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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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
54 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
19 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
42 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
22 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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0answers
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How to optimize this types of problems?

Given that $min [ t_{f} - t_{0} ]$ such that $x(x'(t_{0}),y'(t_{0}),z'(t_{0}),t_{0}) = 0$ $y(x'(t_{0}),y'(t_{0}),z'(t_{0}),t_{0}) = 0$ $z(x'(t_{0}),y'(t_{0}),z'(t_{0}),t_{0}) = 0$ $x(t_{f}) = ...
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0answers
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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
40 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
37 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
34 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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1answer
27 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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0answers
16 views

Help Required in eigenvectors for sparse matrix?

I have a large sparse matrix A(~400000,~400000) . If I randomly remove few rows from the matrix will there be considerable change in the eigenvalues and the eigenvector's compared to eigenvector's of ...
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1answer
24 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
28 views

Radio factory linear program

I need a help with this exercise. I’m supposed to write a liner program for the problem below and then solve it using simplex method, but I’ don’t know how to include all the factors into variables. ...
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0answers
22 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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0answers
46 views

How to Solve this maximization Problem?

You are given two s: N and K. Lun the dog is interested in strings that satisfy the following conditions: The string has exactly N characters, each of which is either 'A' or 'B'. The string s ...
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1answer
31 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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0answers
24 views

How to Solve this Linear Programming Problem?

$$\max[Z(x,y)=x+y]$$ $$-x+y\le 1$$ $$x\ge 0$$ $$y\ge 0$$ What i have done so far ? I tried simplex method , but i can't stop iterating . It really seems like a live lock . So how can i solve ...
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4answers
114 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
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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
43 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
31 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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21 views

measurable selection for almost-minimizers of an irregular functional

I'm faced with the following problem: I have a functional $F$ defined on $H^1$ curves $[0,1] \rightarrow \Omega \subset \mathbb{R}^n$ where $\Omega$ is either a compact subset or the whole ...
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1answer
11 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
56 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
39 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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0answers
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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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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
38 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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0answers
14 views

Maximum of a 3-variable equation with constraints on the variables

So I need find the maximum of the equation $$ (10x + 10)[ (0.03y + 0.1) (0.1z + 0.5) + 1] $$ given that $0 \leq y \leq 30, \; x \geq 0,\; z \geq 0$, and $x + y + z = 100$. I'm not exactly sure how ...
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Reference material on Alternating Minimization Algorithm

I am looking for some good reference material (book/paper) for learning Alternating Minimization Algorithm. Any recommendation from optimization experts will be much appreciated. Thank you.
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There exists a descending chain of symmetry groups from a formal language string down to its smallest grammar.

Background. Let $\tau \in G_i$ be a permutation in the symmetry group of the smallest grammar $g_i$. Then $\tau$ permutes each set of positioned (within $g_i$) symbols $\{x_1^{(1)}, x_1^{(2)}, \dots, ...
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2answers
57 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
61 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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0answers
34 views

Question about duality in nonlinear optimization

Let $f(x)$ and $h(x)$ be functions from $\mathbb{R}^n$ to $\mathbb{R}$ and consider the minimization problem $$ {\rm minimize} ~~~ f(x)$$ $$~~~~~~~~~{\rm subject ~to}~~h(x)=0.$$ Suppose the minimum is ...
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2answers
108 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
71 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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0answers
16 views

Minimizing sum of cubes givien some constraints

I am given a sequence of real numbers $b_1 \ge b_2 \ge \cdots \ge b_n$ and two positive integers $m > k.$ Let $x_2 \ge \cdots \ge x_{m}$ be numbers such that $$b_{n-m+i} \le x_i \le b_{i} \quad ...
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1answer
40 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
89 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
59 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
22 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
23 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
48 views

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 ...
3
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0answers
43 views

Maximum value of an integral.

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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1answer
31 views

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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2answers
118 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
14 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
7 views

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 ...