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 Problem: Karush-Kuhn-Tucker Condition

I am working on the question displayed below. I know that the method of Lagrange Multipliers is used to find the solution for optimization problems constrained to one or more equalities and when our ...
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53 views

Maximum of the function of multivariable?

I need to find the maximum of the function given by $z=x^3+xy$ in $A=[0,1]\times[0,1]$. I think I need to use partial derivatives, but I'm not sure exactly how.
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36 views

Richardson method optimal parameter

The question I am about to ask came to my mind when I was analyzing Richardson method with symmetric and positive definite matrices. But it is really about simple math I somehow can't defeat. Given ...
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146 views

References on constrained least square problems?

I have met some constrained least square problems, for example, my last post. I found that there are various methods for slightly different constraints, and still I often had little clue about how to ...
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2answers
113 views

Local extrema for the function $f(x,y)= x^2+y^2 e^{x^2} + x\sin x$?

I would like to find the stationary points if they exist and so I start by finding the partial derivatives for $x$ and $y$ and equal them to zero and from the second equation I know that $y=0$ but I ...
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54 views

build finite automaton for language minimize states

I want to build a finite automaton that accepts $a^nb^n, n \gt 0, m \ge 0$. I can't do it unless the FA has two final states, i.e.: $delta(q0, a) = q1 delta(q1, a) = q1 delta(q1, b) = q2 delta(q2, ...
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Optimization using Quadratic form [closed]

How do I find the maximum value of $$x^2+xy+2y^2$$ subject to the constraint $x^2+3y^2=16$?
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1answer
37 views

Proving an optimization problem

I have the following problem I would need help with :) : Given $N$ positive values $u_n$, where $n=1, ..., N$, the problem is to determine the unknown parameters $w_n$ to maximize the criterion ...
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55 views

Help with least squares in paper on hough transform

I'm trying to understand this paper. I'm having trouble with a least squares problem. In the paper, equation (4) is $ E^2 = \sum_{r\in R}{\sum_{c \in C}{[ \alpha r + \beta c + \gamma - I(r,c)]^2 ...
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64 views

Optimization problem defined by polynomials only always leads to algebraic solutions?

Let $\Omega$ be a non-empty set in ${\mathbb R}^n$ defined by a set of polynomial inequalities with rational coefficients $P_i(x_1, \ldots ,x_n) \gt 0 (1 \leq i\leq m)$ and $Q_j(x_1, \ldots ,x_n) \geq ...
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83 views

Theory of Moments: Notation

I try to read some papers about Moment Matrices/Optimization over polynomials, but I have some troubles with the following notation: Let $P(V)$ be a power set of some $V=\{1,2,...,n\}$, how does a ...
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Rigorous proof of the “Lagrange-multiplier theorem”

From Marsden's Elementary Classical Analysis: Theorem 8 Let $f\colon U \subset \Bbb R^n \to \Bbb R$ and $g\colon U\subset \Bbb R^n \to R$ be given $C^1$ functions. Let $x_0\in U$, ...
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38 views

Proving an inequality involving multiple constraints

Let $R$ be a discrete set and let $f:{\left[ {0,1} \right]^{\left| R \right|}} \times {\left[ {0,1} \right]^{\left| R \right|}} \to \mathbb{R}$ be defined as $f\left( {{\mathbf{x}},{\mathbf{y}}} ...
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1answer
88 views

Constrained maximization of Leontif utility function $\min(x_1, x_2)$

The maximization problem is: Maximize $u(x_1, x_2) = \min[a_1x_1, a_2x_2]\; \ \text{s.t.}\;\; p_1x_1 + p_2x_2 \leq$ $w$, in which $x_i, p_i$ is the amount and price of good $i$, $w$ is the ...
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228 views

Prove inequality $(x+y+z-2xyz)^2 \le 2$

Problem: Prove inequality $(x+y+z-2xyz)^2 \le 2\ (1)$ with $x^2+y^2+z^2 = 1 \land x,y,z \in \mathbb R$ I tried expand $LHS$ and have: $$(1)\iff 1 - 2 (xy+yz+xz) + 4 xyz(x+y+z)-(2xyz)^2 \ge 0$$ ...
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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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37 views

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 ...
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1answer
97 views

Can a derivative be undefined at a local maxima?

If $c$ is a local maxima x-value for $f(x)$, then can $f'(x)$ be undefined? Or does it always have to be zero? For example, is $x = 1$ a local maxima for $1/x$. Also how do you check if $c$ is a ...
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120 views

Optimization calculus Problem

I have a calculus final coming up and was going over a practice exam my professor gave me and I came across a problem I was struggling with. I would post a picture but I am having trouble posting a ...
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134 views

maximize the expected value of the logarithm of the weighted average of random variables

I'm trying to do the following. $$\max_{m\in\mathbb{R}} \mathbb{E}\left[\log (wA + (1-w)B_m)\right],$$ where $0<w<1$ and $A, B_m > 0$ are correlated random variables. $A$ does not depend ...
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Solve Ax = b, but I have a function that implements A

I have an overdetermined linear system $Ax = b$. I need to choose an $x$. $x$ has about 100 elements in it. If I had the matrix $A$, I would set x equal $A^\dagger b$, the pseudoinverse of $A$ ...
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115 views

Jacobian in Levenberg-Marquardt for 4-Parameter equation

I am trying to fully understand how I can use Levenberg-Marquardt to minimise a 4 parameter equation. There are lots of fancy programs to do this but the documentation about the mathematics is ...
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What optimization method should I use here?

Suppose that we have the following (unconstrained) optimization problem: $$ \min_{\mathbf{w},b} \frac{1}{2}\Arrowvert\mathbf{w}\Arrowvert^2 + C\sum_{i=1}^{l}g_i(\mathbf{w},b), $$ where ...
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281 views

Example calculation of estimating GMM parameters using EM

I'm trying to study expectation maximization and I've almost got the idea. What I'm missing is a concrete example. Could someone familiar with the subject give me a concrete example how one would ...
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1answer
156 views

Prove or disprove the conjecture about the function below.

After thousands of numerical tests we stated the conjecture that their is exactly one local extremum of the function below. $$ {\rm f}\left(w\right) = {1 \over 2}\sum_{i = 1}^{n}\left({1 \over 1 + ...
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Area of inscribed rectangle as a function of right triangle base

I am stuck with trying to find the area of the rectangle $A$ as a function of $x$: (Drawing replicated from my textbook, not sure if more labels would be helpful?) Obviously $A$ is given by the ...
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69 views

a question about Lagrange multiplier?

Q)Given $x_1+x_2+...+x_n=a$ where $a>0$, find the extremum value of $f(x_1,x_2,...,x_n)=x_1^k+x_2^k+...+x_n^k$ Also, find the range of $k$ in which the extremum value of $f$ is a maximum ...
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Help with Optimization Problem: Matrix Calculus

Can someone please help me with this problem? I am clueless :( $$ \left\{ \begin{array}{rclrcl} \min f(u) &=& u^tAu\\ \text{s.$\,$t.} \sum_{j=1}^n u_j &=& 0,& ...
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Method for the minimum of $x\cdot\sin(1/x)$?

I have the function $f(x)=x\cdot \sin\Bigl(\dfrac{1}{x}\Bigr)$ $(\mathbb R^*\rightarrow \mathbb R)$. Is there a method to find the global minimum of $f$? Thanks
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Optimization problem: Finding the maximum value

Can someone please give me a hint on this problem? I want to find the maximum value of y, given the equations:
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(How) Can I prove this sum of simple rational functions does not have more than one maximum?

I want to find out whether functions $$f(x) = \sqrt{\prod^J_{j=1} {a_j x + b_j}} \cdot x^{c_1+1} \cdot \exp(- c_2 x);\; a_j, b_j, c_1, c_2 >0;\; x \geq 0$$ are at most unimodal for positive $x$. ...
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maximize $x^a + y^b$ subject to $p_1x+p_2y=w$ utility max.

This is a utility maximzation problem maximize $x^a + y^b$ subject to $p_1x+p_2y=w$ (utility maximization problem) Anyone has any idea, there are no restrictions on $a$ and $b$, as far as i can see ...
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334 views

How to solve nonlinear constrained optimization in Matlab?

I have to solve a nonlinear constrained function in matlab, and I am not familiar with it's commands. the problem is: minimize $E(b,c)$ constraints: $k1< c\sqrt{b}< k2 ; c/6>k3$ Note: E(b,c) ...
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How to find the minimum of the function?

How to find the minimum of the following function $$ {\rm f}\left(w\right) = {1 \over 2}\sum_{i = 1}^{n}\left({1 \over 1 + {\rm e}^{-x_{i}\,w}} -y_{i}\right)^{2} $$ where $x_{i}, y_{i} \in \left(0, ...
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Maximizing a sum of inner products

Someone asked this question on a French maths forum here and it caught my attention. The question is the following: let $(E, \langle \cdot, \cdot \rangle)$ be a Euclidean vector space. Find the ...
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396 views

Compute the minimum distance between the centre to the curve $xy=4$.

I wish to solve the following problem: Compute the minimum distance between the center to the curve $xy=4$. But I don't know where to start from?
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Optimization Problem

Can someone please help me with this minimization problem? I dunno what to do after replacing p(x) with given s.t.
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Maximal area of a triangle

What would be the most elementary proof of the following: A triangle has been drawn inside the circle with radius $r$ and its area is as large as possible. Prove that the triangle is equilateral. I ...
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3answers
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Minimum value problem

Find the minimum value of $(x+y)(y+z)$ where $x,y,z$ are positive real numbers satisfying the condition $$xyz(x+y+z)=1$$ Hint?
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Optimization, descent direction, neccessary condition

I'm learning about nonlinear, unconstrained optimization. In my book it says that a descent direction $p_k$ must satisfy: $$p_k\nabla f(x_k)^T < 0$$ This seems to mean that $p_k$ must be obtuse to ...
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Perimeter equation

A window is in the form of a semicircle surmounted over a rectangle. Thew total perimeter of the window is 12m. Note:This is a part of a maxima minima question that I was trying to solve. I could ...
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Is the Support Vector Classifier in some sense optimal?

My question is, is the original hard-margin support vector classifier optimal in some sense? If you have an answer that refers to the soft-margin SVC instead, I'd also be interested. I know that the ...
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Selecting a subset from a set such that a given quantity is minimized

Let $A$ be is a set of some $p$-dimensional points $x \in \mathbb{R}^p$. Let $d_x^A$ denote the mean Euclidean distance from the point $x$ to its $k$ nearest points in $A$ (others than $x$). Let $C ...
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Minimization problems (using Lagrange Multipliers/Directional Derivatives)

The first problem asks to find the minimum cost of a rectangular box if the bottom costs \$2, the sides \$5 per square foot, and the top costs $7 per square foot. The volume of the box is given as 20 ...
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Minimization via eigendecomposition of Hadamard matrix products

Let $\boldsymbol{\mathcal{R}}$ and $\boldsymbol{\mathcal{M}}$ be $n\times n$ Hermitian matrices (which are known) and let $\boldsymbol{\mathcal{G}}$ be a rank one $n\times n$ (unknown) Hermitian ...
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448 views

How to find the minimum of $a+b+\sqrt{a^2+b^2}$

let $a,b>0$, and such $$\dfrac{2}{a}+\dfrac{1}{b}=1$$ Find this minimum $$a+b+\sqrt{a^2+b^2}$$ My try: since $$2b+a=ab$$ so ...
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Counterfeit coin problem.

Problem $N$ coins, $N-1$ equal coins and one heavier counterfiet coin. With a balance beam given we want to find the counterfeit coin. We have two different goals: minimise the expected number of ...
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Optimization Word Problem, revenue

A sorority plans a bus trip to the Great Mall of America during Thanksgiving break. The bus they charter seats 44 and charge a flat rate of 350 dollars plus 35 dollars per person. However, for every ...
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212 views

Find the minumum using Newton-Raphson

I have the following function: $f(x) = 100(x_2 - x_1^2)^2 + (1-x_1)^2$ I have to find the minimum of this function using the Newton Raphson method. The point where I have to start is $x = [1.2$, ...
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1answer
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maximum value of $f(x)= 13x^3 + 5y^2 + 6yz + 5z^2$ [closed]

Find the maximum and minimum values of $ f(x, y, z) = 13x^3 + 5y^2 + 6yz + 5z^2 $on the solid ball $x^2 + y^2 + z^2 ≤ 1$.