Mathematics Stack Exchange
Mathematics Stack Exchange

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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Can a function be shown positive without derivative?

In a Finnish matriculation examination was the following problem Let $a_1,a_2,\ldots,a_n$ be reals. For what value the parameter $x$ should be given if one wants to minimize the value of the sum $(x-a_1)^2+\cdots + (x-a_n)^2$. This can be compute…
selfstudying
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Finding minimum $\frac{x+y}{z}+\frac{x+z}{y}+\frac{y+z}{x}$

I would appreciate if somebody could help me with the following problem Q. Finding maximum minimum $$\frac{x+y}{z}+\frac{x+z}{y}+\frac{y+z}{x}(\text{where} ~~x,y,z>0)$$
Young
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Minimizing a Maximum

I want to minimize the function $$f(x) = \max \left\{ \frac{a}{x^2}, \frac{b}{1 - x} \right\},$$ where $a, b > 0$ are constants, and $0 < x < 1$. Is there a way to find an $x$ which will make $f(x)$ as small as possible? When I plot $\frac{1}{x^2}$…
jamaicanworm
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Dual cone and polar cone

Dual cone and polar cone http://en.wikipedia.org/wiki/Dual_cone_and_polar_cone are defined only on $\mathbb R^n$. Has anyone seen the extension to $\mathbb C^n$? Any references for these?
Sunni
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Is Minimax equals to Maximin?

Consider a loss funcation $\ell(x,y)$ with a penalty $g(x,y)$ If I want to consider the worst case robust scenario, that is \begin{equation} \min_x \max_y \ell(x,y) + g(x,y) \end{equation} Is it equivalent to consider the maximin…
Rein
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Looking for mathematical optimizations when translating formulae to code

I'm writing an app which translates formulae into executable code. I've been experimenting with fairly obvious optimizations such as factoring (reducing number of multiplications) in order to make the code run faster. Can anyone suggest any other…
Dmitri Nesteruk
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Distance between point and sine wave

I have a project where I need to know the exact minimal distance between a point $(e, f)$ and a sine wave $y = a + b\cdot\sin(cx+d)$ Is there any way of calculating this? If not, is there a way to approximate this? Thanks in advance! Edit Following…
Wouter Raateland
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Optimal ladder position to maximize height reached

Suppose we have a ladder (with a unit length) and we want to position it to reach the highest point possible on a vertical wall. The terrain is sloped and can be described with a variable k. What is the optimal distance from the wall the ladder…
VPeric
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Filling the area below a decreasing function by rectangles

Suppose $f:\mathbb{R}_+ \to \mathbb{R}_+$ is a strictly decreasing continuous function such that. Let $n$ be a natural number. I want to solve the following maximization problem $$ S_n = \max_{x_1 \leq \dots \leq x_n} x_1 f(x_1) + (x_2-x_1) f(x_2)…
TomH
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Taylor theorem equation

I have one question about Taylor theorem. Originally, Taylor theorem is represented as $$f(x) = f(a) + f'(a)(x-a) + \ldots$$ But my book says Suppose that $$f : \mathbb{R}^n \to \mathbb{R}$$ is continuously differentiable and that…
mangken
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What is the best strategy to find 2 circles whose total area is maximum in a triangle?

I am looking for the best strategy to find 2 circles whose total area are maximum in a triangle. I tried it on an example as seen Figures below. Strategy $1$: If I draw a big circle that touchs to 3 sides and then to draw 3 circles next to the big…
Mathlover
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NP-hard optimization problems for which approximations would be useless?

The optimization problems which I am familiar with (e.g. the Traveling Salesman Problem) are such that approximate solutions to these problems are still quite useful. I'm wondering, however, if there are real-world optimization problems for which…
user541686
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Difference between Newton's method and Gauss-Newton method

I know that the Gauss-Newton method is essentially Newton's method with the modification that the Gauss-Newton method it uses the approximation $2J^TJ$ (where $J$ is the Jacobian matrix) for the Hessian matrix. I didn't understand why we are using…
user161260
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Eleven unit squares inside a larger square

What is the smallest square which contains 11 non-overlapping (except boundary) unit squares? This question is open but I would like to know a method to verify the best known answer at the moment. I'm reading paper at here . In figure 2, there has…
Jaska
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Find max and min of $F=ax^2+2bxy+cy^2$ when $x^2+y^2=1$

Find the Maximum and Minimum of $$F=ax^2+2bxy+cy^2$$ when $$x^2+y^2=1$$ The variables a,b, and c are just real numbers. I have attempted using partial differentiation in order to solve for the given maxima and minima, but I found the algebra used…
Amory
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