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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").

22512 questions
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finding the circle with maximal rainfall

Let's say that I have the rainfall in millimeters for each point on a 2-dimensional map. I want to find the (x,y) coordinate of a circle of radius R that has received the maximum amount of rain. Something like (for a square actually, probably in…
Glasnhost
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Minimising Olympic Medly Times

Just a question inspired by the olympics. Say we are to select a medley team from 4 swimmers and wish to minimise our total time for the race. For example say the pb's for each style and person are: \begin{array}{|c|c|c|c|c|} \hline & \text { Free…
Mathman
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Global minimizer and maximizer for $f(x):=\|A-xx^T\|^2_F$

Consider $f(x):=\|A-xx^T\|^2_F$, where $A \in \mathbb{R}^{n \times n}$ a symmetric matrix and positive eigenvalues $\lambda_1>\lambda_2>\cdots>\lambda_n>0$ with corresponding eigenvectors $u_1,u_2,\ldots,u_n$ I want to compute gradient and Hessian…
Jayden Rice
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Visit 25 rooms - Max Distance

Below are 25 rooms. The centre of the rooms are 1 unit apart from each other - horizontally and vertically. You want to visit all the rooms. You can only visit a room one time. You start in the "red squared" room. Question: If you can "jump" however…
Prim3numbah
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What shape should non-negative vector with unitary $L^1$-norm take to minimize this function?

Imagine that: $\vec{v}$ $\in$ {non-negative vector in $R^N$, such that $||\vec{v}||_1 = 1$ and $v_0 = v_N = 0$}. We define a function on that space: $f(\vec{v}) = \sum_{i=1}^{N}(v_i - v_{i-1})^2$ Question: What form should $\vec{v}$ take to minimize…
Novinator
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hint for minimising $L_p$ norm

$$u^* = \arg\min_{u} \left(\sum_{i=1}^{8} |X(i)-u|^p\right)^{1/p},$$ I need hint solving above equation. I tried deriving with $u$ but I think it's not the right way. where $p\in \{1,\infty\}$, $X(i)=i$
anfjnlnl
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Convexity of argmax

Suppose that the function $F(x,y)$ is supermodular in $(x,y)$. We know that this implies that $x^* =\operatorname{arg\, max} F(x,y)$ is increasing in $y$. Under which conditions (if any) do we have that $x^*$ is a convex function of $y$?
Filomena Garcia
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Maximizing sum of strictly non-negative functions?

In an answer to this question, it was stated that if a function is the sum of two functions $f(x)=g(x)+h(x)$, and ${(\forall{x})(g(x)\ge0)}$ and $(\forall{x})(h(x)\ge0)$, the relative maxima of $f(x)$ can be found by finding the relative maxima of…
rurouniwallace
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Given $x,y,z\geq0$ and $x^2+y^2+z^2+x+2y+3z=13/4$. Find the minimum of $x+y+z$.

Given $x,y,z\geq0$ and $x^2+y^2+z^2+x+2y+3z=13/4$. Find the minimum of $x+y+z$. I tried many method, such as AM-GM, but all of them failed. Thank you.
JSCB
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How do I approach optimizing this multivariable function? What should I learn to be able to do that?

Function to maximize $R(P) = \sum\limits_{i=1}^4 (-\frac{1}{2}p_i^2 + 1000p_i)$ Parameters $P = (p_1, p_2, p_3, p_4)$ Constraints $p_1, p_2, p_3, p_4 > 0$ $\sum\limits_{i=1}^4 (-\frac{1}{2}p_i + 1000) \leq 2000$
m_ocean
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Optimization theory. active set

Consider an optimization problem where you have $$ \min_x f(x)\\ \mbox{s.t} \, \; h(x) = 0 \\ g(x) \leq 0$$ I have understood that the active set consists of the equality constraints together with the part of the inequality constraints that…
Fritz
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max min is less than min max proof

I saw the following proof that max min of a function is $\leq$ than min max of a function on Max Min of function less than Min max of function, pasted below for your reference Let $ f(x_{0}, y_{0}) = \max_x\min_y f(x, y)$ and $f(x_{1}, > y_{1})…
user5965026
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How to maximize functions with strange constraints?

I want to maximize the function $$f(x_1, x_2, x_3) = \dfrac{2\sqrt{3-(x_1+x_2+x_3)^2}}{x_2-x_1}$$ subject to the constraints $$x_1^2 + x_2^2 + x_3^2 = 1$$ and $$0 \leq x_1 \leq x_3 \leq x_2 \leq 1.$$ So normally this function would be unbounded as…
Wyatt Gregory
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What is going on with this constrained optimization?

I'd like to figure out what is going on when trying to maximize a function (below $a_i$ are real numbers) $F = a_1a_2 + a_2a_3 + \cdots + a_{n - 1}a_n + a_na_1;$ When we have active constraints $h_1 = a_1 + a_2 + \cdots + a_n = 0;$ $h_2 = a_1^2 +…
Ranas
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Maximizing possible value of a function given constraint

A few weeks ago I took part in a math competition and had this question through some trial and error I got the correct answer 80 as I didn't have time to use maximization using calculus, is there a way to do this question quickly in a rigorous…
Il Leone
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