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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Find and Evaluate Critical Points

I need to find al the critical points of the following function f(x,y)=$y^2-x^2y-3y+x^4-x^3$ Determine if they are local minima, local maxima, or saddle points, by looking at the Hessian matrices ...
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Maximising with multiple constraints

I have $$Z=f(x_1 ,x_2 ,x_3 ,... ,x_n)$$ function and $$\left[\begin{array}{r}c_1=g_1(x_1 ,x_2 ,x_3 ,... ,x_n) \\c_2=g_2(x_1 ,x_2 ,x_3 ,... ,x_n)\\c_3=g_3(x_1 ,x_2 ,x_3 ,... ,x_n) \\...\\c_m=g_m(x_1 ...
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optimization over equations with max,min

Notation: $\max( x_1, \cdots, x_n )$ denotes the maximal number among $x_1, \cdots, x_n$. $\min( x_1, \cdots, x_n )$ denotes the minimal number among $x_1, \cdots, x_n$. Assumption: $x_i, ...
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Approximate an exponential function

I have an optimization problem, where I would like to minimize $$F=\exp(\mathrm{trace}(A)+\frac{1}{2}\mathrm{trace}(A^2)-\lambda)$$ where $A$ is a non-negative matrix. Is it possible to replace $F$ ...
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Gradient Descent on Non-Convex Function Works But How?

For Netflix Prize competition on recommendations one method used a stochastic gradient descent, popularized by Simon Funk who used it to solve an SVD approximately. The math is better explained here ...
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Extrema with Lagrange Multipliers

I have the following exercise: Define $f:\mathbb{R}^2\to\mathbb{R}$ by $f(x,y)=(x^2y)^{1/3}$. Is $f$ differentiable at $(0,0)?$. Has $f$ absolute extrema in $D=\{(x,y)\in\mathbb{R}^2:x^2+y^2\leq ...
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Finding the widest angle to shoot a soccer ball from the sideline using optimization

I'm trying to do an independent project for my Math class, but I was stuck and couldn't figure out how to use optimization to find position along the sideline that gives the widest angle to shoot. ...
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Controlled system question optimisation

Consider the controlled system $x_{t+1} = x_t + u_t + 3\epsilon_{t+1}$, where the $\epsilon_t$ are independent $N(0,1)$ variables. The instantaneous cost at time t is $x_t^2 + 2u_t^2$. Assuming that ...
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$AB$ is a chord of a circle $C$. Let there be another point $P$ on the circumference of the circle, optimize $PA.PB$ and $PA+PB$

$AB$ is a chord of a circle $C$. (a) Find a point $P$ on the circumference of $C$ such that $PA.PB$ is the maximum. (b) Find a point $P$ on the circumference of $C$ which maximizes $PA+PB$. My ...
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Finding an optimal function subject to some constraints.

I'm looking for hints on a problem I am facing. Not sure where to look to learn to solve such a thing or determine properties of the solution. I need to find an $f(x)$ that maximizes: ...
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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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187 views

Proofs of mathematical optimization theorems

How to show that if x minimizes f over S and x belongs to R, which is a subset of S, then x also minimizes f on R Please help me with this proof. Thank you.
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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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Find $\max \sum_{i=1}^9x_i$ when $\sum_{i=1}^9x_i^3 = 0$ and $|x_i|\leq1$

Given are the real numbers $x_1,x_2,\dots,x_9$ which satisfy the conditions $\sum_{i=1}^9x_i^3 = 0$ and $|x_i|\leq1$ Find the maximum value of $\sum_{i=1}^9x_i$ Intiutively the sum has its max. ...
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Highest (lowest) index of positive time-indexed variable

I have a simple problem involving a variable $x_{it}$ representing the amount of a resource allotted to a task $i$ in time $t$. The quantity of the (renewable) resource is constrained at a value $R$ ...
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The maximum of $xy^2z^3$ on $x+y+z=3$ [closed]

Given that $x+y+z=3$ [$x$,$y$,$z$ are positive real numbers] then prove that maximum value of $xy^2z^3$ is $\frac{27}{16}$.Is there any method possible than AM $\ge$ GM?
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Workflow Optimization

I originally posted the problem in stackoverflow but later on it become clear that it is more of a math problems then coding. Example: We have company doing support work on 3 projects (P1, P2, P3); ...
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Determination of the modulus of continuity

I'm trying to prove the uniqueness of the viscosity solution of an Hamilton-Jacobi-Bellman equation. Thanks to a classical result, I'm left to check if it exist a modulus of continuity $\omega_1$ --- ...
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Is there a name for this optimization algorithm?

I'm a software developer trying to design an optimization algorithm and I'm wondering if what I'm trying to do resembles any of these. There's a daunting number and rather than read each one, perhaps ...
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Closest Pair between 2 sets of powers

Assume $a$ and $b$ are natural numbers, $A=\{a,a^2,a^3,\cdots\}$ and $B=\{b,b^2,b^3,\cdots\}$, find $\min\left|a_i-b_j\right|$ where $a_i\in A$ and $b_j\in B$. For example, if $a=3$, $b=10$, then ...
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Adding a point to shortest path

If there exists a set of n points in a 2D coordinate system and an n-dimensional vector V ...
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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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How to solve for maximum area of a rectangle under a curve?

Having trouble with this optimization question and was hoping I could get some help with it. The function of the curve is $8^{-\frac{x}{5}}$. I would greatly appreciate a full explanation. I already ...
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Constraints in optimization; redundant hardness?

This is not an accurate mathematical problem, and rather a philosophical and ambitious question. As far as I know, unconstrained problems are easier than constrained problems; right? This is mostly ...
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Optimization problem (Sum of distances)

Given an ordered sequence $x_1 \leq x_2 \leq \cdots \leq x_n $ of length $n$ and a cost function $C(i) = \sum_{j}^{n}{\left|x_i-x_j\right|}$. The goal is to minimize the cost function. How do you ...
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Classical Newton method for minimization

For a quadratic convex function Classical Newton method for unconstrained optimization reaches the minimum point in one iteration. It this true? If so, what is the proof ?
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Semicontinuity of the product of two functions

Let $f(x)$ be a left continuous and non-increasing real-valued function. Can I prove that $f(x)x$ is upper semicontinuous?
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Proving $~\sum_{cyclic}\left(\frac{1}{y^{2}+z^{2}}+\frac{1}{1-yz}\right)\geq 9$

$a$,$b$,$c$ are non-negative real numbers such that $~x^{2}+y^{2}+z^{2}=1$ show that $~\displaystyle\sum_{cyclic}\left(\dfrac{1}{y^{2}+z^{2}}+\dfrac{1}{1-yz}\right)\geq 9$
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Finding velocity in optimization problem

Given $s=-16t^2+192t+144$, what is the velocity when $s=0$? This is part of a larger optimization problem which I solved, except for this last part. The critical point occurs at $t=6$, so after ...
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Fast solution to problems involving Lagrange multipliers

Suppose we have a function $f:\mathbb{R^n} \rightarrow \mathbb{R}$ subject to the constraint $g(x_1,...,x_m)=0$ for some natural $m$. We can find the local maxima and minima of $f$ on $g$ by setting: ...
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Circle Packing: Unsolved Problem in Geometry?

Graham and Sloane minimize the second moment of the centres of a number discs in order to maximize their compactness. They use computational geometry techniques to find the optimal packings for ...
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Optimization of electricity costs

I have to solve this exercise for the school and I do not really understand why the teacher solved it like this. Here is the exercise: I want to replace the 60 watt bulbs with 8 watt LED lamps. The ...
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Does having a zero eigenvalue preclude a matrix from being indefinite?

If a $3\times3$ matrix has a positive eigenvalue, a negative eigenvalue, and a zero eigenvalue, is it then, by definition, indefinite? I think so, since the matrix has both a positive and a negative ...
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Is duality theory in optimization as useful as it seems?

I have been reading a lot about nonlinear optimization and duality, and it seems that duality theory is extremely useful. I feel that I am missing some of the negative aspects/difficulties associated ...
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Help show that a second derivative is always negative

How do I show that the second derivative is always negative? I've computed the second derivative to be: $\displaystyle\frac{n}{2\sigma^4}-\frac{1}{\sigma^6}\sum\limits_{i=1}^n(x_i-\mu)^2$ Then I ...
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Propose suitable algorithm for min-max optimization problem

Consider: \begin{equation}\min_{x, y} \max_{\omega} | f(x, y, \omega) |\end{equation} where $(x , y)\in \mathbb{R}\times \mathbb{R} $ and $\omega \in (0, \infty)$. $f$ is the result of dividing ...
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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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Is 0-1 integer programming always NP-hard?

I have the following problem. Maximize $\sum\limits_{m=1}^M\sum\limits_{n=1}^N x_{mn}$ subject to: $\sum\limits_{\substack{m^\prime=1\\ m^\prime \neq m}}^M\sum\limits_{\substack{n^\prime=1\\ ...
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Maximum value of the lowest sum in a set of numbers

Last year in a maths contest held in Catalonia called Cangur it was posed the following qüestion: We write numbers 1, 2, 3, 4, 5, 6, 7, 8, 9 and 10, in a certain order around a circumference. Then ...
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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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Terminology for maximize/minimize choice

I'm writing optimization software where the user needs to decide whether they want to minimize or maximize the value the objective function (where the output will contain many putative optimal ...
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Reasons for the worst-case scenario in robust optimization

When we solve an optimization problem, containing in his objective function an uncertain parameter (i.e. random variable), using robust optimization techniques such as the max-min approach, we first ...
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Optimal Strategy for Chosing Lottery Tickets

You have 2 types of lottery tickets: one that costs $c_1$ and has a probability of winning of $p_1$, and the other costs $c_2$ and has a probability of winning of $p_2$. The goal, as you might expect, ...
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Is it possible to reduce a lambda expression to it's smallest equivalent form?

In the Untyped Lambda Calculus, is it possible to reduce any arbitrary expression to it's smallest equivalent form? (defined as an expression with the smallest number of lambda terms) If so, is there ...
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Find $\alpha$ such that the given point is critical for a implicitly defined funtion.

Can anyone check my solution for this exercise? Let $F:\mathbb{R}^3\rightarrow\mathbb{R}$ be given by $F(x,y,z) = \alpha xz + x\arctan(z) + z\sin(2x+y) -1.$ Prove that a function $z=f(x,y)$ ...
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Find the maximum value

Find the maximum value $$F(y)=\int_{0}^{y}\sqrt{x^4+(y-y^2)^2}dx$$ with $y\in [0;\: 1]$ This problem here
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Maximum area under a curve by calculus of variations

I am asked to find the function that has the maximal area for a given length L when x runs from -a to a. I calculated the integral to be varied as follows: $$ \int_{-a}^{a}\ y + \lambda \sqrt{1 + ...
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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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Packing cannonballs in a tetrahedron

I have a somewhat interesting problem. Assume one has a tower of cannonballs, or spheres as pictured below As in you have a tower of spheres where the first layer has $1$ cannonball, the next ...