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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Intuition about the Cauchy-Schwarz inequality and optimization problems

I'm trying to develop intuition about the Cauchy-Schwarz inequality. Suppose I have positive real vectors x and y. The values of x are already determined and I want to find the values of y that maximize $\sum_i x_iy_i$ subject to $\sum_i y_i^2…
Gaussian0617
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Discrete optimization

I'm having troubles with searching for analytical solution of following problem. Let we work in 3-D space and have the set of points (uniform net at cube's facets): ($-1,\hspace{2mm} -1+j*h,\hspace{2mm} -1+k*h$), ($1,\hspace{2mm} -1+j*h,\hspace{2mm}…
Sh.N.
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Showing a $\mathbb{R}^2 \rightarrow \mathbb{R}$ function attains a global maximum

Given $f:\mathbb{R}^2 \rightarrow \mathbb{R}$ defined by $$f(x,y) = (ax^2+by^2)\exp(-x^2-y^2)$$ where $a > b > 0$, how can I show $f$ attains a global maximum? It is easy to show that it attains a local maximum via the Hessian matrix and I suspect…
David
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Estimating coefficients in a physical system based on observations

I have a physical system which can be modelled as $$Ax+By+C=0$$ I have thousands of measurements of $x$ and $y$ from the physical system (includes some noise). I want to optimize for $A$, $B$, and $C$. More Details: $A$, $B$, and $C$ are composed of…
user2025983
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The optimal value of a simple optimization problem

Let $S_n$ be the simplex in $n$-dimensions, i.e., the set $\{ (x_1, \ldots, x_n) ~|~ \sum_{i=1}^n x_i = 1, x_i \geq 0 \mbox{ for all } i \}$. I am interested in the optimization problem $$ {\rm max} \sum_{i=1}^n c_i x_i $$ subject to the two…
robinson
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Is my Lagrange function correct? How can a point be a maximum and minimum?

minimize $3x_1^2+3x_2^2-2x_1x_2-12(x_1+x_2)+36$ s.t. $|x1+x2|=1$ Determine the maximum and minimum of this problem. The Lagrangian can be set up as: $L(x,\lambda)= 3x_1^2+3x_2^2-2x_1x_2-12(x_1+x_2)+36-\lambda(x1+x2-1)$ From the conditions I infer…
Steven
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Optimizing a positive definite quadratic form with inequality constraints.

I have a positive definite, multidimensional quadratic form: $(x-x_o)^t M (x-x_o)$, where the "${}^t$" indicates transpose and $M$ is a positive definite matrix (in fact, it is a multidimensional guassian). So I know there is a single global maximum…
Luis A. Aguilar
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Optimization of solve

Find the minimum value and the maximum value of the function $$y(x)=\frac{x^3}{x-3}$$ when $4\le x\le5$ I found that $f(x)$ is decreasing on the interval $[4,\frac{9}{2}]$ and increasing on $[\frac{9}{2},5]$ What i actually want is, find way to…
Музаффар Шакаров
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optimization question

Minimize $x_1+6x_2-x_3$ subject to $7x_1+x_2-x_3=6$, $3x_1+x_2+2x_3\leq 6$, $x_1,x_2\in\mathbb{R_+}$. I first tried to represent $x_3$ in terms of $x_1$ and $x_2$, so $x_3=7x_1+x_2-6$, substituting this into the cost function:…
Emir
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group membership assignment by preferences - optimization problem?

I have 30 users and 7 groups. All users will be members of exactly 2 groups. Each group must have between 4 and 6 members. The users have ordered their preferences for group membership (i.e., users have listed in order the groups they would like to…
mjibson
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Ideal amount of piles to sort a stack of 250 cards (magic the gathering)

I'm a hobbyist working on a mechanical sorting machine to sort magic the gathering cards. I'm by no means a mathematician though, and I was wondering if you all wouldn't mind helping me out with a math puzzle to determine the best route to go with…
user13226
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convex set, convex function

I have the following constraints: $$\sum_{1\leq i,j\leq n,\ i\neq j} x_ix_j\geq 0.25$$ $0\leq x_i \leq 1$ for $i=1, \ldots, n$ Is this set convex? I think so, but $0.25-\sum_{1\leq i,j\leq n} x_ix_j$ is a convex function? or not? Note that I have…
user29271
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Saturation Curve

I have an expression which is $\displaystyle\frac x{x+40}$. I'm trying to find a point indicated in the graph. As you can see I drew 2 lines, one tangent to the region which it saturates, the other were it has max growth. where they intersect, I…
Hello
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Minimizing Norm

I have below problem: Find $\bf C$ to minimize $\|\mathbf A-\mathbf B\mathbf C\|_F$. Given ${\bf B} \in \mathbb R^{m \times n}$, ${\bf B}$ has lin. ind. col. A satisfies: ${\bf DA} = {\bf E}$ , ${\bf D} \in \mathbb R^{m \times m}$, ${\bf D}$ has…
dale
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How can I optimize a multi-variable expression with a constant target.

I would like to know what methods are applied for optimizing multi-variable expressions with a defined target. I have a specific example I need help with, but I would like to be pointed into the right direction to apply the theory elsewhere. The…
Daniel Dunn
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