B+C+D+E = 540^\circ$ what is $\min (\cos A+\cos B+\cos C+\cos D+\cos E)$?

Let each of $A, B, C, D, E$ be an angle that is less than $180^\circ$ and is greater than $0^\circ$. Note that each angle can be neither $0^\circ$ nor $180^\circ$. If $A+B+C+D+E = 540^\circ,$ what is ...
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1answer
376 views

Arnol'd's trivium problem #68

I came across this blog that says that its French version has answers to most of Arnol'd's trivium problems, and I figured I'd try my hand at some of the ones they don't have. Number 68 raised my ...