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").

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min : sum of two L2 norms

Is there a closed-form solution of the: $$\min_x \| Ax - b1 \|_2 + \| Ax - b2 \|_2$$ where A = m*n matrix and b1,b2 are m length vectors.
Gopal Chitalia
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What is $\max\limits_{\sum_{i=k}^{n}x_i\leq\sum_{i=k}^{n}y_i \\\forall k=1,2,\cdots,n} \prod_{i=1}^{n} x_i$

\begin{array}{ll} \text{maximize} & \prod_{i=1}^{n}x_i\\ \text{subject to} & \mathrm \sum_{i=k}^{n}x_i\leq\sum_{i=k}^{n}y_i \\\forall k=1,2,\cdots,n\end{array} if $x_1\geq\cdots\geq x_n$ and $y_1\geq\cdots\geq y_n\quad$ ($x_i,y_i\in \mathbb{R}^+$…
Lee
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optimization in image processing

I want to solve the following problem. I know it is an optimization problem, but since I don't know anything about optimization, I would appreciate if you would give me some guidance regarding where to look in order to find the answer (what kind of…
user53785
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What are the prerequisites in general to study mathematical Programming \ Optimization?

What are the prerequisites in general to study mathematical Programming \ Optimization? I'm new to this theory. Please Provide me with detailed Answers.
Ahmed
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Classifying singular points as local min, max or saddle points

I want to determine if a singular point is a local min, max or saddle point. We are dealing with singular points so we cannot use the hessian matrix. What I have written, and I think I must of missed something is : Say we have a function…
Justin
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Gradient vector and Hessian matrix of function containing vector of complex exponentials

I would like to obtain the gradient vector and Hessian matrix for the following function: $$f(\boldsymbol{\theta}) = \mathrm{real}\{u(\boldsymbol{\theta})^H\Gamma u(\boldsymbol{\theta})\}$$ where $u(\boldsymbol{\theta}) = \boldsymbol{u} =…
F. C.
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Dynamic optimization problem.

My starting equations are the following: $V^{e}=w+\beta.((1-s).V^{e}+s.V^{u}(t))\\V^{u}(t)=\begin{matrix} \\max \\a(t) \end{matrix}b(t)-\psi.a(t)+\beta.[\pi(a(t)).V^{e}+(1-\pi(a(t))).V^{u}(t+1)]\\\pi(a(t))=1-exp(-\phi.a(t))\\b(t)=b.t^{-\mu}$ All the…
user53834
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Local Maximum of a function

In a problem, I'm asked to find the local maximum of the function: $$ \rho_v = (\rho^2 - 10^{-4})z\sin(2\phi) $$ over the solid: \begin{align*} 0.005 &\leq \rho \leq 0.02 \\ 0 &\leq \phi \leq \frac{\pi}{2} \\ 0 &\leq z \leq…
osmano807
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Finding the largest sphere that fits inside a polyhedron

If we are given a three-dimensional polyhedron $P \subset \mathbb{R}^3$ defined by $m$ linear inequalities: $$P = \{\textbf{x} \in \mathbb{R}^3\ |\ A\textbf{x} \leq \textbf{b}\}$$ where we let $\textbf{a}_i^T $ denote the $i^{th}$ row of $A$, then…
User1996424
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Proof that $E_n = \int_0^1 \left| \ln t - P_n(t) \right|\mathrm{d}t = 1/(n+1)^2$

Assume one wants to minimize the distance between $f(x)=\ln x$ and $P_n(t)$ where $P_n$ denotes a polynomial of degree $n$. Etc $P_1 = ax 0+ b$. One way to judge whether the polynomial is a good approximation is to minimize the integral $$ E_n =…
N3buchadnezzar
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maximize $\prod_{i=1}^{N}{\left(1+x_{i}\right)}$

I am trying to find the set of $x_{i}$ that maximize $\prod_{i=1}^{N}{\left(1+x_{i}\right)}$ given that $\prod_{i=1}^{N}{x_{i}} = c$ and $0\leq x_{i} \leq 1$ for all $i$. As far as I can tell unless the solution set is $x_{i}$ is equal to…
StrongBad
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Conceptual question about solving for critical points

I am hoping someone can please explain a few lines of working to me regarding critical points. I am really confused as to what I am actually doing in each step. I have the following function: $$ \\f(x,y)= 2x^2-y^2+6y$$ I am asked to find the…
melm
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Farthest Possible Points in a Rectangular Room

A rectangular room has dimensions 12x12x24. That is, the floor and ceiling and both the side walls are 12x24 and the 2 end walls are 12x12. A point M corresponds to one of the vertices on the floor. If we define the distance between any 2 points on…
laser01
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maximisation of value of a function under constraints

Let $x,y$ and $z$ be positive real numbers . What is the maximum value of $xyz$ under the constraints $x^2+z^2=1$ and $y-x=0$ Please explain the process.
Khamba
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Objective function vs fitness function.

What is the difference between an Objective Function and a Fitness Function? Are Objective Functions only used in Linear Programming? Are Fitness Functions only used in Genetic Algorithm? Do we need to apply any operation on a mathematical function…
user366312
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