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.

learn more… | top users | synonyms (5)

4
votes
0answers
86 views

How to minimize $\min_k k \frac{b^k/n}{\lfloor b^k/n \rfloor}$

This problem looks familiar, but I don't remember its solution: $$ \min_k \ \ \frac{b^k/n}{\lfloor b^k/n \rfloor}k $$ subject to $$ b^k \ge n \\ b,n,k \in \mathbb{N} $$ Does it have a name? What's ...
0
votes
0answers
125 views

Optimization problem given a known solution space

Here is my problem. I have to find four points in 3D (x1,y1,z1; x2,y2,z2; x3,y3,z3; x4,y4,z4) that satisfy some given quadratic constraints. In addition, I have a solution space in the form of a set ...
2
votes
1answer
112 views

Determining the Existence of Global Minimum/Maximum

Determine whether the function defined as $$f(x,y,z)=x+y+z$$ has a maximum or a minimum value on the set $xy+yz=1$, $xz+yz=4$, $x>0$, $y>0$, $z>0$. It is clear to me that it does have a ...
1
vote
1answer
34 views

Symbolic evaluation of an optimization problem

I'm looking at the following problem: Minimize $\sum_{i=1}^{m} \frac{x_i}{x_{i-1}}$ under the constraints $-x_0 \le -1$, $x_{i-1} - x_i \le 0$, and $x_m \ge N$ where $N>0$ and $m>0$ are some ...
0
votes
1answer
41 views

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 ...
0
votes
1answer
137 views

Minimizing total cost function

In today's test (question c) I had to minimize equation $(3)$ and solve for N*. I did it through deriving, setting to $0$ and solve for N (no doubts about that). My question is, in this image it ...
1
vote
1answer
150 views

Property of strictly convex polynomial

I have some difficulties in the following problem. Thank you for all comments and helping. Let $f:\mathbb{R}^n\rightarrow \mathbb{R} (n\in \mathbb{N})$ be a polynomial. Suppose that $f$ is strictly ...
3
votes
1answer
95 views

Simple optimization trick

Let $f,g:X\to\Bbb R$ be two functions where $X$ is any set. Then $$ \left|\sup_x f(x) - \sup_x g(x)\right|\leq \sup_x|f(x) - g(x)|. $$ This fact is fairly easy to prove, but it seems to be a ...
0
votes
2answers
78 views

Anyone saw this interesting function before?

Say $\theta\in\Re^n$ and $\theta_i\in(0,1)$ for all $i$. Define $$ f(\theta) = \frac{1}{n}\sum_i^n\{(1-\theta_i)\log(1-\theta_i)+\theta_i\log\theta_i\} $$ It is easy to see the minimizer of ...
2
votes
1answer
127 views

Proof there exists a vector under certain constraints

This question is a basic optimization problem, also a linear algebra question: Let $x$ be a feasible point for the constraints $Ax=b$, $x\geq0 $ that is not an extreme point. Prove that there exists ...
3
votes
1answer
109 views

Optimization Problem.

I'm working on some calculus homework, which deals with optimization problems, we have the solution posted for us and when looking over it I got a bit confused. Here's the question: An open ...
0
votes
1answer
84 views

Minimizing the function $\sqrt{2x} - \left\lceil \frac{\sqrt{1+8x}}{2} \right\rceil$ for $x > 0$

How can you find the minimum of $\sqrt{2x} - \left\lceil \frac{\sqrt{1+8x}}{2} \right\rceil$ for positive integer values of $x$?
1
vote
1answer
64 views

Maximize $x_1$ and $x_2$

I have the following question to tackle: Maximize $x_1$ and $x_2$ for: $$ x_1, x_2 \geq 0$$ $$ -x_1 + x_2 \leq 5$$ $$ x_1 + 4x_2 \leq 45$$ $$ 2x_1 + x_2 \leq 27$$ $$3x_1 - 4x_2 \leq 24$$ So I ...
3
votes
2answers
66 views

In Search of a More Elegant Solution

I was asked to determine the maximum and minimum value of $$f(x,y,z)=(3x+4y+5z^{2})e^{-x^{2}-y^{2}-z^{2}}$$ on $\mathbb{R}^{3}$. Now, I employed the usually strategy; in other words calculating the ...
1
vote
2answers
97 views

Finding absolute max and min values of function

Function given as $f(x,y) = 3x^2 + 2xy^2$. If $(x,y)$ lies in the region inside including edges of the triangle in the first quadrant given by $x\ge0, y\ge0, y\le2-x$. Reduce $f$ to a single variable ...
2
votes
1answer
167 views

Minimise $\sum_{i=1}^n \frac{1}{a_i + x_i}$ subject to constraint using Lagrange multipliers

Minimise $\displaystyle \sum_{i=1}^n \frac{1}{a_i + x_i}$ subject to $\displaystyle \sum_{i=1}^n x_i =b$, $x_i\geq 0$ for $i=1,\cdots , n$, where $a_i >0$ for $i=1,\cdots , n$ and $b>0$. I know ...
1
vote
3answers
455 views

A constrained linear least Frobenius norm problem:$\min_{X} \|A-XB\|_F$ subject to $Xv=0$?

Assume we are given two matrices $A, B \in \mathbb R^{n \times m}$ and a vector $v \in \mathbb R^n$. $\|\cdot\|_F$ is the Frobenius norm of a matrix. How can we solve $$\min_{X \in \mathbb R^{n ...
2
votes
1answer
84 views

Cubes, squares and minimal sums

I have trouble solving the following task: i need to find positive integers a and b such that 1) $a \neq b$ 2) $ \exists c \in \mathbb{N} : ~ a^2 + b^2 = c^3$ 3) $\exists d \in \mathbb{N}: ~ a^3 + ...
0
votes
1answer
48 views

inequalities for optimization over psd matrices with constraints

Consider two p.s.d. matrices $A$ and $B$ both in $\mathbb{R}^{d \times d}$. Define $$a = argmax_{x \in \mathbb{R}^d} x^\top A x $$ and $$b = argmax_{x \in \mathbb{R}^d} x^\top B x $$ both subjected to ...
3
votes
3answers
191 views

Max and Min of $f(x,y)$

Let $f(x,y)=x(y \log y-y)-y \log x$. Find $\smash{\displaystyle\max_{\frac{1}{2} \leq x \leq 2}}(\smash{\displaystyle\min_{\frac{1}{2} \leq y \leq 1} f(x,y)})$.
-1
votes
1answer
100 views

Find the biggest sum from sequence of number which within a range

I need help. How do I find the greatest sum from sequence of number within a finite range, for example: Given sequence {2,5,4,3,6} and the range is 11, so how to find the number within the sequence ...
2
votes
3answers
130 views

Packing radios into cartons - why is my solution wrong?

A manufacturer of car radios ships them to retailers in cartons of $n$ radios. The profit per radio is $\$59.50$, minus shipping cost of $\$25$ per carton, so the profit is $59.5n-25$ dollars per ...
2
votes
2answers
562 views

satisfy the Euler-Lagrange equation

Two circles of unit radius, each normal to the line through their centers are a distance d apart. A soap film is formed between themas shown below; energetic considerations require the filem to ...
3
votes
1answer
56 views

Some basic questions about minima of a real-valued functions

The following theorem is basically from the Fermat's Theorem page of wikipedia. Let $X$ denote a subset of $\mathbb{R}$, and suppose $f : X \rightarrow \mathbb{R}$ attains a global minimum at $x ...
2
votes
4answers
425 views

Minimize $\sum a_i^2 \sigma^2$ subject to $\sum a_i = 1$

$$\min_{a_i} \sum_{i=1}^{n} {a_i}^2 \sigma^2\text{ such that }\sum_{i=1}^{n}a_i=1$$ and $\sigma^2$ is a scalar. The answer is $a_i=\frac{1}{n}$. I tried Lagrangian method. How can I get that ...
0
votes
2answers
62 views

Please help to make me understand why I cant optimize this function: $U=x^{1/3}*y^{2/3}$ ?

If I want to maximize a production the function of which is given by $$L=-x^2+10x-2y^2+12y$$ I know I have to take the partial derivatives of of the function in respect to X and Y, so $$\frac ...
0
votes
0answers
60 views

Minimum of some functions

Denote $U=\{(x_1,x_2,...,x_n):0<x_j<1 (1\leq j\leq n),\sum_{j=1}^nx_j=1\}$. Let $f_i=f_i(x_1,x_2,...,x_n)$ ($1\leq i\leq n-1$) be $n-1$ real functions which satisfy: ...
5
votes
5answers
254 views

Minimum of $x^2 + 3x - 1$ on $[0,1]$ and $[-2,2]$

Consider the problem of finding the absolute minimum of the function $f : [0,1] \rightarrow \mathbb{R}$ that satisfies $f(x)=x^2 + 3x - 1$ everywhere. Suppose we suspect, by graphical methods, that ...
1
vote
4answers
245 views

How to get from high school math to optimization?

What are the math subjects that a person with high-school math background needs to learn to reach the point of learning and understanding different techniques of mathematical optimization? It would be ...
0
votes
1answer
57 views

Distributed Robust Optimization

Consider the following constrained optimization problem $\mathcal{P}$. $$ \min_{x \in X \subseteq \mathbb{R}^n} f(x) \ \text{sub. to: } g(x,y) \leq 0 \ \forall y \in Y \subseteq \mathbb{R}^m $$ ...
1
vote
1answer
130 views

HELP please with Optimization with constrain using lagrangian

I am reading this book on optimization and they present the following problem: Lisa wants to maximize her utility U(q1,q2) subject to a budget constrain, budget constrain is $p1*q1+p2*q2=I$. Ok , I ...
1
vote
1answer
76 views

Distance between a point to a $2d$ ellipse in $3d$ ambient space

Suppose we are working in the 3D Euclidean space. We are given an arbitrary point $p$ and a 2d ellipse: $$E=\{x:x^TQx\leq1,x^Tq=0\},$$ where $Q$ is a positive definite matrix and $q$ is an ...
1
vote
1answer
63 views

Non-trivial solution for $2*a^k = b^k + c^k$

I have a data set where I want its median to be the arithmetic average of maximum and minimum by multiplying every value with a factor $k$ and then applying the exponential function. This leads to the ...
4
votes
2answers
204 views

Wolves and chicks puzzle

This problem is from the handheld video game, Professor Layton and the Curious Village. I think the solution is very cool, but more than that, I want to know how to show that the minimum number of ...
2
votes
1answer
598 views

Algorithm to find optimal cuts of pipe

I have varying lengths of pipe in inventory. When a customer requests various lengths I want to find the optimal way of cutting what I have in inventory. I need to make a program that does this. This ...
0
votes
1answer
82 views

Find a number that minimizes distance to a vector of sets of numbers

Assumptions $V$ is a vector of sets $V_1,V_2,...,V_n$ of numbers: $V=[V_1, V_2,..., V_n]^T, \forall_{i=1..n}V_i\subset\mathbb{R}$ $c\in\mathbb{R}$ is constant $d(V,c)$ is an error metric: ...
1
vote
1answer
1k views

Why do we use gradient descent in the backpropagation algorithm?

The common approach for training neural networks, as far as i know, is the backpropagation algortihm, which uses gradient descent to reduce the error. (i) why should one use a fixed learning rate / ...
1
vote
2answers
137 views

Finding the length and width of a house that maximize its area

A house is built in the shape of a rectangle, with $3$ rectangular interior sections separated by parallel walls, using fencing. The owner has $900$ feet of fencing, and he wants to enclose the ...
1
vote
1answer
46 views

calculus of vatiations; finding the minimum value

Find the minimum value of the quadratic form: $$x^{2}+2y^{2}+3z^{2}+2xy+2xz$$ subject to the condition: $$x^{2}+y^{2}+z^{2}=1$$
1
vote
1answer
118 views

Optimal cuts from pipe

I need to make a program that calculates the optimal way to cut pieces of pipe to what a customer wants. My advanced math skills are bad but I know this is the cutting stock problem. My first problem ...
0
votes
2answers
81 views

Finding the minimum of $6a^3+9b^3+32c^3+\frac{1}{4abc}$ for positive $a,b,c$

If $a,b,c$ are real positive numbers. How to find the minimum for: $$6a^3+9b^3+32c^3+\frac{1}{4abc}$$
1
vote
1answer
101 views

Simplex method: Utter, extreme confusion

We want to maximize $ z = 30x_1 + 20x_2$ with $$2x_1 + x_2 \leq 140$$ $$x_1 + 2x_2 \leq 160$$ $$x_1 + x_2 \leq 90$$ $$ x_1, x_2 \geq 0$$ So my book says the first step is writing these to ...
1
vote
1answer
34 views

Existence of Minimum Value

Assume $x\ge0$, show that the function $f(x,y)=(2xy+y^2)e^{-x}$ has a minimum value. Note that actual minimum value is $-4e^{-2}$.
9
votes
1answer
418 views

Largest rectangle in a convex polygon

What is the least $k > 0$ such that every convex polygon of area $k$ contains a rectangle of area 1? I can prove that $k \le 8$, but surely this can be improved. Let $\mathcal{C}$ be a convex ...
1
vote
1answer
62 views

An optimization problem

I need to prove the following result: Given a real sequence $a=(a_n)_{n\in\mathbb{Z}}$ and a number $A>0$ then $||a||_{1}\leq A$ if and only if there exists $b_n$ such that $-b_n\leq a_n\leq ...
4
votes
3answers
53 views

$\max_{\|x\|=1}f(x)=\max_{\|x\|\le 1} f(x)$

I'm reading the book "Matrix Analysis and Applied Linear Algebra". On page 450, eq(5.15.5), I think I found an error made by the author. So I post it here. If I'm wrong, please correct me. The ...
3
votes
2answers
62 views

What kind of optimization is this problem?

I'm not asking for a solution, I just need to know what type of optimization is this problem?. Find $\mathbf{q}$ that minimizes the following: $$\min_\mathbf{q}{|\mathbf{BXq|^2}}$$ $$\, \mbox{s.t}\ ...
4
votes
1answer
319 views

Maximizing distance between points

I asked a similar question on SciComp, but it is a little out of the domain, so I thought I'd give it a try here as well. Give n points, I would like to place them in a periodic box (periodic such ...
0
votes
0answers
95 views

Optmizing sum of two vectors

I apologize in advance for the title, but I don't know how to express exactly what I want to do. So, here's my problem: I have 66 vectors, each one with 8 values, those values can be positive or ...
0
votes
1answer
341 views

Minimizing a function using gradient (example from Wikipedia)

This example is from Wikipedia (http://en.wikipedia.org/wiki/Gradient): The gradient of function $f(x,y,z)=2x+3y^2-sin(z)$ is $\nabla f= \frac{\partial f}{\partial x} i + \frac{\partial f}{\partial ...