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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Minimize a non-convex function subject to linear dynamics constraint

I want to solve the following problem: $$\min\limits_{\bf u} \frac{\bf c^T {\bf x} (T_f)}{\| \bf c\|\|{\bf x} (T_f)\|}$$ subject to $$\dot{\bf x} (t) = A {\bf x}(t) + B {\bf u}(t)$$ $$x(0) = x_0$$ ...
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1answer
88 views

If g(x) is the maximum value of f(t)

Let f be continuous on [a,b] and define a function g(x) on [a,b] as follows g(a)=f(a) and for a $\lt\ $x $\le\ $b then g(x) be the maximum value of f(t) on [a,x]. Prove that g(x) is continuous of ...
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5answers
549 views

I am trying to maximize an exponential function [closed]

I am looking for the value of $x$ that will maximize $y$ in the following equation $$ y=e^{-(x-a)^2/b} $$ where $a$ and $b$ are constants. Any help is appreciated
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1answer
69 views

linear approximation with respect to L1 norm

I am trying to solve this problem: Find the best $L^1$ linear approximation of $e^x$ on [0,1] i.e. minimize $\int_0^1|e^x-\alpha-\beta x| dx$ any hints how to proceed
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1answer
34 views

Explanation of strategies in infinite horizon dynamic programming problem

My question is regarding the Bellman equation regarding strategy $\sigma^{(1)}$ on the last 2 lines (I have attached pictures of the book below). If we know that all future states will have value of ...
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1answer
74 views

The meaning of 'worst case'

When giving bound on convergence rate, complexity and so on, people sometimes will specify it by 'worst case'. What is the meaning of 'worst case'?
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1answer
33 views

Finding a solution to matrix equation occurring inside an optimization problem

As a part of an optimization problem (while equating the derivative of the cost function to 0), I'm getting the following expression. $$-2XX^TC + 2XX^TACC^T + \gamma GA = 0,$$ where, $X, C, G$ are ...
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1answer
626 views

Can Three Equilateral Triangles with Sidelength $s$ Cover A Unit Square?

A previous question on the site asked for a short proof of the fact that three equilateral triangles with unit side length cannot be arranged to cover a square with unit side lengths. Given the truth ...
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1answer
46 views

Rate of Convergence of complicated sequence with interactions

I have been working on a problem where the sequence turns out to be so complex that i am unable to find its convergence rate with necessary and sufficient conditions on the parameters.After working ...
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1answer
293 views

The Euler-Poisson equation

$$\int_{0}^\pi (x''^2+4x^2) dt$$ $$ x(0)=x'(0)=0; x(\pi)=0;x'(\pi)=sinh(\pi)$$ This is The Euler-Poisson equation, i found: $$\frac {\partial f}{\partial x}-\frac {d}{dt} \frac{\partial f}{\partial ...
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2answers
132 views

How to maximize shipping box volume

Earlier last week I realized I needed to ship a large volume of things domestically. Of course, I decided that I wanted to do so as cheaply as possible. I first looked at USPS standard post rates. I ...
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37 views

lagrange method, linear constraints and unique global maximum

My book in linear programming states two things that I do not understand. We are working with the lagrange method with linear constraints.: From multivariate calculus we have that at a critical ...
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1answer
73 views

Maximum Volume of a rectangular box in ellipsoid

This is the problem I am working on: Find the maximum volume of a rectangular box that can be inscribed in the ellipsoid: $x^2/25 + y^2/4 + z^2/49 = 1$ with sides parallel to the coordinate axis I ...
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0answers
72 views

Hammersley–Chapman–Robbins bound for Rice distribution

I am trying to evaluate the Hammersley–Chapman–Robbins bound for the variance of an unbiased estimate $\hat{\alpha}$ of $\alpha$ (for a given $\sigma$) for the Rice distribution: $$p(x|\alpha,\sigma) ...
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1answer
36 views

Finding Maximum and Minimum for f(x,y)

The problem I am working on is: Find the maximum and minimum values of the function: $f(x,y) = -3x^2 - 14xy - 3y^2 -8$ on the disk: $x^2 + y^2 \leq 4$ The $-14xy$ term is severely throwing me for a ...
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2answers
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How to minimize $x^2+4xy+5y^2-4x-6y+7$ without using calculus

I would like to find the smallest possible value of the function $$f(x,y)=x^2+4xy+5y^2-4x-6y+7$$ without taking any derivatives. My thoughts were to complete the square on both $x$ and $y$ and ...
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4answers
217 views

Maximizing the area of a triangle with its vertices on a parabola.

So, here's the question: I have the parabola $y=x^2$. Take the points $A=(-1.5, 2.25)$ and $B=(3, 9)$, and connect them with a straight line. Now, I am trying find out how to take a third point on ...
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0answers
35 views

Quasiconcavity of $g(x)=xf(K-x)$

The function $f(x)$ is strictly increasing, finite, positive and twice continuously differentiable on the compact interval $[0,K]$, and $f(0)=0$. I'm trying to either find a counterexample to, or a ...
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1answer
58 views

Question of proof of maxima related to quadratic form

Suppose $\bf{A}$ is a symmetric positive-definite matrix and now we want to maximize function $f(\bf{x})=\bf{x}^\rm{T}\bf{A}\bf{x}$ with restriction $\bf{x}^\rm{T}\bf{x}=\rm{1}$. Using Lagrange ...
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1answer
42 views

Showing CP-rule is not optimal for $P \mid p_j = 1, \text{ intree} \mid \sum C_j$.

We are asked to find a counterexample that shows that the Critical Path rule is not optimal for $P \mid p_j = 1, \text{ intree} \mid \sum C_j$. However, after trying for two hours, I don't think I'll ...
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1answer
23 views

Simply function F and find alpha for which F will be min

I have point coordinates like [x, y], where x and y are positive natural numbers. I need to ...
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2answers
64 views

Find all minima and maxima of $f(x) = (x+1)^{1/3}(x^2-2x+1)^{1/5}$

Find all minima and maxima of : $f(x) = (x+1)^{1/3}(x^2-2x+1)^{1/5}$ I feel kind of ashamed by posting such a question but I am not able to get the right answer. I've tried to do it the Fichtenholtz ...
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1answer
78 views

Find max/min of $f(x,y,z)$ on closed unit ball $B$ in $\mathbb{R^3}$

$f(x,y,z) = 3x - 2y + z$ Let $B$ be a closed unit ball in $\mathbb{R^3}$, find the max/min of f on $B$. We first need to observe $(a)$ the behavior of $f$ in $B^0$ $(b)$ the behavior of $f$ on ...
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Combinatorics : Minimization of the number of common objects between subsets

Let's consider the following setup. I have access to $N$ objects. Thanks to these objects, I can build up sub-packets containing $k$ such objects. I know that there exists a total of $\displaystyle ...
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3answers
51 views

Can we not determine the minimum and maximum value of a function by just obtaining the value of the function at the critical points?

Can we do the above or is it neccessary to find the sign of the second derivative. I find no problem with any of the above but the answer of the following question left me confused. "The real number ...
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0answers
65 views

find min max of function on unit ball

$g(a,b,c)=3a-2b+c$, B is a closed unit ball in $\mathbb R^3$. Find the max/min of g on B. What is the behavior of $g$ on the open unit ball, and the boundary of the unit ball? I think the unit ball ...
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2answers
162 views

What is the derivative of a matrix w.r.t itself?

what is the derivative of \begin{equation}\partial \frac{x^TVx}{\partial V} \end{equation} where V is a matrix and x is a vector. In general what is the right way to calculate matrix derivatives w.r.t ...
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1answer
623 views

Solving non-linear optimization using generalized reduced gradient (GRG) method

Consider the following elementary maximization problem: \begin{align} f{=}\mathrm{argmax}_{y_{l,c}, p_{l,c}}~\sum_{l=1}^{L}\sum_{c=1}^{C} y_{l,c}\text{log}_2\left(1+\frac{p_{l,c}}{I_{l,c}}\right) ...
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160 views

Optimization of parameter for recursive Cauchy sequence

I have the following recursive sequence I'm analyzing: $$V_0 = 50, V_1 = (1-10k)V_0,$$ $$V_{n+1} = (1-10k)V_n - 5kV_{n-1}$$ where $k > 0$ is a parameter that I'm investigating by running ...
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1answer
78 views

Find minimum of $P=\frac{\sqrt{3(2x^2+2x+1)}}{3}+\frac{1}{\sqrt{2x^2+(3-\sqrt{3})x +3}}+\frac{1}{\sqrt{2x^2+(3+\sqrt{3})x +3}}$

For $x\in\mathbb{R}$ find minimum of $P$. $P=\dfrac{\sqrt{3(2x^2+2x+1)}}{3}+\dfrac{1}{\sqrt{2x^2+(3-\sqrt{3})x +3}}+\dfrac{1}{\sqrt{2x^2+(3+\sqrt{3})x +3}}$ Source : Viet Nam national test for high ...
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1answer
64 views

Optimal solutions of x and y for $\max_{x,y}~\min (f(x,y),~g(x,y))$

Can someone help me to find analytical solutions for optimal values of $x$ and $y$ which satisfy the following optimization problem? \begin{align} \max_{x,y}~\min & ...
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0answers
109 views

StackEgg optimal algorithm

What is the minimum number of days that is needed to complete the StackEgg game? (It's on the right if anyone didn't notice.) There are four markers (Questions, Answers, Users, Quality) I believe each ...
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1answer
100 views

Minimize Product of Sums of Squared Distances

The Question Given two sets of vectors $S_1$ and $S_2$,we want to find a unit vector $s$ such that $$\{\sum_{u\in S_1}(\|u\|^2-\langle u, s \rangle^2)\} \cdot \{\sum_{v\in S_2}(\|v\|^2 - \langle v, ...
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1answer
43 views

Is this function jointly convex in its variables?

I have a function which I suspect is jointly convex, but have a difficult time proving it, especially since the Hessian is messy. The function is $f(y_i,i=1.2,\ldots,N)=\sum_i l_i w_i + y_i$, where ...
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1answer
28 views

If $x = \operatorname{argmin}_{x \in X} \lvert Ax - y\rvert^2$ does it mean that $Ax = \operatorname{Proj}_X(Ay)$?

Suppose that $A$ is an invertible matrix and $$x = \operatorname{argmin}_{x \in X}\lvert Ax - y\rvert^2,$$ then does it mean that $Ax = \operatorname{Proj}_X(y)$ like in the definition of ...
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2answers
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How to minimize the function $f(x,y) = x^2 + \frac{9}{4}y^2 + 3xy -2x-2y \;\;\;$ s.t. $x,y \geq 0$

My task is to minimize the function $$f(x,y) = x^2 + \frac{9}{4}y^2 + 3xy -2x-2y \;\;\;$$ subject to $x,y \geq 0$. Do I need to use Lagrange multipliers in this problem? I tried simply taking the ...
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4answers
151 views

Maximum value of the integral $\int_0^1e^{|t-x|}dt$ for $0 \leq x \leq 1$

Define $$f(x)=\int_0^1e^{|t-x|}dt$$ I have to find the maximum value of $f(x)$ when $0 \leq x \leq 1$. To remove the modulus, I wrote $$f(x)=\int_0^xe^{x-t}dt + \int_x^1e^{t-x}dt$$ ...
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Minimize the cost of a 3 cubic unit volume box, given the price of each of six sides per unit area

I was wondering if anyone could help verify my answer of a question, or if it is incorrect to maybe let me know my mistake? The questions asks to minimize the cost of a 3 cubic unit volume rectangle ...
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148 views

Triangle containing most points from a set

Given a point set in $\mathbb{R}^2$, I need to find a triangle connecting three points of the set that contains the most points of the set. Points that lie on the connecting lines don't count. The ...
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1answer
32 views

Conditions for global max of symmetric function to lie on diagonal

Assume $f:[0,1] \times [0,1]$ is symmetric, i.e. $f(x,y) = f(y,x) \;\;\forall x,y \in [0,1]$. Assume further that $f$ is smooth, and that for every $x \in [0,1]$ the map $\phi_{x}(y):=f(x,y)$ attains ...
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0answers
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Approximation for the minimal test cover / minimal group test problem

There are multiple approximation methods I find for the minimal test cover, where approximation is with respect to the size of the test set. However I am looking for approximation which starts with a ...
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1answer
33 views

How to set up matrix to compute best coefficients

Suppose we're given a non-linear spring with the following relationship between the applied weight ($x$) and displacement ($y$): $y = ax + bx^3$. I've done a sequence of $m$ tests measuring the ...
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2answers
76 views

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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Optimization, find the dimensions of the poster with the smallest area

The top and bottom margins of a poster are 4 cm and the side margins are each 2 cm. If the area of printed material on the poster is fixed at 380 square centimeters, find the dimensions of the ...
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3answers
105 views

Spectral Theorem / Quadratic Form Minimization Problem

Here is the problem: Let $A$ be an $n \times n$ symmetric matrix. Let $S = \{ \mathbf x \in \mathbb R^n : ||\mathbf x|| = 1 \} $ denote the unit sphere. Let $Q(\mathbf x) = \mathbf x ^TA\mathbf x $ ...
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1answer
59 views

Modelling Problem in Linear Programming Standard Form

I'm having a hard time setting this up, so that's what I need help with. The solving I understand. We’re making a drink with the following requirements: at least 500 calories, at least 20 mg. of ...
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2answers
45 views

Another optimization problem

I am having trouble figuring out a next step in an optimization problem the question is to find the max and min values of $f(x,y)=\frac{x+y}{2+x^2+y^2}$ I calculated $f_x$ and $f_y$ and set both ...
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1answer
37 views

Confirmation of maximization problem

Hey all I am working on a problem, and my numbers are coming out not so nice so I think It is possible that I am making a mistake and if so Id be really interested in learning how to do it correctly. ...
4
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2answers
371 views

Minimal distance between points on two graphs

What is the minimal distance $d$ between the graphs of $y_1 = \sin x$ and $y_2 = 2 + \sin x$? The trivial observation is that $d \le 2$ (just set $x = 0$ in both equations), but numerical computations ...
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1answer
142 views

how to write largest circle inscribed inside a triangle as an optimization problem?

can someone show me how to write this problem as a convex optimization problem.Find the largest disk that can be bounded by $X \geq 0$ , $Y \geq0$ and $X+2Y\leq1$. My institution is to cast to ...