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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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Subtour elimination constraint in Travelling Salesman Problem

I am trying to understand travelling salesman problem, the Dantzig, Fulkerson, Johnson(1954) formulation. In the general formulation given below I am having trouble to implement subtour elimination in a practical problem. $Min $$\sum\sum…
madetolast
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Extreme values of a function

$x_1,x_2,...,x_m\in R^n$ are given. How to find $u\in R^n$ such that $\sum^{m}_{i=1} (d(u,x_i))^2$ is minimal. I tried this way: If $J(u)=\sum ^{m}_{i=1} (d(u,x_i))^2$ $x_i=(x^{1}_{i},x^{2}_{i},...,x^{n}_{i}),…
user23709
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Maximum value of a product

How to write the number $60$ as $\displaystyle\sum^{6}_{i=1} x_i$ such that $\displaystyle\prod^{6}_{i=1} x_i$ has maximum value? Thanks to everyone :) Is there a way to solve this using Lagrange multipliers?
user23709
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Extreme values of a function with conditions

What is a way to find extreme values of a function $u(x,y,z)=xy+yz+xz$ with conditions $x+y=2, y+z=1$?
user23709
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Maximizing area of a triangle from a parabola equation

The figure shows a right triangle in the first quadrant. One side of the triangle is along the x-axis and the hypotenuse runs from the origin to a point on the parabola $y= 4−x^2$. Find the $x$ and $y$ coordinates that maximize the area of the…
caseyschlenker
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Optimal paths between two closed lines

Imagine I have an outer shape and an inner shape (that may overlap like the following picture) Is there any algorithm or mathematical property I can use to find a third set of points which will be in the middle of the two?
graphtheory92
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Positioning Problem

Please help me with how to start with this question: A particle of unit mass moves along the x-axis subject to a force $u(t)$. We want to transfer the particle from rest at the origin, to rest as x=1 in unit time, so that the effort involved ,…
Natalie
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Scalability in the context of optimization

I am having a hard time trying to make sense of the phrase "Scalable optimization", be it dynamic or not. I read from a few papers discussing that some global optimization solvers like BARON, ANTIGONE, etc. do not scale well with the number of…
mathsnovice
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Softmax inverse for distance measure

I have a distance measure going from 0 to infitiny, where 0 indicates closeness. I need to perform a transformation in order that the distances sum up to 1. Therefore, softmax seems appropriate, especially that the gap between the values is not…
giac
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Mathematically, what is the difference between averaging before or after an optimization?

Say I have an optimization problem where I want to maximize $x$ such that $f(x; p)$ is largest. Here, $p$ is a parameter, $x$ is a variable, and $f$ is a function that depends on $x$ and $p$. However, the issue is that I don't know what $p$ is, I…
Mail
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Number of Iterations for the Preconditioned Conjugate Gradient

I've been stuck on this problem for few days now. Given a quadratic function of the form $f(x) = \frac{1}{2} \langle x,Qx \rangle + \langle c, x \rangle $, where $ Q = B + \sum_{j=1}^s a_j a_j^T $ is of dimension n and B is positive definite, I need…
Fedzchen
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What is a dual norm, and how to compute it?

I came across a concept called dual norm in my optimization course, which I am not familiar with. I am trying to understand what it is and how to compute it. from Wiki, the definition of the dual norm is, Let ${\displaystyle \|\cdot \|}$ be a norm…
dawen
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Slater's Condition for Dual Problem

I am reading the paper "A Semidefinite Optimization Approach to Quadratic Fractional Optimization with a Strictly Constraints" by Maziar Salahi & Saeed Fallahi. I this paper, they tried to prove that if the Slater condition holds for the primal SDP…
Stephen Ge
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Optimization with polynomial objective and constraints

Let us be given the following optimization problem. \begin{equation} \begin{aligned} & \text{minimize} && u \cdot x^2 + v \cdot y^2 + w \cdot z^2, \\ & \text{subject to} && x + y + z = 1, \\ & {} && 0 \leq x < y < z \leq 1,\\ & {} && u \cdot x +2 v…
Richie
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How to prove the dual optimal solutions are bounded

Consider following optimization problem: \begin{equation} \label{eqn:primal} \begin{aligned} \min_{x\in \mathcal{X}} f_p(x)\\ \text{s.t.}~g(x)\leq 0\\ h(x) = 0 \end{aligned} \end{equation} with non-empty domain $\mathcal{X}$ and finite optimal value…
Stephen Ge
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