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 weighted and connected problems in an optimal way

We need to solve several problems, with these properties: Each problem is in a category $A$, $B$, $C$, .... We label the problems $A_1$, $A_2$, ..., $B_1$, $B_2$, etc. For each problem $X$, we have an estimate $d_X$ of its difficulty, i.e. of the…
Bass
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What is non-linear assignment problem?

I am reading wikipedia article about assignment problem https://en.wikipedia.org/wiki/Assignment_problem but cannot understand the definition of 'linear' assignment problem. What is an example of non-linear assignment problem ? thanks
Guldam Kwak
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How to maximize $\sum_{i=1}^N \log(1+a_i x_i^2) \text{ s.t.} \sum_{i=1}^N x_i= 1,x_i \geq 0$

How to maximize $\sum_{i=1}^N \log(1+a_i x_i^2) s.t. \sum_{i=1}^N x_i= 1,x_i \geq 0$? The objective function is non-convex. Is there any tutorial or solution about how to deal with such problem ? Thank you!
Kris Prokins
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maximizing lagrangian with matrix contraints.

$\max_\limits{S} \sum_{j=1}^kS_j^TKS_j $ $s.t\ S^TS = I$ where $S = (S_1, .., S_k),\ S_i\ (nx1)$ Not sure how to solve this problem since it cannot form a lagrangian equation due to conformability issues.
John Karasev
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Optimal spacing between breaking points for a linearised function

This is my problem: I am looking for a way to optimally define the number of breaking points I need in order to approximate a quadratic function to a linear one. Right now, I have selected previously the number of breaking points ($m$) involved in…
frrndn
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Find matrix $M\in\mathbb{C}^n\times\mathbb{C}^n$ that minimizes $||v^*(M-M^2)v||$ for some $v\in\mathbb{C}^n$

Let $v\in\mathbb{C}^n$, want to find two symmetric matrices $M,N\in\mathbb{C}^n\times\mathbb{C}^n$ such that $M+N=I$ $$ is minimized This boils down to finding matrix $M\in\mathbb{C}^n\times\mathbb{C}^n$ that minimizes $||v^*(M-M^2)v||$…
Tian He
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If $m(α)$ is global minimum of $ \ln(x^2 + αx + α)$, find $α$ such that $m(α)$ is maximal

For any α in the interval (0, 4) let $m(α)$ be the global minimum of the function $$\ln(x^2 + αx + α)$$ For what value of α does $m(α)$ attains its maximum? I'd prefer if the solution was as simple as possible, thanks.
Joe Jewels
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Minimization of sum of squares

I'm having trouble figuring out how to minimize the expression: $$(k_1 + 2)^2 + (k_2 + 2)^2 + \cdots + (k_m + 2)^2$$ given that $k_1 + k_2 + \dots + k_m = 17$. Any help would be appreciated!
user730203
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Finding a minimum

I would like to minimize the function $f(x_1,\dots,x_k) = x_1 + x_2 / x_1 + \dots + x_k / x_{k-1} + M / x_k$ where $M>0$ and $x_k \ge \dots \ge x_1 \ge 1$. First I looked at a simpler function $g(x_1,x_2) = x_1 + M/x_2$. Computing the partial…
math noob
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Constrained optimization of an inequality

If I am trying to solve an optimization problem of a $f(x,y)$ subject to the following constraints: $$0\le x \le y$$ $$ 0\le y\le 1$$ Then do I setup the Lagrangian as follows?…
nvm
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How to find corners / vertices of Polyhedron

Polyhedron $\mathcal{P}\subseteq\mathbb{R}^2$, given by the inequalities: $x_1+2x_2\geq 1,\qquad x_1\geq -1,\qquad x_1-x_2\geq -3,\qquad x_2\geq 1,\qquad -2x_1-x_2\leq 0 $ Giving matrix $A$ and vector $b$, such that $Ax\leq b$: $$ A=…
user619755
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How to show that $\mathcal{R}=\{Dx+c:x\in\mathcal{M}\}$ is a polyhedron

Let $\mathcal{M}\subseteq\mathbb{R}^n$ a polyhedron, $D\in\mathbb{R}^{m\times n}, c\in\mathbb{R}^m$. How do I show that $\mathcal{R}=\{Dx+c:x\in\mathcal{M}\}$ is a polyhedron? I know that a polyhedron can be defined as the convex hull of a finite…
user619755
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Adjoint method versus Chain Rule

Currently i have come across adjoint method for optimization problems; and i have found that this method can be used as an alternate for chain rule. This made me confused. If anyone explains the mechanism, with example, of adjoint method for…
NeuralAI
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Optimizing a n-sided regular polygon prism.

I am trying to come up an equation for the maximum volume for a prism with a base of a regular polygon, given that the surface area is 100 units squared. I ended up with these two equations, but I'm not sure how to optimize as there are three…
mbzht
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Optimization over restricted value

How to get the $\mu_1 (x_1)$? I tried to use the derivative and equal it to zero or simply solve for u but I never get $\mu_1 (x_1) = 0$. I plotted the function and it does not seem to ever be equal to $0$ in this interval. It seem like $\mu_1…
ncmq
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