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").

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Maximum of $ f(x,y) = 1 - (x^2 + y^2)^{2/3} $

For the function $ f(x,y) = 1 - (x^2 + y^2)^{2/3} $ one has to find extrema and saddle points. Without applying much imagination, it is obvious that the global maximum is at $ (0,0)$. To prove that, I set up the Jacobian as $$ Df(x,y) = \left(…
bonifaz
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Finding Maximum Under Constraint

Suppose $a$,$b$,$c$ satisfy $a+b+c=1$ and $a$,$b$,$c\in [0,1]$ Find the maximum value of $(a-b)(b-c)(c-a)$
chloe_shi
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Calculating the optimal fireballdamage with limited skillpoints to allocate in a Game (Diablo 2)

I have written a program for calculating the best skillpointallocation with limited points to maximize the fireballdamage in Diablo 2. But there is a case where it's slightly off, because of the way you have to put points into…
SAJW
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Efficient way to maximise stock profits

I got a message from my future self. Rather than a sports almanac, it contained minute-by-minute data for tomorrow's prices on a few select stocks. I want to use this to earn a bit of money. Is there an efficient way to decide when to sell and buy…
Arthur
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Optimizing a limited number of select/copy/paste operations to end up with the most characters possible

I was given this question by a friend and tooled around with it a bit in excel. I'm sure I could brute force it, but I wanted to see if people smarter than me could help me optimize this and find a mathematical way to solve it. The problem is this:…
ReconFirefly
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Proving that for finite set $M$: conv$(M)$ is a polytope

I want to prove that for a finite set $M$, the convex hull conv$(M)$ is a polytope. I often see this written, but I haven't actually seen a proof for this yet. The definition I use for polytope is, that it is a bounded polyhedron and a polyhedron is…
user619755
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What does min max in optimization problems mean?

With $A$ and $B$ being different decision variables, does $$\min_{A} \max_{B}$$ in front of an objective function specify a specific order of optimization? In other words, is it saying "minimize $A$ while maximizing $B$", or "minimize $A$ while…
develarist
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If $a+b+c=k$ and $a^2+b^2+c^2 =2k$ what is the maximum value of $k$?

$a,b,c$ are real numbers and they satisfy the following equations. $a+b+c=k$ $a^2+b^2+c^2=2k$ Find the maximum value of $k$. I tried substituting for k in the second equation from the first and got $a^2+b^2+c^2=2(a+b+c)$ Rearranging the terms I…
ooiioooiooiiooioooiiooii 123
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How to shift sequences of real numbers so that the distance between them is minimized?

I understand that I am using the terminology incorrectly because distance is defined between two objects while my question refers to $m$ objects. However, I don't know the correct name for this, so I use the word distance. Corrections are…
Vladislav Gladkikh
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sufficient (first-order) condition for optimality

Consider the problem minimize $f(x)=||Ax-b||_2^2$, where $A$ is an $m\times n$ matrix with $m\geq n$, and $b$ is a vector of length $m$. Assume that the rank of $A$ is equal to $n$. We can write down the first-order necessary condition for…
Ian
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How to maximize this problem

$$ \max_{\pmb x} \cfrac{\pmb x^H \pmb P \pmb P \pmb x} {\pmb x^H \pmb P \pmb x + \sigma^2}$$ s.t. $$ ||\pmb x ||^2_2 < N$$ where $\pmb x$ is a vector,$\pmb P$ is symmetric positive definite matrix, $\sigma^2$ and $N$ are both constants. Any…
Palace W
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Change of variable in optimization problem

I have an optimization problem of the form: \begin{alignat}{4} &\text{minimize}\quad &&f(x) \\ &\text{subject to} \quad && \|x\|_2 \leq \varepsilon \end{alignat} with variable $x \in \mathbb{R}^n$ where $f$ is Lipschitz continuous. I want to convert…
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Find the values of $x$, $y$ and $z$ minimizing $\frac{r^2x}{y+z}+\frac{s^2y}{x+z}+\frac{t^2z}{x+y}$

$$\frac{r^2x}{y+z}+\frac{s^2y}{x+z}+\frac{t^2z}{x+y}$$ $r$, $s$, $t$ are positive coefficients. Find the values of non-negative variables $x$, $y$ and $z$ so that the above expression is a minimum.
Anonymous
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Jacobi-Method with Projection onto Box Constraint

I'm solving the constrained least squares problem $\underset{u \in [0,1]^N} \min \lVert Au-f \rVert_2^2$ with $u \in \mathbb{R}^N$, $A \in \mathbb{R}^{N \times N}$ and $f \in \mathbb{R}^N$ by using the following modified Jacobi-Method: $u^{k+1}=\max…
TestTestGuest
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How can I estimate the minimum distance between 2 sets of coordinates?

I am creating stimuli for an experiment. The stimuli consists of 2 distinct sets of 9 points in a 2D plane. Each set of points can be described by 9 pairs of coordinates. (i.e. http://learnmem.cshlp.org/content/5/6/420/F1.expansion) I would like to…
Sarah S
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