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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partial derivative of a vector with respect to a variable

I have a vector in the following form $\mathbf{w}^T = [a_1*w_1, a_2*w_2, \dots, a_d*w_d]$ what is the partial derivative of $\mathbf{w}$ with respect to $w_j$ ? (1 or 2) $\frac{\partial ...
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Max no. of piece in k cut

Suppose I have large piece of rectangular sheet. Cutting is allowed only vertically and horizontally. My approach is if no. of cut is even then max. no of piece is (n/2)*(n/2) if no of cut is odd ...
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65 views

Solving LP with two $L_1$ inequality constraints

Is there a "fast" way to solve the following LP formulation with the following constraints: $$ \max_{\mathbf{f}} \mathbf{f}'.\mathbf{g} \\ \mathbf{1}'\mathbf{f}=1\\ \|\mathbf{f}-\mathbf{h}\|_1\le ...
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Maximum Area of a Triangle when 1 Side, Perimeter Known

This is an example of a "quantitative comparison" question the GRE would test. Suppose the following information is known: one side of a triangle has length 12 the perimeter of the triangle is 40 ...
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squaring the equality constraints

When creating an unconstrained optimization problem from an equality constrained one, the usual way to build the Lagrangian, is by adding a term consisting of a multiplier, multiplied by the equality ...
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Proving boundedness of a function (part 1).

Consider the function \begin{eqnarray} f(x_1,x_2,\cdots, x_n) = \frac{\sum_{i}^{n}a_ix_i}{\sum_{i}^{n}b_ix_i}, \end{eqnarray} over the set $S = \{x := (x_1,x_2,\cdots, x_n):-1 \leq x_i \leq 1,\; ...
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Multivariable calculus max/min [closed]

Find and classify the critical points of this function: $f(x,y)= (x^y)-(xy)$ in the domain $x>0, y>0$. I am having trouble treating x and y as constants when taking partial derivatives.
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Optimization on area , rectangle with fixed length on 3 sides.

I've stumbled with the problem below "Some unused land is adjacent to a straight canal. A gardener wants to use 200 meters of fence in order to create a rectangular garden, by using the fence ...
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fence a circular land and a square land.

With a wire mesh of 1000 mts divided into two parts , we want to fence a circular land and a square land. a)Calculate the lengths of each of the parties such that the total area enclosed is ...
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Maximize/minimize $1/3 x^3 + y$ with constraint $x^2 + y^2 = 1$?

I keep running around in circles when I use the Lagrangian multiplier method getting $x = 1/y$ But then when I substitute $(1/y)^2 + y^2 = 1$ I then get $1/y^2 + y^2 = 1$ and this doesn't give me ...
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Google Code Jam's Cookie Clicker Program…

Today, the Google Code Jam's cookie clicker problem was something like this. Problem In this problem, you start with 0 cookies. You gain cookies at a rate of 2 cookies per second, by ...
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Solving Problem by different Method ( non-induction)

I have this problem , which I was able to prove it by induction, but I wonder could be solve by direct method ( for example combinatorial method). I want to find number of solution for $$0 \le ...
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Maximizing the Magnitude of the Resultant Vector

Given a set of $n$ two-dimensional unit vectors: $\left\{ \mathbf{v}_1, \dots, \mathbf{v}_n \right\}$, I want to find the coefficients $\left\{ \alpha_1, \dots, \alpha_n \right\}$, $0 \leq \alpha_i ...
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find maximum value of a function involving factorials

Find the maximum value of $\large \frac{35^n}{n!}$ for any positive integer $n$. How I can solve this problem using calculus? Is there any other method to solve this kind of problem?
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Finding the Absolute Maximum and Minimum of a 3D Function

Find the absolute maximum and minimum values of the function: $$f(x,y)=2x^3+2xy^2-x-y^2$$ on the unit disk $D=\{(x,y):x^2+y^2\leq 1\}$.
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If $0\le x\le 1$ and $0\le y\le 1$, find $\max\{(x^2y-y^2x)\}$

If $0\le x\le 1$ and $0\le y\le 1$, find $\max\{(x^2y-y^2x)\}$ My work: Though I could not approach the problem, I tried to find out a few facts. So,I defined the above expression as ...
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Prove that so and so is $O(x^4)$

Given $f(x) = x^3 + 20x + 1$, how would I prove this is $O(x^4)$? By definition, the function is $O(x^4)$ iff $f(x) <= cn^4$, where $c$ is some constant. However, I'm not sure where to go from ...
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A question on minimizing $\| . \|_2^2$ vs $\| . \|_2$

Suppose we are in $\mathbb{R}^n$ Is the problem of $d(x,Y) = \inf\{ \| x - y\|^2 : y \in Y\}$ equivalent to $d(x,Y) = \inf\{ \| x - y\| : y \in Y\}$ Pardon me, let us keep it simple and just stick ...
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106 views

Bounded logarithmic function

I am trying to find any function that it grows logarithmically up to a certain point, and after that point it remains constant. Can anyone help me with that
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Help finding local extrema of $f(x)=\frac{x}{\sqrt{2}}-3\sin\frac{x}{2}$

Find the local extrema of $f(x)=\dfrac{x}{\sqrt{2}}-3\sin\dfrac{x}{2}$ on the interval $0 \leq x \leq 2\pi$ $f^{\prime}=\dfrac{1}{\sqrt{2}}-3\cos \left(\dfrac{x}{2}\right) \left ( ...
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51 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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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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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 $x^2 + y^2 + z^2 ≤ 1$.
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Finding minimum value of multi-variable expression without partial derivatives

Minimize where $a$ and $b$ are positive real numbers $\sqrt{a^{2}\; +\; 4}\; +\; \sqrt{\left( 3-a \right)^{2}\; +\; \left( b-2 \right)^{2}}\; +\; \sqrt{25\; +\; \left( 6-b \right)^{2}}$ I could take ...
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100 views

$f$ does not have extrema at the Lagrange multiplier value

I have the function $2x^2+y$ and one constraint $x-y^2=1$ and want to find maximum value by lagrange multiplier. Intuitively, I see the point $(2,1)$ satisfies $c$ and have value of $f(2,1)=9$. ...
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479 views

Absolute maximum and absolute minimum of f(x)= ln x on [1,2]? [closed]

Can somebody help me with this one. Find the absolute maximum and absolute minimum of $f(x)$ = $ln(x)$ on $[1,2]$.
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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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how to solve this equation containing “min”?

$$ xy=128,x+y=\min $$ How to find $x$ and $y$ with the minimum sum? This example is simple and can be done by brute forcing but I want to know what is the proper way of solving it. How to solve ...
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Second Order Differential Equation - finding maximum and minimum values of particular integrals

Given that $y$ = $\frac{3}{4}\cos3x + \frac{1}{4}\sin3x$ is a particular integral of the differential equation $$\frac{d^2y}{dx^2}+ 4\frac{dy}{dx}+13y = 6 \cos3x-8\sin3x$$ how do I show that ...
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Minimum number of money to make each element in list greater than or equal to 0?

Given list with positive and negative integers.We have to make each element greater than or equal to zero.There are two types of moves first increase all elements by 1 requires P unit of money, second ...
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302 views

Newtons method and finding stationary points

I have an equation $l(x) = \sqrt{(x - 0.2)^2 + (x^2 - 2.7)^2}$. Now I basically want to find at which x coordinate that $l(x)$ will be it's smallest. I have differentiated the equation to find ...
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finding maximum perimeter of a triangle

So, here we are given task to find maximum perimeter of a triangle with a given base 'a' and given vertical angle 'x' , now how should I proceed in given problem its confusing me Now supposing ...
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Minimization problem

For Which positive value(s) of $x$ the following function is most minimum $f(x) = x^2 + ax +c$ [ where $a ,c > 0$ ] [note : I know there is no positive $x$ for which $f(x)$ is minimum but I ...
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129 views

find rotation needed to minimize distance of pairs of vectors

Given two sets of vectors, how can I find the rotation needed to rotate one set onto the other? The sets are ordered and of same size, with vector n from set 1 corresponding to vector n from set 2. ...
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458 views

How to prove equality from poincare inequality?

Let $$D = \{y \in C^1(0,1) : y(0) = y(1) = 0\}$$ Suppose there exists a $C_0$ such that $$\int_{0}^{1} y^2 \ dx \leq C_0 \int_{0}^{1} (y')^2 \ dx$$ for all $y \in D$, and for all $C < C_0$ ...
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Finding the sequence that maximizes a constrained sum

Let $0 < a < 1$ and let $S_k$ be a unknown sequence of such that $S_k > 0$ and $$ S_n + S_{n-1} + \ldots + S_1 = C = constant. $$ What should be $S_k$ so that the sum $$ S_n + aS_{n-1} + ...
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The closes point to a curve in space.

I am working on the following problem. Find the point closest to the origin, of the curve of intersection of the plane $2y+4z =5$ and the cone $z^2 = 4(x^2+y^2)$ I was able to see that the ...
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Show that $\min\{a_1,a_2,…,a_n\}$ is maximum when $a_1=a_2=…=a_n$.

Given $a_1,a_2,...,a_n\in\mathbb R$, and $a_1+a_2+...+a_n=A$. Show that $\min\{a_1,a_2,...,a_n\}$ is maximum when $a_1=a_2=...=a_n$. I feel this is quite a common sense but I don't know how to ...
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151 views

Maximizing triangle area

Here is the problem: We start with a triangle ABC with area 1. We choose a point (F) on side AB, then someone else chooses a point (G) on side BC. We then choose the last point (H) on side CA. Our ...
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754 views

Optimization and window area

A Norman window has the shape of a rectangle with a semi circle on top; diameter of the semicircle exactly matches the width of the rectangle. Find the dimensions of the Norman window whose perimeter ...
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Prove a set in $\mathbb{R}^2$ is convex.

Let $$\Omega = \{(x_1,x_2)\in\mathbb{R}^2:x_1^2-x_2\leq 6\}$$ Prove that $\Omega$ is a convex set from first principles using the convex combination. edit: Thanks Ewan for that, but I am trying ...
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319 views

extremum values of a function in three variables

Consider the function $f:\mathbb{R}^3\rightarrow \mathbb{R}$ is given by $$f(x,y,z)=y^2+xyz+x^6$$ Does the function have a local maximum or a minimum at the origin? My question is is there a ...
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51 views

Maximising an expression

We are to maximise $x^{2}y-y^{2}x$, where $x,y \in [0,1]$. I've tried using AM-GM to find another (easier to maximise) expression, which gave me $xy(x-y) \le \frac{1}{2}(x^{2}+y^{2}) (x-y)$ but that ...
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3answers
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Linear Programming Problem Using the Two-Phase Method

I have been given the following LP problem and asked to use the two phase simplex method to solve it. I believe there isn't a solution, but would anyone be able to confirm this for me? Thanks. max ...
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358 views

Numerical optimization with nonlinear equality constraints

A problem that often comes up is minimizing a function $f(x_1,\ldots,x_n)$ under a constraint $g(x_1\ldots,x_n)=0$. In general this problem is very hard. When $f$ is convex and $g$ is affine, there ...
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928 views

Proving the Kantorovich inequality

Nevermind! I worked it out. This question seems really silly now. I am working through the proof of the Kantorovich inequality on pages 6 and 7 of the following lecture notes: u.mich optimisation ...
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138 views

Maximizing Volume [duplicate]

Possible Duplicate: Maximizing volume of a rectangular solid, given surface area Maximize the volume of a rectangular solid, given that the sum of the areas of the six faces is 6a^2 for a ...
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362 views

General method to find inf, sup, maxs and mins of a function

Could someone explain how to find inf, sup, max and min values of a function (real-valued functions of real variable, generally continuous/differentiable, with some possible points of discontinuity)? ...
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45 views

How do I solve a LP problem when constrains have different inequalties?

How can I solve this LP problem: Maximize p=x subject to : x+y <=30 x-2y <= 0 2x+y >=30 x>=0 , y>=0 using simplex method?
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808 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) ...