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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Solution of a concave objective function

We have to minimize $f(x,y,z) = a x + b ln(y) + c ln(z)$ with linear constraints of the form $\ d < x,y,z < e $, where $\ a, b, c,d $ and $\ e$ are constants. $\ f(x,y,z) $ is a strictly concave function because its Hessian matrix turns out to be…
user323707
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How to compute the min-cost joint assignment to a variable set when checking the cost of a single joint assignment is high?

I want to compute the min-cost joint assignment to a set of variables. I have 50 variables, and each can take on 5 different values. So, there are 550 (a huge number) possible joint assignments. Finding a good one can be hard! Now, the problem is…
yoda
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Projection onto a convex set is a monotone map

We know a map $ T: \mathbb{R}^n \to \mathbb{R} $ is monotone if $ \forall \; x, y \in \mathbb{R}^n $, $ \; (T(x)-T(y))^T(x-y) \geq 0 $. How can we show that projection onto a convex set is a monotone map? Thank you!
hawkins23
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Minimizing the difference between 2 numbers

Formulate as a linear optimization problem. For given numbers a and b find two numbers x and y whose difference is at most 1, such that |x −a|+|y −b| is minimal. So far I know that |x −y| $\le$ 1 but I am not sure where to go next.
stackdsewew
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Extreme values of a function on a set

I have a function $\ f(x,y,z)=xyz$ on a set $\ M=\{x,y,z:x+y+z=3\} $ and have to find extreme values of the function on set $M$. I made Lagrange's function $$ L(x,y,z,\Lambda) =xyz+\Lambda x+\Lambda y+\Lambda z-3\Lambda$$ Then I took partial…
ratrt13
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How can I find the analytical solution for this optimization problem?

How to find the analytical solution for this optimization problem? $$ \begin{align} & \underset{x,y,z}{\text{maximize}} & (1+\frac{x}{1+z})(1+\frac{y}{1+z})\\ & \text{such that} & 0\leq x\leq x^+\\ && 0\leq y\leq y^+\\ && 0\leq z\leq z^+\\ &&…
kik
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Given $f(x)=4-e^{-\cos(x-2)}$, find the maximum value of $f(x)$ in the range $[-2,0]$.

Given $f(x)=4-e^{-\cos(x-2)}$ Find the maximum value of $f(x)$ in the range $[-2,0]$. $\forall a \in \mathbb R$, $e^a>0$ Hence, the maximum of $f(x)$ will occur when $e^{-\cos(x-2)}$ is a minimum. $e^a$ will be minimised when $a$ is minimised due to…
Danxe
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confusion regarding Lagrange's multiplier

I have confusion regarding Lagrange's multiplier. I was referring to this wiki article http://en.wikipedia.org/wiki/Lagrange_multiplier. It says that if I have the two contours of the original function $f(x,y)$ to be minimized and the constraint…
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Regarding direction of gradient of a function

I was reading this book related to optimizing a function f(x,y) with constraints g(x,y) = 0. The book says that let us suppose g(x,y) define a surface, then gradient of the g(x,y) will be orthogonal to the surface. But I am a bit confused how is it…
user31820
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Optimization: Simplifying the function from an expression in derivative and again taking derivative

I am trying to maximize a function. When I took the first derivative, I found an expression very similar to one in original function. My question is that can I use the expression in the original function to simplify it and then again take derivative…
Eln
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How to approach this complicated optimization problem?

I have a 4x4 matrix of constants $A$ and an unknown column vector of the length 4 $B$. The sum of $B$'s elements is also a given constant $c$. We multiply $A×B$ and get a resulting column vector $P$ of length 4. My task is to set optimal element…
Byte Commander
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What's the difference between conjugate gradient and scale congjugate gradient algorithm?

I am now reading an image processing paper, which uses scale congjugate gradient algorithm to minimize a linear system object function. This paper is the first one that proposes the idea of scale congjugate gradient algorithm. After reading the…
feelfree
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Why are the first order conditions, when applying the method of Lagrange, set equal to zero?

As stated in the title, I'm wondering why we need to set the FOC equal to zero in case we have a constrained optimization problem. I know how it works but I don't really get the intuition behind it. I know that might eb a stupid question but I…
Peter
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How to write this constraint?

I have an integer variable $q_i$ and a binary variable $x_{ ij }$. How can I write the following constraint? if $x_{ ij } = 0$ then $q_i = 0$ else if $x_{ ij } = 1$ then $q_i > 0$
drzbir
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inexact line-search

I have to read up in convex optimization - and at the moment I stuck at inexact line search. The method of exact linesearch (like golde section, etc.) is pretty obvious and I have an "graphical idea". But with inexact line search, I have a problem…
Kenni
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