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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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How to describe this relation

The constraint I want to place is $Mx_i-y_i<0$,that is to say when $y_i = 0$ ,then $x_i$ must equal to $0$. However the constraint is only needed when $a>b$ ($a,b$ are both the parameters). Can I describe the constraint as below? $$Mx_i-y_i < 0,\…
frank-z
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Why the local optimum of this problem is always global?

I am reading the paper "Continuous methods for extreme and interior eigenvalue problems" by G.H. Golub and L.-Z. Liao. The papers says for the following problem, (Lemma 2.1 (i), $0>\lambda_i-c\ge \lambda_1-c$, $i=1,...,n$ and $c$, $\lambda_i$s are…
user112758
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Order of evaluations

I have a list of objects with $N$-dimensional criteria (actually hotels in my case). Now I'd like to optimize the order of the criteria by which I filter to spent the least time on some evaluation process per object. Each property $i$ takes a fixed…
Gere
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Can I call my algorithm as EM algorithm? Can I claim convergence guaranty?

I am trying to minimize the following problem: $$\hat{x} = \underset{\bar{x}}{\operatorname{argmin}} \left( \left\|y- A_{MV}^{EPG} \bar{x}\right\|^2 + \|\mu_{T} \bar{x}\|^2 + \|D_{s} \bar{x}\|^2 \right)$$ Where $\bar{x}$, a column vector, is my…
Dushyant Kumar
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In Nonsmooth Optimization, whats the point of $x \in G$ being a local minimum results in $0 \in \delta\ f(x)+N_G(x)$

I'm reading about nonsmooth optimization and specially the book 'Nonsmooth Optimization: Analysis and Algorithms with Applications to Optimal Control' by Marko M. Mäkelä, Pekka Neittaanmäki. At page 72 one finds the following Theorem 5.1.6: If $f$…
John Doe
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Minimization of vector sum through rotation

I'm curious if there's an algorithmic way to find the minimal vector sum of $N$ vector magnitudes by applying a rotation $\Phi$ to each individual vector. As an example in two dimensions, if I have three vector magnitudes $||x||, ||y||, ||z||$ that…
Brady S
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Stuck formulating constrained optimization problem with Simplex

I have an exercise to solve, and it is a constrained optimization problem. Here it is: "A company makes large championship trophies for youth athletic leagues. At the moment they are planning production for fall sports: football and soccer. Each…
Vento
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Bilinear Optimization Problem

How could I solve the following optimization problem using MATLAB or an other way? Given ${E}^{1}, {A}^{21}, {A}^{22}, {C}^{1}, {A}^{12}$ $ \underset{{C}^{2}, {E}^{2}}{min} {\left \| {C}^{2}{E}^{1} - {A}^{21} \right \|}_{F}^{2} + {\left \|…
Royi
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Maximum of $xy+y^2$ subject to right-semicircle $x\ge 0,x^2+y^2\le 1$

Maximum of: $$ xy+y^2 $$ Domain: $$ x \ge 0, x^2+y^2 \le1 $$ I know that the result is: $$ \frac{1}{2}+\frac{1}{\sqrt{2}} $$ for $$ (x,y)=\left(\frac{1}{\sqrt{2(2+\sqrt{2})}},\frac{\sqrt{2+\sqrt{2}}}{2}\right) $$ But I don't know how to get this…
Diana
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Lagrangian Multiplier for liner problem

I have a (probably) stupid question but I can't find the answer. I have the following problem (my problem is much more complicated but as an example) : \begin{equation} \begin{matrix} \displaystyle \min_{\substack{q_1,K}} & J= CK + \sum_{i=1}^{10}…
NoName
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Find the Min of P(x,y)

Find the Minimum of the following function : $$P(x,y) = \frac{(x-y)}{(x^4+y^4+6)}.$$ This is a math problem I found in an internet math competition but it is really complex to me !!!
lomasomu
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Irrational equation for a maximization problem

I have the following maximization problem $ \max_h m_1 + 10 (h)^{1/4} + h + m_2 - 2 (h)^{1/4} + m_3 - (h)^{1/4} $ where $m_1, m_2, m_3$ are three fixed values. The FOC for a maximum is $ \dfrac {10}{4}…
Luigi
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Maximize $f(\textbf{x},\textbf{y}) = f_1(\textbf{x}) + f_2(\textbf{x},\textbf{y})$

(Sorry if I'm not formal enough) Let $$ f(\textbf{x},\textbf{y}) = f_1(\textbf{x}) + f_2(\textbf{x},\textbf{y}) $$ a real function of vector variable $\textbf{z} = (\textbf{x},\textbf{y})$ , that admits max. You can assume both $f_1, f_2$ have max…
user8469759
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Maximum value of $(ab)(a+b)$ given $a^2 + b^2 = 100$

Can anyone help me with finding the maximum value of $$y = {(ab)(a+b)}$$ With that condition that $$ 100 = a^2 + b^2$$ and both $a,b$ are positive numbers. Any help appreciated. Thanks.
Bootstrap
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Optimizing community learning.

I'm playing with the problem of community learning. I think this is pretty generic as it can be applied to economic development, investments and others. I want to know if there's a name for this and if there is previous work before delving…
Axiverse
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