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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Effecient way to find optimal solution in a 2 player game

I have a function: \begin{equation*} f(a_1,\ldots,a_7,b_1,\ldots,b_4)=-14-7 a_1+30 a_1 a_2-7 a_4-2 a_4 a_5+21 a_6+21 a_7+16 a_1 b_1-24 a_1 a_2 b_1+6 a_4 b_1-6 a_4 a_5 b_1+6 a_1 b_2-6 a_1 a_2 b_2+8 a_4 ...
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35 views

Estimate the number of Local Minima

I am asking this question about local minima, but actually I started by trying to find the global maximum/minimum over a compact set, of a smooth function (the objective). The function has a random ...
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4answers
1k views

Calculating the max and min of $\sin(x)+\sin(y)+\sin(z)$

I took the partial derivatives of $\sin(x)+\sin(y)+\sin(z)$ and it didn't work out, so I am trying to use Lagrange's method (with the constraint: $x+y+z=\pi$)... I am not sure how to set this up. ...
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0answers
156 views

Multivariable calculus: optimizing for shortest path along a curvy plane?

I want to write a computer program which can help me spend the least amount of energy and time walking between locations on my university campus. My campus is very hilly, and it is also extremely hot ...
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1answer
34 views

Can we minimize $m\log_m{n}$, given $n$?

If we are given $n$, a positive real, can we find a the positive real $m$ that minimizes the function: $$m\log_m{n}$$ I'd prefer to find the function that gives a value for $m$, but I'm also ...
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1answer
85 views

Linear System with constrained solutions

After a model my problem I found a rectangular linear system : $$Ax=b$$ I can easely solve it with a least square with QR/SVD... But the model include constrains for each solution $x_i$, the $\vec{x}$ ...
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337 views

Looking for a significant example that highlights the suboptimality of the greedy algorithms

A week from now, I'll have to present my work to a bunch of coworkers who aren't used to the optimisation world and terminology. One of the main algorithms I implemented uses a greedy type algorithm ...
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81 views

Methods to minimise multilinear functions with trilinear, quad-linear and higher-linear terms?

My goal is to minimize functions such as $$f_1(\mathbf{p})=p_1p_3p_7+p_1p_4p_7+p_2p_3p_7+p_2p_4p_7-p_1p_3p_5p_6-p_1p_4p_5p_6-p_2p_3p_5p_6-p_2p_4p_5p_6$$ and ...
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5answers
67 views

Can we minimize $3^m 2^{n/m}$, given $n$?

If we are given $n$, a positive real, can we find a the positive real $m$ that minimizes the function: $$3^m 2^{n/m}$$ I'd prefer to find the function that gives a value for $m$, but I'm also ...
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2answers
53 views

Limits/partial derivative

Please help me with this question. It's topic on limits and partial derivative. Find the maximal value of $f(x,y)=xy((1-x^2/a^2-y^2/b^2))^{0.5}$ for $a=74.4$, $b=64.8$. Round off your answer to $4$ ...
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72 views

Numerical Methods

Assuming I am given a Program which can calculate the value of a continuous, infinitely differntiable (we cannot calculate these derivatives), real, positive function of two real variables which has ...
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2answers
59 views

minimizer of c/x

It seems to me that the solution to the optimization over $\lambda \in \mathbb{R}^n$ \begin{align} & \underset{\lambda}{\arg\min} \sum_i c_i/\lambda_i\\ \textbf{s.t } & \sum_i \lambda_i = 1\\ ...
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1answer
283 views

Find max/min points for multivariable functions

I have a question about the general procedures to find the max/min points for multivariable functions, would really help if somebody could please clarify my doubts. So for single variable function, ...
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0answers
79 views

Conditions to satisfy trigonometric inequality

I'm looking for sufficient (and necessary would be good too) conditions on $a,b,c$ such that \begin{align} a\cos\phi + b \cos 3\phi + c \cos 5\phi \geq -1 \hspace{20pt} (\forall \phi) \end{align} ...
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497 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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1answer
52 views

What is the name of this method used to solve a nonlinear problem?

The lecturer taught this method in my Optimization and Control Theory Class and I wasn't quite there when he named it. Could you help me out? He gave the following example of the method in class: ...
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3answers
99 views

Is there another method?

If $x$ and $y$ are positive numbers such that $x + y = 1$, find the maximum value of $x^4y + xy^4$. I could do this problem my simplifying the expression to $xy(1-3xy)$ and taking $k=xy$, forming ...
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0answers
47 views

Least square with constraints

I want to solve the least squares problem $(Ax-b)^2$ with no intercept term for linear regression with the constraint that the sum of the params/weights is equal to 1. I am trying to get the closed ...
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0answers
126 views

Lattice Reduction Problem: Minimizing the “Longest” Basis Vector

Suppose we have a basis for an integer lattice formed by the vectors $\vec v_1, \vec v_2, \ldots,\vec v_n$. Then let $A$ be the augmented matrix $( \vec v_1| \space \vec v_2| \cdots |\space \vec ...
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3answers
355 views

What is the global maximum of $x^{1/x}$

Let the following function be defined as such: $$F_x: \Bbb R \to \Bbb C, x \mapsto x^{1/x}, \forall x \ne 0$$ What I want to know is $$\max_{x<0}\Re\left(F_x\right)=\,?$$ and ...
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1answer
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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114 views

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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170 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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647 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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2answers
326 views

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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1answer
845 views

I Need Help Finding the Area of the Largest Trapezoid that can be inscribed in a circle

Im currently learning how to maximize areas. theres a question that I'm stuck on Find the largest trapezoid that can be inscribed in a circle of radius 2 and whose base is the diameter of the ...
0
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1answer
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 ...
4
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1answer
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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112 views

How to solve this minimization (maximization)?

I'm facing this problem: $$ \large \min_{x \in \mathbb{R}_+^3} \max \left\{ { \sum_{i=1}^3 x_i^2-2 x_1 x_3 \over \left(\sum_{i=1}^3 x_i \right)^2} , { \sum_{i=1}^3 x_i^2 + 2 (x_1 x_3 - ...
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1answer
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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717 views

How to find maximum and minimum volumes

I would appreciate if somebody could help me with the following problem: Q: Let $S$ be the region bounded by the curves $y=\sin x \ (0 \leq x \leq \pi)$ and $y=0$. Let $V(c)$ be the volume of the ...
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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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1answer
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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1answer
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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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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Software for Binary Integer Linear Programs

I am aware that there is good software out there to solve integer linear programs (ILPs). However, is there (preferably free or low cost) software I could use to solve large binary integer linear ...
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142 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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36 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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2answers
300 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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29 views

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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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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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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1answer
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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1answer
157 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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1answer
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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1answer
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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1answer
593 views

How to justify the solution of this problem?

Assume $\mathbf{x} \in \mathbb R_+^N$ with support $P=\{p_1,p_2,\cdots,p_K\}$ ($P$ is unknown). We already know that $$f_1(\mathbf{x}) = f_2(\mathbf{x}) = \cdots = f_{N-1}(\mathbf{x})$$ where ...
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126 views

Optimal tax rate

Suppose you have two countries A and B, with a tax rate $T_A$ and $T_B$, respectively. The tax is redistributed to all people equally. Hence if you live in A and you make $I$ as income then you will ...
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1answer
870 views

Local extreme value & saddle point: multi variable calculus

I am asked to find all local extreme values & saddle points of $$f(x,y) = 2x^2 + y^2 - xy - 7y + 8$$ $$f_x(x, y) = 4x-y, \qquad f_y(x,y) = 2y-x-7$$ $$f_x(x,y) = 0, \qquad y = 4x$$ $$f_y(x,y) ...
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1answer
904 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 ...