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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Mathematical areas that are applied to Rubiks Cube solution

I had seen about Group Theory being applied to rubik's cube and infact the solution algorithms are also based on group theory... I want to know whether other mathematical fields like "optimization" or ...
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168 views

Polyline - smoothing and extracting specific turning points

I have a problem where I am given multiple polylines constructed from data points. I have to analyse these lines to fine a specific pattern. I am looking for a rise followed by a plateau of values ...
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644 views

Finding the local extrema of this trigonometric, multivariate function

QUESTION Find all extrema and their places for $$ f(x,y) = \mathtt{sin} x + \mathtt{cos} y + \mathtt{cos} (x-y)$$ for $ 0 \le x \le \frac{\pi}{2}$ and $ 0 \le y \le \frac{\pi}{2}$ ATTEMPT I go ...
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Finding the minimum value of a rational function.

Prove that if $x$ is real and $a>c$ & $b>c$ the minimum value of $$\frac{(a+x)(b+x)}{(c+x)} ;Given\space( x>-c)$$ is $$({\sqrt{a-c}+\sqrt{b-c \space}})^2$$ I tried using minima condition ...
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496 views

Explain a surprisingly simple optimization result

The following optimization problem came to my attention as an idealization of the silly browser game Cookie Clicker, but is representative of a range of strategy games: You have an initial ...
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Issues with CVX package for optimization

I am trying to use the cvx package for optimization. However, I am having some issues with it. I have a variable X which is a matrix but I cannot add $X^{-1}$ in the objective function. What should I ...
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110 views

Mathematical formulation in operations research

Does anyone know how I would enforce the following constraints using a mathematical formulation? Any help or feedback is appreciated. a) If person A is given project 1, then person D must be given ...
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63 views

Monotonically Increasing Mapping?

$\mathbf{h}_1, \mathbf{h}_2\in\mathbb{C}^{n}$ are given column vectors and $a>0$ is a given constant. Consider the matrix ...
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92 views

Learn about reproducing kernel Hilbert spaces?

Why are reproducing kernel Hilbert spaces an important topic to learn? What is possibly achievable with that theory that is not reachable with just standard Hilbert space theory?
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37 views

Making the Smallest Number of Mistakes Possible

I have the following problem. I have a set of $k$ labelled points, $\left\{\mathbf{x}_i, y_i\right\}_{i=1}^{k}$, where $\mathbf{x}_i\in \mathbb{R}^{2}$, and $y_i\in\left\{-1,1\right\}$. I want to ...
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50 views

how to solve this optimization problem with functions?

Suppose $\theta\in[0,1]$ and $s(\cdot)$ is strictly increasing in $\theta$ with $0\leq(0)<s(1)\leq1$ and $s(0)+s(1)>1$ How could I solve the program, $\max_{s(\cdot)\in ...
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67 views

Can a condition for a global maximum (of some specific function) be given?

Suppose we have a twice continuously differentiable function $h(x) := \frac{g(x)}{1 - \delta + \delta F(x)}$, $0<\delta<1$, defined on the interval $[0, a]$ (where $a$ may be infinite). The ...
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173 views

$a,b,c>0,a+b+c=21$ prove that $a+\sqrt{ab} +\sqrt[3]{abc} \leq 28$

$a,b,c>0,a+b+c=21$ prove that $a+\sqrt{ab} +\sqrt[3]{abc} \leq 28$ I have tried to use AM-GM inequality, but get no result as follows: $$a+\sqrt{ab}+\sqrt[3]{abc}\leq ...
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436 views

Finding minimum sum of absolute differences between heights of $n$ boys and $m$ girls

Given two sets $A$ and $B$, $A$ has heights of $n$ boys and $B$ has heights of $m$ girls, $m \ge n$. We have to find one solution of pairing up $n$ boys with $n$ (out of $m$) girls so that the sum of ...
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77 views

Speeding up solution of a binary integer program

To solve the problem of making a "good" schedule for a tournament between N teams, using memories from my (long gone) student days, I expressed it as a binary integer program. With the current set of ...
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414 views

Using gradient descent and Newton's method combined

I have this function $f(\mathrm{X})$ where $\mathrm{X=A+B+C}$ where $\mathrm{A}$ is a diagonal element with variable $a$ on its diagonal. $\mathrm{B}$ is another diagonal matrix with variable $b$ on ...
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298 views

Optimizing buying and selling point for a stock

I am working on a problem and I need help getting started. Any pointers would be greatly appreciate it My problem: Given a $50,000 purse and 20/20 hindsight, and a particular stock, what are the best ...
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Complexity of Earlist Avaible Due Date for Scheduling Problem 1|ri, pi=1|Lmax

Let us consider the scheduling problem 1|ri,pi=1|Lmax (basically, this means there is one machine on which we have to schedule n jobs (all with identical procssing time 1) in such a way that the ...
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74 views

(Proximal) subgradient inclusion property proof

I'm having a bit of trouble proving what seems to be two fairly straightforward statements for a nonlinear optimisation class I'm taking. We're studying properties of the proximal subgradient, ...
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35 views

Convex optimization issues

I have to optimize a function $f(a,b,c_{ij})$ which consists of a terms like matrix $\mathrm{X = A + B + C}$ where $\mathrm{A}$ is a diagonal matrix with the diagonal elements equal to $a$. ...
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141 views

Issues in optimization with positive definite constraints

I have this function $f(\mathrm{X})$ such $\mathrm{X}$ is a positive definite matrix which is equal to $\mathrm{A+B+C}$. $\mathrm{A}$ is a diagonal matrix with variable $a$ on the diagonal elements. ...
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147 views

Dividing a set of points into two sets of roughly equal diameter

Let $S$ be a finite set whose cardinality is more than 1 and $d: S\times S\rightarrow\mathbb R$ be a positive symmetric function (that is, $d$ is a distance without the axiom of triangle inequality). ...
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61 views

Minimization problem convex set

I'm trying to minimize the function: $$f(w)=w^T\mu+k\sqrt{w^T\Sigma w}$$ where $w$ is a vector in $W=\{x \in \mathbb{R}^n|x_1+...+x_n=1 , x_i \geq 0 \forall i\}$. The vector $\mu \in \mathbb{R}^n$, ...
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18 views

Minima problem?

This is a question in my textbook which I can't solve. Any help would be appreciated, thanks. "A piece of wire 10 metres long is cut into two portions. One piece is bent to form a circle, and the ...
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156 views

Interchange of max and min

Let $f_1(x)$ and $f_2(x)$ be two functions of $x$. Is this true \begin{align} \max_{x\in \mathbb{R}}~\min_{i}~f_i(x) = \min_{x\in \mathbb{R}}~\max_{i}~-f_i(x) \end{align} (UPDATE: I am not asking if ...
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Determine the minimum of $a^2 + b^2$ if $a,b\in\mathbb{R}$ are such that $x^4 + ax^3 + bx^2 + ax + 1 = 0$ has at least one real solution

I just wanted the solution, a hint or a start to the following question. Determine the minimum of $a^2 + b^2$ if $a$ and $b$ are real numbers for which the equation $$x^4 + ax^3 + bx^2 + ax + 1 = 0$$ ...
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59 views

Checking whether a solution to MIP is optimal

Consider a binary integer program \begin{align} \min \quad &\sum _{j \in J}f_j x_j +\sum _{i \in I} c_i y_i \notag \\ \mbox{s.t.} \quad &\sum _{j \in N_i} x_j \ge 1-y_i, \quad \forall i\in I ...
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896 views

Max- Min Optimization problem

I am a noob in mathematic, so I would need your help in solving the optimization problem below \begin{array}{l} \max\limits_{\bf l} \min \left( \left| {\bf g}_1 {\bf Ml} \right|^2, \left| {\bf g}_2 ...
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Algorithm of projection

Suppose $S$ is a compact surface in $\mathbb{R}^{3}$ defined by a sufficiently smooth level set function $f$, that is, $S=\{s: f(s)=0\}.$ I am studying an algorithm that projects a point $x_{0}$on ...
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On the bounds of the objective function in a standard LP

Consider a standard linear programming (LP) such as: \begin{align} \sum_{i=1}^{N}\frac{a_{i}}{b_{i}}x_{i}\end{align} \begin{align}\text{s.t. }\left ( \sum_{i=1}^{N}x_{i}=1 \; , \; ...
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235 views

Estimate Euler angles between rotated coordinate system via Newton-method based on position vectors

I've got $N$ position vectors $\mathbf{a}_i = \begin{pmatrix} a_{i,x} \\ a_{i,y} \\ a_{i,z} \end{pmatrix}$ in one coordinate system and $N$ corresponding position vectors $\mathbf{b}_i = ...
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60 views

How is it possible to use normals in the definition of a linear programming constraint?

I'm trying to calculate the center of a feasible region in a system of linear inequalities using linear programming techniques. After a bit of research, it looked like defining the center as a ...
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130 views

Local Extrema and Global Extrema

When we have a convex function we know that a local minimum is a global minimum, and similarly for a concave function. What are some other situations where finding local extrema can yield global ...
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85 views

Maximum of a trigonometric Polynomial

Given $$x+y+z=\pi$$ $$3\sin(x)+4\sin(y)+18 \sin(z)=A$$ Question:find maximum of $A$. I spend so many time on this question. answer is $ 35\sqrt{7} /4$, but why?
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Maximizing a continuous recursive function

So I've been working at this for a while and have so far been unable to find any resources on maximizing a particularly strange function that I've been trying to deal with. The function is of the form ...
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Optimizing with Many Langrangians

I'm working on a problem where I've got to minimize the following: $\sum\limits_{i=1}^n(a+c\sqrt{(y_i^2+1)}+dy_i-v_i)^2$ with the following constraints: $0\leq c \leq 4e$ $|d| \leq c \mbox{ and } ...
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Confusion related to derivative

What is the gradient of $||x + b||_1$ with respect to $x$? Here $x$ is the variable and $b$ is the constant.
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Confusion about the implementation of thresholding operation

I was reading this paper. I didn't get the application of thresholding operator here I didn't get how the -c part came in the solution $\mu = -c + S(c-b/a, \lambda/a)$
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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 ...
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258 views

optimization of a non-differentiable, component-wise step function

I would like to estimate the (local) minimum of a function $c:R^N \mapsto R^+$ where: $c$ is only differentiable almost everywhere, there exists a component $j$, such that $\frac{\partial ...
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28 views

Is monotonicity of $g$ necessary and sufficient for preserving critical points of $f$ in $g(f(x))$?

Problem 1. I want to find the argminof $f(x)$. Suppose solving $f'(x)=0$ is too difficult. Instead, solve Problem 2: optimize $g(f(x))$, (presumably by solving $g'(f(x))\cdot f'(x)=0$?). For what ...
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Looking for a window containing the solution of an equation

I need to solve billions of times equations $\,f(x)=0\,$ with $$f(x) := \sum_{i=1}^N \frac {z_i}{c_i + x}$$ All $z_i$ are positive and add to $1$. Among the $N$ coefficients $c_i$, $M$ are negative ...
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Prove or disprove this argument

Let $L>0$ and let $\Omega$ be the set of all integrable functions from $[0,L]$ to $]0,+\infty[$. For all $\varphi, \psi \in \Omega$ define $\left \langle \varphi,\psi \right ...
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354 views

Optimizing trigonometric and nonlinear functions

First, Please, keep in mind that I'm a programmer not mathematician, and I have a fair mathematical background. I used optimization in Java to fit some observations to a trigonometric function, I ...
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76 views

online learning to maximize profit

I have a software which takes input as investment and gives the output as return on a particular stock. Now profit metric $x_i$ is defined as the ratio of return $g_i$ to maximum possible return ...
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87 views

A stochastic programming with a chance constraint

Let $X$ be a bounded positive variable with an unknown probability density function (PDF) and $f(X)$ be a differentiable positive function. $$\begin{align*} &\min/\max ...
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88 views

Minimize $\sum_i \arccos(v_i\cdot x)^2$ subject to the constraints $\|v_i\|=1$ and $\|x\|=1$?

Some background: (skip to the end for the actual question) Recently I have been trying to define some notion of an average of points on the surface of a sphere. My original idea was to ignore the ...
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Optimal Configuration for a Set of Points

Consider a set of $n$ points on the plane with positions $\mathbf{p}_1,\dots,\mathbf{p}_n$, such that each point $i$ has at least one neighbor $j$ at a distance of no more than $\lambda$ away from it ...
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118 views

A maximization problem in Sobolev space

For $k>0$, let $f_k$ be a sequence of positive functions in $H_N^1(0,1)$, where $H_N^1(0,1):=\{u\in H^1(0,1)|u^{'}(0)=0=u^{'}(1)\}$, $H^1(0,1)$ is the usual Sobolev space consisting of $L^2(0,1)$ ...
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What is an approach for optimizing the values of a matrix?

My apologies if I get some terminology wrong, I don't have a formal math background; half my problem is articulating what I'm trying to do and identifying the domain of math that deals with this kind ...