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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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Projection theorem for compact differentiable manifold

By Hilbert projection theorem, if $x\in\mathbb{R}^n$ and $D$ is a closed subspace of $\mathbb{R}^n$ then the optimization problem $$\underset{y}{\min} \|x-y\| \ s.t. \ y \in D \quad\quad (P1)$$ has an unique solution, namely, the $\bar{x} \in…
shamisen
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Minimize function with constraint

I have a Markowitz problem : Min $x^T*C*x$ $x : {x_1 , x_2 ... x_n}$ is a vector of size $N$ $C$ is a known matrix $[N \times N]$ 1) $∑ x_i$ = 1 2) $x_1 < 0 $ I can minimize the function with the first constraint with Excel Solver. I find the…
user159854
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Basic Optimization Problem

I sat for an exam a few days ago. I managed to answer every question except for question $1$c in the calculus paper. Provided that I got question $2$d correct (my answer was $m=0.5$), the absence of an answer for question $1$c should not in any way…
Charles Salmon
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Extremum of a function under constraints

I have a function $f : E \subset R^n \to R$. $E$ is compact and $f$ is continuous so the extremums exist. But $E$ is not defined by an equation but an inequality, so i can't use the Lagrange method ... How do i do ? Exemple : $n=2$, $E=$closed disc…
user113865
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About a minimum-norm problem.

I am studying on the optimization via vector method. The reference book is Optimization by Vector Method by Luenberg. I have trouble in understanding the following statement [p.123]; We consider the unknown $x^*$ in a dual space $X^*$ and express…
user142054
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Finding the minimum in a given direction

What's the manual way of finding function minimum in a given direction? The function is: $\min f(x)=x_1^2+x_2^2+x_1x_2 -x_2-x_1$ with the starting point in $x^0=[1,0]^T$ in a direction $d^0=[2,1]^T$.
tymsam
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Multivariable calculus max/min

Find and classify the critical points of this function: $f(x,y)= (x^y)-(xy)$ in the domain $x>0, y>0$. I am having trouble treating x and y as constants when taking partial derivatives.
Zach
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the coordinate axes has minimum area

How can i find the point in the first quadrant on the parabola $$ y = 4-x ^ 2 $$ such that the triangle tangent to the parabola at the point and the coordinate axes has minimum area. Some help to interpret the equation so minimize in this exercise…
Rosa Maria Gtz.
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Optimization of an Ellipse

The question is: Find the points on the ellipse defined by the equation $x^2 + 4y^2 = 4$ nearest the point $(1,0)$. I'm having a hard time coming up with the change in x half of the distance formula.
Jay3
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Optimization on area , rectangle with fixed length on 3 sides.

I've stumbled with the problem below "Some unused land is adjacent to a straight canal. A gardener wants to use 200 meters of fence in order to create a rectangular garden, by using the fence for three sides and the canal as the fourth. For…
Joel Hernandez
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optimize volume of box

A rectangular box with a square base is made of 48 square meters. What dimensions should have the box to have the maximum volume? Is that correct my equation that i must maximize?
Rosa Maria Gtz.
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find the minimum and maximum distance between both objects as function of $r$

Suppose that an object $O$ moves in the plane $x,y$ along a path with respect to time $ t $ of the form $O (t) = (x (t), y (t)) = (2 \cos (t) , 2 \sin (t))$ and another $ P $ object while moving along a path $ P (t) = (z (t), w (t)) = (r \cos (t +…
Rosa Maria Gtz.
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Steepest Descent for a Quadratic

I have the formula for calculating the step-size for steepest descent for a given quadratic. However, the formula says that Q is positive definite. My Q is not. Does the same formula apply? Admins, how do I get this question answered. It seems that…
user100503
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maximizing distance in a polyhedron

How can I prove that the point that is maximally distant to a specific point in a convex polyhedron must be in a vertex of the polyhedron?
Anne
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The minimum of $\dfrac{x(1+y)+y(1+z)+z(1+x)}{\sqrt{xyz}}?$

Can we use $AM\geq GM$ inequality to find the minimum of $\dfrac{x(1+y)+y(1+z)+z(1+x)}{\sqrt{xyz}}?$ I can find out that minimum is $6$, but can we use $AM\geq GM$ to show this?
Silent
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