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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Solving Linear Inequalities for Optimization

I want the max of: $100-(2x_1+3x_2+4x_3+5x_4+6x_5+7x_6)$ I am given 5 inequalities: $x_1+x_4\le6$ $x_2+x_5\le8$ $x_3+x_6\le7$ $x_1+x_2+x_3\le9$ $x_4+x_5+x_6\le11$ and $x_1+x_2+x_3+x_4+x_5+x_6\ge0$ It seems obvious that i want the most of $x_1$, and…
Jimmy Pitts
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Lagrange's method with infinite number of constraints?

In economics, one often face the following maximization problem: $ \max_{{\{c(t)\}}_{t=1}^\infty}\sum_{t=0}^\infty \beta^tu(c(t)), $ where $\beta$ is the discount factor, $u(c(t))$ is the utility of consumption at time $t$. Very often we have one…
Kun
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Maximizing an Expected Value

I have a simple program that helps with purchasing decisions. The problem being solved is how to most profitably select products for a grocery shelf using competing products in different varieties. I'm looking for some pointers on ways to help me…
netmau5
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Confidence Interval of Nelder Mead Method

I'm using the Nelder-Mead method to minimize a quantity with 4 variables for a given input. I want to know how I should calculate the confidence interval, ie. to answer the question: with what confidence am I getting the true minimum? The…
Ary
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argmin as projection in the dual averaging algorithm

I am struggling to understand the dual averaging algorithm as presented in the paper Dual Averaging for Distributed Optimization: Convergence Analysis and Network Scaling. More precisely the update of the parameters given as $$\Pi^\psi_\chi…
vkoe
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Understanding how to state the Karush-Kuhn-Tucker Conditions for a given problem

I'm trying to understand an example given by Nocedal & Wright (1999), pg 329, Example 12.4. According to a definition given earlier in this book: At a feasible point x, the inequality constraint $i\in I$ is said to be active if $c_i(x)=0$ and…
Paul
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Maximize sum of logs subject to constraints

I have the optimization problem $$\begin{array}{ll} \text{maximize} & \displaystyle\sum_{i=1}^n \log(c_i + x_i)\\ \text{subject to} & \displaystyle\sum_{i=1}^n x_i = 1\\ & x_i\ge0\end{array}$$ where $c_i>0$. How could I solve it?
pepelac24
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Is there a reason that the maximal volumes of rectangular prisms with multiple open faces but constant surface area follows this pattern?

Suppose a rectangular prism has a surface area of $12 \text{ m}^2$. The maximal volume of this prism is well known. If the side lengths of the prism are $x$, $y$, and $z$, then the surface area is given by $$A=2xy+2xz+2yz = 12$$ and the volume is…
Apoapsis
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Ask an optimization problem

Consider $x,y,z$ are $n$-dimensional variables, and $a,b,c$ are three nonnegative vectors. Let $f(x,y,z)=a^T x+b^T y+c^T z-\left(\|x\|_2^{2/3}+\|y\|_2^{2/3}+\|z\|_2^{2/3}\right)^3$. How to find the maximum of $f(x,y,z)$ over $x,y,z\ge 0$?
Michael Timback
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Equivalence of following statements about shortest path problem

We formulate the shortest path problem as follows: We have a directed graph $D=(V,A)$ with length $c_{j}$ for each arrow $e_j$ in $A$ and two special points $s,t\in V$. The node-arc incidence matrix $A$ is defined as $a_{ij}$ is $1$ if the arrow…
user235238
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Maximum of $x+y$ with constraint

What is the maximum value of $x+y$ given that $x^2-4xy+4y^2+\sqrt{3x}+\sqrt{3y}-6=0$? $x,y$ are real numbers. Notice that it has terms $\sqrt{x}$ and $\sqrt{y}.$
winterfly
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Finding an optimal sequence

It's my first time on this site:) I have to find a strictly increasing finite sequence $\{x_k\} _{k=1, \dots, n}$ with $x_1=c^2$ that will minimize the following expression $$\sum_{i=1}^n\sqrt{x_{i+1}-x_i} \frac{c}{\sqrt{x_i}},$$ subject to the…
Albert Fletcher
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Box Optimization: find best boxes for multiple products

I have a company that sells all kind of products and we have many issues with boxes. We are going to buy wholesale volumes, so we need to get the sizes right. We are trying to develop an algorythm for optimizing box sizes. I have the data of all my…
Santiago Valdés
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Naive question: max {x - y}

I am sorry if this question is an extremely naive question. Consider I have an optimization problem with an objective function as follows: $\max_x \{f(x) - g(x)\}$ Does that mean, As $g(x)$ has a negative sign, I am maximizing $f(x)$ while I want to…
Mohsin
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Minimum without using of differential calculus

Find minimum of $$x + y^5$$ where $x>0,y>0 $ $xy=1$ without using of differential calculus.
user110661
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