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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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l1 Quadratic Programming

Within a SQP- algorithm it can happen that the constraints of the quadratic sub- problems are infeasible. In order to overcome this infeasibilities, a l1 penalty method can be used according to Nocedal and Wright. The quadratic program has then the…
Karsten
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find min max of function on unit ball

$g(a,b,c)=3a-2b+c$, B is a closed unit ball in $\mathbb R^3$. Find the max/min of g on B. What is the behavior of $g$ on the open unit ball, and the boundary of the unit ball? I think the unit ball can be defined by the equation $x^2+y^2+z^2=1$. …
hobbitty
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Find ALL local maxima numerically

Is there an algorithm that given a function and its derivative gives me all local maxima (in an interval)? All optimization algorithms I know of focus on finding one local or one global maximum. I could run such a traditional optimization algorithm…
phischu
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Shortest distance between 2 points

Ok, this problem might be trivial, but i'm a bit stuck. A string with length 3m is shaped to an "L". Find the shortest distance between the 2 endpoints. So what i've done is to draw the figure, and i can see that pythagoras is key here. But how do…
kirkegaard
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Maximize area in $1000\times 1000$ array using two non-overlapping squares centred at two points, p1 and p2.

Maximize area in $1000\times 1000$ array using two non-overlapping squares centered at two points, p1 and p2. The following conditions must be met: The area (square 1 area + square 2 area) should be a maximum for the given points, p1 and p2 The…
Francis
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Given change in proportions and assuming minimum movement and a direction, calculate minimum proportion moving in that direction

Let $x_1, \dots, x_n$ with $\sum x_i = 1$ be proportions of a discrete distribution. Suppose the distribution changes and let $y_1, \dots, y_n$ be the subsequent proportions (and so $\sum y_i = 1$ too). Let $d_i = y_i - x_i$ be given. For example…
Hugh
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How to solve the constraint nonlinear least-square problem?

I read a paper which says it can be solved by Gauss-Newton type method: I cannot understand why bsin(theta) appears.It seems so starange. Also it is very kind of you to recommend some math books to solve the problem above.
amos
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How to deal with a very small line search step in optimization?

The Armijo type line search is to find an $a_k > 0$ such that $$ f(x^k + \alpha_kd^k) \leq f(x^k) + \sigma_1 \alpha_k \nabla f(x^k)^Td^k $$ given $\sigma_1 \in (0, 1/2)$. We know that for sufficient small positive $\alpha_k$, the inequality…
Saddle Point
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Product of numbers in any two cells sharing a side is $2$

In a $3\times 3$ square, every cell has a positive number. The product of numbers in any two cells sharing a side is exactly $2$. What is the minimum sum of all the numbers? We may color the cells in chessboard fashion, and write $a$ in black and…
Alexi
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Optimization Word Problem AP Calculus Final

A large window consists of a rectangle with an equilateral triangle resting on its top. If the perimeter of the window is 33 feet, find the dimensions of the rectangle that will maximize the area of the window.
Charity
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Stationary points and gradient?

Wolfram Alpha tells me that $xy^2$ only has the stationary points (0,0).... but why? We get the gradient $$(y^2 , 2xy)$$ and this is surely 0 as long as $y=0$ meaning $x$ can be whatever it wants to be and it'll still be a stationary point?
Danielsensq
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Positivity constraints in optimization

How do you enforce positivity constraints in non-linear optimization (e.g. a constraint $x > 0$)? I remember there being a good reason for why most models use non-negativity constraints.
Jacob
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$L(x) \leq U(x)$ & $L'(x)$ and $U'(x)$ exist. Assume there's a pt c st. $U(c)=L(c)$. Consider $U(x)-L(x)$ and show that $c$ is a min of this function

Question and attempt at question are in the photo below. I have gotten halfway through but I am confused how to show the rest of the question (mainly part c) Thanks for your help in advance :)
erin
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Deleting 0's from a random mod 2 matrix

I am fairly new to optimization problems, so please forgive my lack of knowledge. That said, I'm trying to write a program that takes an NxM matrix randomly filled with 0's and 1's, then reduces this matrix by selectively deleting rows and columns.…
Curator
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Optimization problem: rowing across a lake

A woman at a point A on the shore of a circular lake with radius $r=3$ wants to arrive at the point $C$ diametrically opposite $A$ on the other side of the lake in the shortest possible time. She can walk at the rate of 10 mph and row a boat at 5…
ahorn
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