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 vector that maximizes $f(x) = 2x_1^2+2x_2^2-x_3^3+2x_1x_2$

Find the vector with $||x||^2=x^Tx=1$ that maximizes the following function. $f(x) = 2x_1^2+2x_2^2-x_3^3+2x_1x_2$ I have rewritten the quadratic form as $f(x) = \frac{1}{2}x^T \begin{bmatrix} ...
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Maxima and Minima of Functions of Two Variables $ f(x,y) = e^{x+y^2}\cdot y $ and $ f(x,y) = e^{x^2-y^2}\cdot y $

I'm having trouble finding the local minimum and maximum of the next functions: $$1. f(x,y) = e^{x+y^2} \cdot y $$ $ f_x'= (e^{x+y^2}\cdot y) ; $ $ f_y'= (e^{x+y^2}(1+2y^2)) $ $$ 2. f(x,y) = e^{...
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Minimization of a combinatorial function

The following gamma function depends on the overall sum of $x_n,x_j,x_k$ $$\gamma(X)=\sum_{x_n+x_j+x_k=X}\left [ \left ( \prod_{i=1}^{s}(x_{ni}-1)!C_i^{x_{ni}} \right )\times \binom{x_j}{x_{j1},x_{...
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Largest Arithmetic Sum of Relatively Prime Numbers Under 30

Pick however many integers in the range $[1,30]$ (inclusive). The only constraint is that all of these numbers must be relatively prime to each other. What is the largest possible arithmetic sum ...
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171 views

Minimizing a summation?

I have absolutely no idea how to approach this problem. I've been looking through notes, and I think I missed this when my professor discussed this in class. $$ \text{Consider the data}\\ i\: x_i\: ...
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Do we really need the constraint qualification?

I can't keep my fingers off Nocedal/Wright's Numerical Optimization (1999,1E) and I apologize. But maybe YOU can shed light on the question: Why does a point $x \in \mathbb{R}^n$ need to satisfy the ...
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104 views

Modeling, Measuring, and Maximizing “Mixedness”

Possible key-terms: combinatorial optimization techniques; simulated annealing; genetic algorithms; Kirkman's schoolgirl problem; Steiner triple systems; orthogonal regrouping. Background: My class ...
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101 views

Maximize the following sum

Let $a, b, c, d, e$ be nonnegative integers such that $625a + 250b + 100c + 40d + 16e = 15^3$ . What is the maximum possible value of $a + b + c + d + e$? Quick arithmetic gives: $15^3 = ...
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Functional Lifting in Optimisation - Reference Request

I'm looking to learn about the use of (functional) lifting applied to a non-convex optimisation problem to give a (larger) convex problem. Unfortunately, I'm having a great deal of trouble finding ...
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Finding extrema.

Find the minimum distance between point $M(0,-2)$ and points $(x,y)$ such that: $y=\frac{16}{\sqrt{3}\,x^{3}}-2$ for $x>0$ . I used the formula for distance between two points in a plane to get: $$...
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Complexity of finding M nodes in a graph to maximize the pairwise minimum distance between nodes

I want to know the complexity of finding a set of M nodes, $\{U_1,\dots,U_M\}$, in a given graph $G$, to maximize $d(U_i,U_j)$ over all pairs $i\neq j$, where $d(\cdot,\cdot)$ is the length of the ...
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Simple question on maximizing a family of concave functions

Given $x \in [0,1]$, let $f_k(x)$ be concave function in $x$ for all $k= 0,1,...,N$. Is the following two maximization formulations equivalent? $$ \max_{0 \le k \le N} f_k(x) =?= \max \{f_0(x), f_1(...
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Simple verification: is this equivalence always true?

I have a constrained optimization problem and I am trying to reduce the number of contraints and am afraid to be losing information by doing so. If we have two constraints as the following $$A \geq B \...
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If the iteration $x^{k+1}=x^k-t_kH_k^{-1}\nabla f(x^k)$ converges superlinearly to a stationary point $x^*\ne x^k$, then $t_k\to 1$

Let $f\in C^2(\mathbb{R}^n)$ $(H_k)_{k\in\mathbb{N}_0}\subseteq\text{GL}_n(\mathbb{R})$ $x^0\in\mathbb{R}^n$ and $$x^{k+1}:=x^k+t_k d^k\;\;\;\text{for }k\in\mathbb{N}_0\tag{1}$$ with $(t_k)_{k\in\...
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What is the minimum point of $x\mapsto x^Ty$ for $|x|\le 1$ and a fixed $y\in\mathbb{R}^n$?

Let $y\in\mathbb{R}^n$. I want to minimize $$f(x):=x^Ty\;\;\;\text{for }|x|\le 1$$ The minimum point should be $$-\frac{y}{\sqrt{y^Ty}}\tag{1}$$ However, how can we derive $(1)$ analytically? Since $f$...
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100 views

General process to find global extrema of a function?

I have been reading and watching videos about local and global extrema, but all of this material covers the topic just graphically, and nobody really explicitly cares on how to find the global maximum ...
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61 views

Operations Resarch Optimal Scheduling

Consider the following problem: A car manufacturing company needs to transport car frames, which are $10$ cubic units each, and wheels, which are $2$ cubic units each, across the Atlantic ocean. ...
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Constrainted optimization involving logarithms

The problem is to minimize $ f(x_1, x_2 ,x_3, x_4):= - \Big[ \log ({\frac{1}{4} + x_1}) + \log ({\frac{1}{2} + x_2})+ \log ({\frac{1}{5} + x_3})+ \log ({\frac{3}{4} + x_4}) \big]$ such that $...
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77 views

Algorithms For Large-Scale $\ell_{\infty}$ Minimization

The general problem I want to solve is well studied: $$ \min_x \Vert Ax\Vert_\infty \;\;\; \mathrm{s.t.} \;\;\; Bx=c, $$ which is equivalent to the following linear program: $$ \min_{t,x} \, t \;\;...
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Is this reducible to a standard optimization problem?

There are $N$ agents who needs to be allocated $K$ discrete resources. There is a bottleneck threshold utility $R$ per agent. The $i$th agent has utility $r_{ij}$ if he is allocated $j$th resource. ...
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171 views

Global maximum and minimum of $f(x,y,z)=xyz$ with the constraint $x^2+2y^2+3z^2=6$ with Lagrange multipliers?

The global maximum and the global minimum of the function $f(x,y,z)=xyz$ with the constraint $x^2+2y^2+3z^2=6$ can be found using Lagrange multipliers. $\nabla f = \lambda \nabla g$ $g(x,y,z)=x^2+2y^...
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Derivatives - optimization (minimum of a function)

For which points of $x^2 + y^2 = 25$ the sum of the distances to $(2, 0)$ and $(-2, 0)$ is minimum? Initially, I did $d = \sqrt{(x-2)^2 + y^2} + \sqrt{(x+2)^2 + y^2}$, and, by replacing $y^2 = 25 - x^...
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345 views

ADMM formalization

I found lots of examples of ADMM formalization of equality constraint problems (all with single constraint). I am wondering how to generalize it for multiple constraints with mix of equality and ...
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Effect on Minimizer of Tightening Constraints

The Statement of the Problem: Consider the minimization problem $f(x,y)=14x+20y$ under the constraints $x+2y \ge 4 $, $7x+6y \ge 20$, and $x,y \ge 0$. Don't use the simplex method! (i) Draw the ...
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To show $f(x)$ has ONLY one Max in $x\in[0,1]$

I have function $$f(x)=\left(\frac{1-x}{2-x}\right)x^{p-1}~\text{where}~p>1;~x\in[0,1]$$ I want to show that $f(x)$ has ONLY one maximum in $x\in[0,1]$. I get the second derivative as $$f"(x)=\...
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Why is the gradient of the objective function in the Lagrange multiplier theorem not $= 0$?

A special case of the Lagrange multiplier theorem may be stated as: Let $S, T \subset \mathbb{R}^{n}$ be open. Let $f: S \to \mathbb{R}$ be differentiable on $S$ and $g: T \to \mathbb{R}$ ...
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maximizing a coordinate of $x^T A^T A x \leq r^2$

Given a vector $\mathbf{x} \in \mathbb{R}^n$, a scalar $r\gt 0$ and an invertible matrix $\mathbf{A} \in \mathbb{R}^{n\times n}$, I'd like to maximize one of the components $x_\alpha$ constrained by $\...
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How to find max value without Lagrange

I am trying to find the maximum and minimum values of the function $$f(x,y,z)=2x-y+4z$$ on the unit sphere $$x^2+y^2+z^2=1$$, but without using langrange multipliers or gradient. I would like to do ...
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Optimisation to solve for trigonometric expression?

I have a question that requires the use of optimisation to solve for the following expression: $$\cos ec{(\cos^{-1}{(-\frac{\sqrt{3}}{2})}+\sin^{-1}{(-\frac{\sqrt{3}}{2})})}$$ I'm a bit baffled, as ...
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Can I optimize area of cylinder with no givens?

I have a problem which should be very easy (as the rest of them are on this worksheet) but this one has me stumped. The question reads: A metal can is in the form of a cylinder. It has a bottom ...
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Fraction of area covered by three circles

Take a square with edges of size $10$. Now take take three circles of radius $5$. Prove that you can't cover the square with these three circles. Find the maximum proportion of the area of the ...
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Find $3$ numbers whose product is $27$ and whose sum is minimal

Find $3$ numbers whose product is $27$ and whose sum is minimal I'm thinking one might have to use langrange multipliers. The answer is $(3,3,3)$, I am not sure how to get there though.
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How to test if a feasible solution is optimal - Complementary Slackness Theorem - Linear Programming

I have this linear program $$\begin{cases} \text{max }z=&5x_1+7x_2-3x_3\\ &2x_1+4x_2-2x_3&\le8\\ &-x_1+x_2+2x_3&\le10\\ &x_1+2x_2-x_3&\le6\\ &x_1,\,x_2,\,x_3\ge0 \end{...
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Solve $\text{Minimize} \max\{|x-a_i|, i=1,..,n\}$

I have to solve $$\text{Minimize} \max\{|x-a_i|, i=1,..,n\}$$ For $a_1 \leq a_2 \leq ...\leq a_n$ My intuition says that this x is a point in the middle of the $a_i's$ but I am not sure that it is ...
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Stationary points of general function

When studying the stationary point(s) of the following $$ Q=\frac{K(x)}{x} $$ I find $$ \frac{dQ}{dx}=\frac{x\frac{dK(x)}{dx}-K(x)}{x^2}=0 $$ and hence $$ \frac{K(x)}{x}=\frac{dK(x)}{dx} $$ I'm ...
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To find the maximum and minimum value of x such that it satisfies a polynomial

Find the maximum and minimum value of $x$, where: $x+y+z=4$ $x^2+y^2+z^2 =6$ I thought I could use these values to form a equation having $x,y,z$ as roots and the sum of roots and $\sum{xy}$ but ...
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Prove a property of primal-dual problems

When I was studying the computation aspects of quantile regression, I consulted some linear programming book and found an interesting property as follows: If the primal problem have unbounded ...
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does the volume of a ball remain constant under deformation?

I'm a psychology student and was reading Piaget, he says that the volume of a sphere (ball of clay) remains constant if we deform the sphere into a roll for example, If you take the limit case of the ...
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minimizing linear combination of inner products

$\mathbf{x},\mathbf{y_1},\mathbf{y_2}\in \mathbb{R}^{m}$ and $\alpha_1,\alpha_2 \in \mathbb{R}$. Also $\|\mathbf{y_1}\|_2 = \|\mathbf{y_2}\|_2 = 1$ and $\alpha_1\geq\alpha_2\geq0$. How should we ...
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Problems with vector vector derivative in optimization

I have a loss function of the followoing form: $L(\mathbf{a}) = \|\mathbf{b} - \mathbf{a}\|_2^2$ Where, $\mathbf{a}$ and $\mathbf{b}$ are vectors of dimension $d\times 1$. I need to calculate $\...
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Could someone point me in the right direction for this complex analysis equation?

I'm supposed to show that the maximum value of $|z^2+1|$ on the unit disk $|z|\leq1$ is 2. My teacher's hint was "triangle inequality". I've been racking my brain how to tie the triangle inequality ...
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Find the minimum of $f(x) = x^2+\sin(x)$

I need to find the minimum of $x^2+\sin(x)$ but I can't get an answer. So far I've done this: The first derivative is $f'(x)=\cos(x) + 2x=0$ and the second derivative $f''(x)=-\sin(x) +2$ From the ...
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122 views

Maximization under Kronecker product vectors

I need some hints to solve the optimization problem on $\mathbf{x}$ and $\mathbf{y}$ \begin{equation} \begin{array}{c} \text{max} \hspace{4mm} (\mathbf{x}\otimes \mathbf{y})^TA(\mathbf{x}\otimes \...
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Optimization—Finding the Area of the Largest Isoceles Triangle

I managed to solve $(a)$. Since the area of a triangle is determined by $\frac{1}{2}$ base $\times$ height, and we already know the height, we just have to solve for the base. Using Pythagorean ...
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Parts of the whole: Which base begets the largest percentage of fractions expressible as a terminating decimal?

Update: It appears the question I actually meant to ask was quite different. As Robert Israel explained in his answer I was calculating the wrong thing. After writing some ugly code (may take a sec to ...
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linear programming problem - how much additional resources should I buy?

I have the following linear optimization problem: Maximize $$\sum_{i=1}^{n}x_{i}B_{i}$$ subject to the constraints $$a_{11}x_1+a_{12}x_2+\cdots+a_{1n}x_n \le l_1$$ $$...$$ $$a_{m1}x_1+a_{12}x_2+\...
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How to find $\theta$ at which $d$ is the maximum possible?

I have an equation: $$d=\dfrac{v\cos \theta}{g}\left(v \sin \theta + \sqrt{v^{2} \sin^{2}\theta + 2gh} \right),\ g≈9.81 \dfrac {m}{s^{2}}$$ How to find $\theta$ at which $d$ is the maximum possible?
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Optimization fence problem with twist.

Suppose you have a 10x15 foot dog house and you wish to build a fence in a yard in a L shape to the north and east of the dog house. If you have 75 feet of fencing material available, what dimensions ...
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Help with a homework problem involving $\textbf{H}$-conjugate vectors

My problem is the following: Let $\textbf{H}$ be a symmetric $n\times n$ matrix. Are the following claims true? Why? a) If the vectors $\textbf{d}_1$ and $\textbf{d}_2$ and vectors $\...
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Given a fixed volume for a solid cylinder, is it possible to find the minimum or maximum curved surface area?

Given a fixed volume for a solid cylinder, is it possible to find the minimum or maximum curved surface area $ 2 \pi r h $ of this cylinder?