Mathematics Stack Exchange
Mathematics Stack Exchange

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").

22512 questions
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Minimize Rosenbrock Function With Conjugate Gradient Method

I want to minimize $$ f(x,y) = (1-x)^2 + 100(y-x^2)^2 $$ For the conjugate gradient method I need the quadratic form $$ f(\mathbf{x}) = \frac{1}{2}\mathbf{x}^{\text{T}}\mathbf{A}\mathbf{x} - \mathbf{x}^{\text{T}}\mathbf{b} $$ Is this even possible…
Philipp
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Is this a Known/Solved Optimization Problem?

Background: There are n entities. Each entity has the same m attributes. Each attribute can vary between several values. Problem: Select the smallest subset of entities such that you have at least one entity for each value of each…
Robert Robison
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Minimum energy configuration of N charged particles in a circular enclosure

I came across this paper which describes minimum energy configuration for N equal point charges in a circular enclosure which seem to form shell like structure. For lower values of N local optimum is point charges spaced equidistant from each other…
Avinash Kumar Ranjan
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Lagrange multiplier application question

This is the application of Lagrange multiplier. Is this solution true? If there exist a little bit mistake in detail of the answer, please tell me to correcting this. Sorry for adding photo. I dont have enough ability to write this as MathJax.…
Bstr
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Optimization over subsets of $\mathbb{R}$

I have a problem which has the form $$ \begin{align*} \min_{\mathcal{A}} \quad & \int_\mathcal{A} f(x)\; dx \\ \text{s.t.} \quad & \int_\mathcal{A} g(x)\; dx = C \\ & \mathcal{A} \subseteq [a, b] \\ & f, g \text{ integrable over }…
sirallen
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Balancing learning rates when optimizing multiple variables

I have the following optmization problem. There are values $p_1', p_2', ..., p_n'$, which can be approximated using the following equations: $$ p_1 = F_1 (g(\vec{a}, b)) \\ p_2 = F_2 (g(\vec{a}, b)) \\ ... \\\ p_n = F_n (g(\vec{a}, b)) $$ $\vec{a}$…
Mathnoob
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Find two integers such that the sum of the differences is minimum.

I have a list of non repeating positive integers $(a_1,\dots,a_n)$. I need to find integers $x, y$ such that the value of $\sum_{i=1}^n\min\{|a_i-x|,|a_i-y|\}$ is minimal. For example: If $a = [2,3,6,7]$ (my list of numbers) then $x$ and $y$ for…
Rick
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Optimization problem [proof of generalized cauchy schwarz]

Could you help me solve this problem please ? Maximize $x^ty$ with constraint $x^tQx \leq 1$ (where $Q$ is definite positive) What I tried : I tried using KKT but I don't know why I get $-\sqrt{y^tQ^{-1}y}$ as the maximum instead of…
array.02
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how to make the objective value of primal program close to zero

\begin{equation} \begin{aligned} \min_{t_1, t_2 \in R} \quad & t_1t_2 + \frac{1}{t_1t_2^2}\\ \textrm{s.t.} \quad & t_1, t_2 > 0\\ \\ \end{aligned} \end{equation} How do I make the objective value of the given primal program arbitrarily close to…
jun
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Minimizing error with Newton-Gauss optimization

I have a function which I'm trying to estimate: $U(x, y) = \begin{bmatrix} a_0 + a_1x + a_2y \\ a_3 + a_4x + a_5y\end{bmatrix}$ I have an initial estimate of the parameters. I want to iterate n times and identify a $U_n$ such that $E(U) =…
Utkarsh Sinha
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Maximize a function

Resolve the following problem. Let $\alpha, \beta > 0, p_{1},p_{2} > 0$ fixed. Maximize $x^{\alpha}y^{\beta}$ subject to $p_{1}x + p_{2}y = 10$ $x,y > 0$.
Rem
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How is ADMM Separable?

I'm learning about ADMM by reading Boyd's paper Distributed Optimization and Statistical Learning via the Alternating Direction Method of Multipliers. The paper says that ADMM is an improvement over the Method of Multipliers because the latter…
user7652576
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Maximize product with constraint

$A_i$ and $x_i$ are positive numbers, I want to solve the following: \begin{align} {\text{maximize}} &\hspace{3mm}& \prod_{i=1}^N (1 + A_{i}x_{i}) \\ \text{subject to} &\hspace{3mm}& \sum_{i=1}^N x_{i} = 1 \end{align} I was trying to guess the…
Wai
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Number of paths in asymmetric capacitated vehicle routing problem

This seems like a natural question, but I was not able to find an answer. My question is this. Consider a capacitated vehicle routing problem (CVRP) where $n$ customers have to be serviced by $m$ vehicles on any given day. All the vehicles have the…
arunisnowhere
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Maximising the minimum of two functions

I am trying to find the values $x \in [0, 16]$ and $y \in [0, 12]$ that maximise $$ f(x,y) \equiv \min \{\ln(x) + 2\ln(y), \ln(16-x) + 2\ln(12-y) \}$$ Intuitively, it is pretty clear that the solution is $x = 8$, $y = 6$. However, I am having a bit…
afreelunch
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