Mathematics Stack Exchange
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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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Maxima and Minima: $y = x^4$

Given a stationary point, I was taught to test if it was a maximum or a minimum using the concavity test, i.e. If $f''(x)>0$: concave up (thus a local minimum) If $f''(x)<0$: concave down (thus a local maximum) But if we used the concavity test on…
J-J
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Maximize product with sum constraint

Suppose I have $N$ numbers $x_i \geq 1$, $i = 1, ... ,N$, and want to solve the following: \begin{align} {\text{maximize}} &\hspace{3mm}& \prod_{i=1}^N x_{i} \\ \text{subject to} &\hspace{3mm}& \sum_{i=1}^N x_{i} = A \end{align} where $A$ is any…
user173690
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An intuitive explanation of the primal-dual algorithm

I am a beginner to optimization, and I am trying to understand the primal-dual algorithm. However, all explanations I have seen so far are very maths-heavy. So it's very difficult to understand the details without having a broader appreciation of…
Karnivaurus
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Why do we transform constrained optimization problems to unconstrained ones?

In methods like Lagrange multipliers or augmented Lagrangian methods we transform a constrained optimization problem into an unconstrained one and then solve it. For example in Lagrange multipliers we might have: maximize $f(x, y)$ subject to $g(x,…
Bar
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What algorithm to use for distribution of pieces

Hi am working for for a manufacturing company whose operation works in this manner We get rolls of a material in a particular size our supplier lets say 8000 meteres per roll. Then we get orders from different customers of smaller sizes such as 2000…
user992010
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Monotonicity and optima of functions

It is said that the logarithm is a monotonically increasing function, hence the logarithm of a function achieves its maximum value at the same points as the function itself. Is there a similar property for monotonically decreasing functions and…
Adam I.
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Calculating the Dual Problem to a Minimisation Problem - confusion

Let $A$ be an $m$x$n$ real matrix. Let QP denote the problem: minimise $f(x,y) = \frac12 x^Tx$ such that $x \ge 0, y\ge 0$, and $Ax-y=b$. I want to prove that the dual of this problem QP* is: maximise $ f^*(\lambda, \mu, x) = \lambda^Tb -…
Paul Slevin
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Escalators: Stand right, walk left. When can this rule be ignored?

On my daily commute to work, I find a sign on all escalators, stating Stand right, walk left The idea behind this, obviously, is that you should leave space for people in a hurry, so they can run up the escalator and catch the train, while the…
Dave Vogt
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How can I find the maximum value of $x_6-x_1$ subject to the two constraints $\sum_{j=1}^{6} x_j^2$ and $\sum_{j=1}^{6} x_j = 0$

I currently have six variables $x_1, x_2, x_3, x_4, x_5, x_6$. I am trying to determine how large I can make the difference $x_6-x_1$ while satisfying the constraints: $\sum_{j=1}^{6} x_j^2 \leq 1$ $\sum_{j=1}^{6} x_j = 0$ So far I have tried to use…
user136503
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Prove circle packing solution is optimal

Background: This is a follow on from this question of how to maximise the area of two non overlapping circles of arbitrary radii packed into a rectangle of arbitrary width and height. I proposed a solution based on greedy selection of the first…
Nic
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Maximization of $x+y$ when each is greater than $1$ and $xy = 16$.

The product of two numbers $x$ and $y$ is $16$. We know $x\ge 1$ and $y\ge 1$. What is the greatest possible sum of the two numbers?
Kamal
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Minimum of $f(x,y)=|ax-by|$ subject to box constraints

$$ \begin{array}{ll} \underset {x, y} {\text{minimize}} & f(x,y) := | a x - b y | \\ \text{subject to} & 0 \le x \le C_1 \\ & 0 \le y \le C_2 \end{array} $$ What is the minimum? I found a similar question, Minimum of $|az_x-bz_y|$, but their $x$ and…
user173737
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Gradient descent vs. Newton's method -- which one requires more computation?

Broadly speaking, when numerically minimizing a d-dimensional objective function: Gradient descent generally requires more iterations, but each iteration is fast (we only need to compute 1st derivatives) Newton's method generally requires fewer…
user541686
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Two-way matrix optimization

I have run into a problem like this. Looks a bit unusual, but I think should be doable. Find $U$ achieving $$\min_U \left( \| A - UW \|_2^2 + \| RU - H \|_2^2 \right)$$ $A,U,W,R,H$ are all matrices of dimension so that the operations above make…
PA6OTA
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Finding the maximum and minimum

Can't understand how to find the maximum and minimum with the given definitions (with both x and y).. can someone explain step by step?
Elin
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