Questions tagged [optimization]

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.

In mathematics, computer science, economics, or management science, mathematical optimization (alternatively, optimization or mathematical programming) is the selection of a best element (with regard to some criteria) from some set of available alternatives.

An optimization problem can be represented in the following way: given a function $f:A\to\mathbb{R}$ from some set $A$ to the real numbers, we want to find an element $x_0\in A$ such that $f(x_0)\le f(x)$ for all $x \in A$ ("minimization") or such that $f(x_0)\ge f(x)$ for all $x \in A$ ("maximization").

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What is the optimal value for x in a 1-1000 random number generator 2 stage algorithm where x is the "split point".

I would like a mathematical answer to this. If I have a random number generator algorithm (let's say 1 to 1000), and I simply pick a random number over and over until I see each of the 1000 possible numbers at least once, as we get closer and…
David
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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 I need to use Lagrange multipliers to do this, but I'm…
user55225
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Convex maximization problem (existence and uniqueness)

Short summary: Let $n > k \geq 2$. I am given an arbitrary full column rank $n \times k$ matrix $W$ whose rows sum up to $1$. I am trying to find a square matrix $R$ such that $W R$ has certain properties: component-wise, it should be …
Lithiesque
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How to fit ordinary differential equations to empirical data?

For some biological systems, there exists ordinary or partial differential equations that allow one to simulate their activity/behavior over time. Some of these models even produce data that is very difficult to tell apart from real data. What I…
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Special linear matrices with minimal norm

If $A$ is matrix in $\mathcal M_n(\Bbb R)$ we define the norm $$\Vert A\Vert = \sqrt{\operatorname{Tr}(A^TA)}$$ Now I'm wondering is it true that over the set of special linear matrices $M$ i.e. such that $\det(M)=1$ the matrices with minimal norm…
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Portfolio optimization

I prepare Markowitz optimization model for 4 different local companies with Excel functions and Solver. And I get confusing result for me, maybe somebody will explain it. So, from historic information I calculated daily returns and from that daily…
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Closed Form Solution for a Sum of Linear Least Squares Problems of the Same Argument

I have the following minimization problem of: $$ \min_W \sum_{n=1}^{N} \| l_n - P_n W x_n \|^2_2$$ where $l_n$ is a $C$ dimensional vector, $P_n$ is a $C \times L$ dimensional matrix, $W$ is a $L \times D$ dimensional matrix and $x_n$ is a $D$…
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Derivative of quadratic with Hadamard product

I proposed a similar question involving logarithms, but the problem is about scalar. I am trying to solve the more generalized form: $$ \min_{\mathbf{x} \in \mathbb{R}^N_+} \left( \sum_i \left( h_i^T(\mathbf{x}\circ\mathbf{x}) - \mathbf{c_1}\log…
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Prove that $\sqrt{2a^2+b+1}+\sqrt{2 b^2+a+1}\geq 4$ when $a+b=2$.

Suppose that $a,b$ are non-negative numbers such that $a+b=2$. I want to prove $$\sqrt{2 a^2+b+1}+\sqrt{2 b^2+a+1}\geq 4.$$ My attempt: I tried proving $$2a^2+b+1\geq 4$$ but this seems wrong (try $a=0,b=1$). How can I prove the above result?
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linear optimization of multiplication coefficients

There are $n$ multiplication coefficients for which optimized values are searched. There is a restriction that the multiplication coefficients are not allowed to be negative and must have a sum of $1.0$: $\sum_{i =0}^n c_i = 1.0$ The coefficients…
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Matrix Differentiation of Generative Neural Network

I wish to construct a neural network (restricted Boltzmann Machine) with network parameter set $\Omega$ that will minimise the cost function, \begin{equation} \mathcal{C}(\Omega) = -\text{Tr}\big[A\log(B_\Omega)\big] \end{equation} where $B_\Omega$…
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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 moves for a solution has to be 11. I also want to…
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$\max_{\|x\|=1}f(x)=\max_{\|x\|\le 1} f(x)$

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 statement from the book: If $f:…
Shiyu
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Convex functions lack saddle points?

I am reading "Deep Learning" by Ian Goodfellow. At page 86, the author explains how to use the Hessian to evaluate whether a point of a multivariate function is a maximum or a minimum At a critical point, where $ \nabla_x f(x)=0 $, we can examine…
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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: $$\min_\mathbf{q}{|\mathbf{BXq|^2}}$$ $$\, \mbox{s.t}\ q_i^Hq_i=1, i=1,2,...,N$$ $\mathbf{B}$ is given and…
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