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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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About the existence of a certain optimal schedule for the $1|\text{ } | \sum w_j T_j$ problem

There's a homework exercise I have to do for my scheduling course about which I have some questions. The exercise is this: Consider the $1|\text{ }|\sum w_jT_j$ problem. (This means that we look at one machine which has to process a number of jobs…
Stijn
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Transformation of fractional constraint to convex/linear constraint in optimization problem

I have this constraint in an optimization problem: $$\frac{a}{x_1} + \frac{b}{x_2} + \frac{c}{x_3} = d$$ where $x_i$ are the variables. Other than this my problem is linear. Is it possible to transform this constraint into a linear form?
Jakob Loegstrup
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What is a cone, possibly of polynomials?

I'm reading texts about optimization when given a set of polynomial constraints, like $f_1=\cdots=f_m=0$ and $q_1\ge 0,\ldots,q_\ell\ge 0$, over the reals. I have encountered statements about "the cone" of a polynomial or a set of polynomials. For…
Jack
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Minimising subject to inequality constraints

$$\begin{array}{ll} \text{minimize} & e^{x-y}\\ \text{subject to} & e^x + e^y \leq 20\\ & x \geq 0\end{array}$$ My attempt: To minimize the objective function, minimise $(x-y)$, i.e., maximize $y$ and minimize $x$. Given the 2nd constraint, the…
LUCIFER
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Optimization: Cut wire in to two pieces one square and one circle

A straight piece of wire $40$ cm long is cut into two pieces. One piece is bent into a circle and the other is bent into a square. How should wire be cut so that the total area of both circle and square is minimized? my work I found the function to…
Pwer331
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Sum of Norms to Root of Function

Let $\alpha$ be a line defined by $\alpha(t): [0,1] \rightarrow \mathbb{R}^3$ and $x_1, \ldots, x_n \in \mathbb{R}^3$ be $n$ fixed points. Let $A(t) = \sum_{i=1}^n \frac{1}{\|\alpha(t) - x_i\|_2}$ the sum of the distances from any point in $\alpha$…
Johnny
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Finding the stiffest wooden beam from inside a log. (Rectangle inside circle)

Problem The stiffness of a wooden beam of rectangular cross section is proportional to the product of the width and the cube of the depth of the cross section. Find the width and depth of the stiffest beam that can be cut out of a circular log of…
Tuki
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Local minimum of a function

Let $f:\mathbb R^n \mapsto \mathbb R$ be a differentiable function. Let us assume that the point $\bar x$ is a local minimum of the function f on every line passing through $\bar x$ what means that the function $g(\alpha) = f(\bar x + \alpha d)$ has…
treskov
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Proving minimum of distances exists

My first stackexchange post! I'm ready to be flamed haha. ;) My problem: For some n points, prove that there exists a point p that minimizes the sum of the distances between p and each of the n points. At first, I thought this would be easy to…
ekim
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maximum sum of dot product - optimal pairs of vectors

Suppose I have a set of vectors $A_{i}$ and another set $B_{j}$ where $A_{i}$ and $B_{j}$ do not necessarily contain the same number of elements. What is the optimal pairing of vector $A_{i},B_{j}$ such that you produce the largest sum of dot…
user3071643
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Minimize $f(x)=\lambda|x|+1/2(x-a)^2$

I have so far: $x-a+\lambda = 0,x>0$ $x-a-\lambda = 0,x<0$ Is there an explicit solution w.r.t. $a$ and $\lambda$?
jubueche
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Prove that $C_1$, $C_2$ are convex cones $\rightarrow$ $x\in C_1\bigoplus C_2$ is a convex cone

Let $C_1$ and $C_2$ be convex cones in $\mathbb{R}^n$. Show that $C_1\bigoplus C_2$ is also a convex cone and that $C_1 \bigoplus C_2 = $ conv$(C_1 \cup C_2)$. My attempt: Need to show that $\boldsymbol{x}\in C_1$, $\boldsymbol{y}\in C_2$ implies…
user391838
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A simple problem in two dimensional optimization

I have the following problem: given $$f_1=\frac{-12d_1}{\pi d_2-12}$$ and: $$f_2=\frac{\pi d_1d_2}{\pi(d_1+d_2)-12}$$ I need to find the maximum of the following function: $$F(d_1,d_2)=|f_1|+|f_2|$$ with the following constrains: $$|f_1|\ge…
Riccardo.Alestra
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Nonlinear Programming by Wiley 3rd ed, 1.2 a,b (I don't understand obj function)

The following is from page 30, chapter 1 of Wiley's Nonlinear Programming: Theory and Algorithms (third edition). This is for a graduate course in Optimization Theory. My problem is that I do not understand how they came up with the objective…
user391838
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sum of a geom series declaying at exp(-kx)

Resource allocation problem Given T, total amount of resources N, targets of resource allocation T > 0; N > 1; T < N Allocate resources amongst targets, s.t. $$0 <= allocation_{i} < 1$$ $$allocation_{i+1} < allocation_{i}$$ $$\sum_{i=1}^N…
Dinesh
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