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.
1
vote
1answer
58 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 ...
0
votes
0answers
27 views
Non linear optimization Gradient method
Let $f(x)= x^2 -\frac{x^3}{3}$. Ok so i found the local min is at 0 and i was given $x_0=1,\alpha = \frac{1}{2}$, I dont understand how i am suppoused to find $x_k$ such that $x_{k+1}=x_k-\alpha ...
3
votes
2answers
62 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
43 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
78 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
108 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
$$\min_{X \in \mathbb R^{n ...
2
votes
1answer
67 views
Cubes, squares and minimal sums
I have trouble solving the following task: i need to find positive integers a and b such that
1) $a \neq b$
2) $ \exists c \in \mathbb{N} : ~ a^2 + b^2 = c^3$
3) $\exists d \in \mathbb{N}: ~ a^3 + ...
0
votes
1answer
24 views
inequalities for optimization over psd matrices with constraints
Consider two p.s.d. matrices $A$ and $B$ both in $\mathbb{R}^{d \times d}$. Define $$a = argmax_{x \in \mathbb{R}^d} x^\top A x $$ and $$b = argmax_{x \in \mathbb{R}^d} x^\top B x $$ both subjected to ...
3
votes
3answers
103 views
Max and Min of $f(x,y)$
Let $f(x,y)=x(y \log y-y)-y \log x$. Find $\smash{\displaystyle\max_{\frac{1}{2} \leq x \leq 2}}(\smash{\displaystyle\min_{\frac{1}{2} \leq y \leq 1} f(x,y)})$.
2
votes
3answers
50 views
Packing radios into cartons - why is my solution wrong?
A manufacturer of car radios ships them to retailers in cartons of $n$ radios. The profit per radio is $\$59.50$, minus shipping cost of $\$25$ per carton, so the profit is $59.5n-25$ dollars per ...
3
votes
1answer
49 views
Some basic questions about minima of a real-valued functions
The following theorem is basically from the Fermat's Theorem page of wikipedia.
Let $X$ denote a subset of $\mathbb{R}$, and suppose $f : X \rightarrow \mathbb{R}$ attains
a global minimum at $x ...
2
votes
4answers
123 views
Minimize $\sum a_i^2 \sigma^2$ subject to $\sum a_i = 1$
$$\min_{a_i} \sum_{i=1}^{n} {a_i}^2 \sigma^2\text{ such that }\sum_{i=1}^{n}a_i=1$$ and $\sigma^2$ is a scalar.
The answer is $a_i=\frac{1}{n}$.
I tried Lagrangian method. How can I get that ...
0
votes
2answers
53 views
Please help to make me understand why I cant optimize this function: $U=x^{1/3}*y^{2/3}$ ?
If I want to maximize a production the function of which is given by $$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
27 views
Example of delayed column generation
Can someone point me to a small example of how delayed column generation works for the cutting stock problem. I have found several sources that describe it abstractly but I still don't understand ...
0
votes
0answers
56 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
237 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
148 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
37 views
Distributed Robust Optimization
Consider the following constrained optimization problem $\mathcal{P}$.
$$ \min_{x \in X \subseteq \mathbb{R}^n} f(x) \ \text{sub. to: } g(x,y) \leq 0 \ \forall y \in Y \subseteq \mathbb{R}^m $$
...
0
votes
1answer
57 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
39 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
56 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
120 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
60 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
0answers
34 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: ...
0
votes
1answer
33 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
41 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
34 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
36 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
0answers
28 views
how to determinte 3 parameter by best fitting?
I have a bunch of experimental data given by someone else which should fitting into the following form
$$
y = A\exp(-b/(x-\mu))
$$
where $A$, $b$ and $\mu$ are constant but not known. I am ...
0
votes
2answers
49 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
53 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
29 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 $-4e^{-2}$.
6
votes
0answers
154 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
32 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
50 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
43 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
54 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
42 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
49 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 ...
1
vote
1answer
58 views
Tricky algebra for minimization
Find the local minimum for $f(x, y) = 2x^4 + y^2 - 4xy + 5y,\:x,y \in \mathbb{R}$ find the local minimum.
Okay this seems easy enough, the necessary condition dictates that candidates are of the form ...
0
votes
1answer
54 views
Lipschitz constant for optimization of multivariate function
I intend to implement an optimization algorithm which requires the computation of the Lipschitz constant. My function is a multivariate function with more than 50 variables. I am wondering whether ...
0
votes
1answer
68 views
Show that the infimum is convex
Let f be bounded convex function on a convex subset $A\times B \in \mathbb{R^m}\times\mathbb{R^n}$. Define $g(x)=\inf{f(x,y):y\in B}$ Show that g is convex on $A$.
Okay, let $x = (x_1,x_2), ...
0
votes
1answer
98 views
If $f$ is strictly convex in a convex set, show it has no more than 1 minimum
Suppouse $A \in \mathbb{R^n}$ is convex. If $f:A\to\mathbb{R^n}$ is strictly convex, show that the set of minimizers if either a singleton or empty.
Ok, Suppose there exist more than one minimizer, ...
2
votes
0answers
32 views
maximal m-elements of the matrix inversion
Suppose the $n\times n$ matrix $A$ is invertible, and all its elements are between 0 and 1. The existing matrix inversion operation of $A^{-1}$ will take $O(n^3)$ time. Now I just want to find the ...
1
vote
1answer
51 views
How would you solve a Tikhonov Regularized Least Squares problem with nonnegative constraints?
For a Tikhonov Regularized Least Squares problem with nonnegative constraints, what are some methods that solve it?
Are methods solving a Least Squares problem with nonnegative constraints and the ...
2
votes
2answers
64 views
Are there constraint problem calculators?
So I just remembered Lincoln Logs exist, so I found ten giant sets of them on ebay for Buy It Now, and I'm trying to decide what combination of purchases gives me the most logs for the least money if ...
1
vote
2answers
83 views
Is $y=x+\frac{1}{x}$ a hyperbola?
How do we confirm or disprove that? And is there a name for this kind of function? $$f(x)=c(x-a)+\frac{d}{x-a}+b$$
If we restrict that $x-a>0$ and $c,d>0$, an observation is that the minimum ...
3
votes
1answer
109 views
Why are additional constraint and penalty term equivalent in ridge regression?
Tikhonov regularization (or ridge regression) adds a constraint that $\|\beta\|^2$, the $L^2$-norm of the parameter vector, is not greater than a given value (say $c$). Equivalently, it may solve ...
1
vote
2answers
80 views
Max and min value of $7x+8y$ in a given half-plane limited by straight lines?
So, there are four inequalities:
$$\begin{eqnarray*}
y &\geq &-3x+15; \\
y &\leq &-11/3x+56/3; \\
x &\geq &0; \\
y &\geq &0.
\end{eqnarray*}$$
If we draw all those ...
5
votes
0answers
62 views
$\int_{-\infty}^{+\infty}\left|f(g(t))-f(h(g(t)))\right| \,dt=0$ How to find $f$?
Let $h,g$ be given entire functions.
Consider
$$\int_{-\infty}^{+\infty}|f(g(t))-f(h(g(t)))|\, dt=0,$$
where $|\cdot|$ means modulus.
How do I find non-polynomial analytic solutions for $f\,$?
...
