Mathematics Stack Exchange
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Questions tagged [constraints]

In mathematics, a constraint is a condition of an optimization problem that the solution must satisfy.

In mathematics, a constraint is a condition of an optimization problem that the solution must satisfy. Reference: Wikipedia.

There are several types of constraints—primarily equality constraints, inequality constraints, and integer constraints. The set of candidate solutions that satisfy all constraints is called the feasible set.

1019 questions
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Ordering people with friendship constraints

$n$ people have to be ordered in a line, one after the other. We say that: A person X sees a person Y, if X stands behind Y. A person X hears a person Y, if X stands at most $k$ positions in front of Y. Each person has two friends (friendship…
Erel Segal-Halevi
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How to set up Lagrangian for matrix constraints?

Suppose we have a function $f: \mathbb{R} \to \mathbb{R} $ which we want to optimize subject to some constraint $g(x) \le c$, where $g:\mathbb{R} \to \mathbb{R}$. What we do is that we can set up a Lagrangian $$ L(x)=f(x)+\lambda(g(x)-c) $$ and…
Boby
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Minimizing $\frac{x_1+x_2}{1+x_1x_2}+\frac{x_2+x_3}{1+x_2x_3}+\dots+\frac{x_n+x_1}{1+x_nx_1}$ for non-negative $x_i$ satisfying $x_1+x_2+\dots+x_n=1$

An "Olympiad type" inequality: Let $x_1,x_2,\dots,x_n$ real numbers in $[0;+\infty)$ with $x_1+\dots+x_n=1$ and $$f(x_1,\dots,x_n)=\frac{x_1+x_2}{1+x_1x_2}+\frac{x_2+x_3}{1+x_2x_3}+\dots+\frac{x_n+x_1}{1+x_nx_1}$$ Find the minimum of $f$ under the…
Alexique
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Non-negative non-decreasing polynomial lower than $x^N$ somewhere in $0 \le x \le 1$

Is there a degree $N$ polynomial $f(x)$ such that $f(1) = 1$, $f(x)$ is non-negative on $0 \le x \le 1$, $f(x)$ is non-decreasing on $0 \le x \le 1$, and there is a value of $x$ in $0 \le x \le 1$ such that $f(x) < x^N$? My gut feeling is that…
Olli Niemitalo
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Conic by three points and two tangent lines

With the exception of degenerate situations, a conic is uniquely determined by five points lying on it. Likewise, five lines tangent to a conic uniquely define that conic. With four points and one tangent line, there are in general two conics…
MvG
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Extremising $\int_0^1 f(x) f(1-x) \ \mathrm{d}x$ subject to length of $f$ and endpoints

I have recently learnt some Calculus of Variations and was trying to apply this to a question I made: Over all functions $f: [0, 1] \to \mathbb{R}$ satisfying $f(0) = f(1) = 0$ with fixed curve length $\ell \geq 1$ (i.e. $\int_0^1 \sqrt{1 +…
Sharky Kesa
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$a \geq b \geq c \geq 0$ and $a + b + c \leq 1$. Prove that $a^2 + 3b^2 + 5c^2 \leq 1$.

Positive numbers $a,b,c$ satisfy $a \geq b \geq c$ and $a + b + c \leq 1$. Let $f(a, b, c) = a^2 + 3b^2 + 5c^2$. Prove that $f(a, b, c) \leq 1$. One observation is that the bound is met: $f(1, 0, 0) = f\left(\frac{1}{2}, \frac{1}{2}, 0\right) =…
Alex
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Proving that a sphere has a minimal surface to volume ratio using Calculus of Variations

I know the problem is traditionally solved via the isoperimetric inequality, but I was hoping to solve it by minimizing a surface of revolution subject to a volume constraint. The surface area of a surface of revolution…
JAustin
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the relation between cardinality, L1-norm and L2-norm of a vector

For every $u\in \mathbb{R}^n$, $\textbf{Card}(u)=q$ implies ${\lVert u \rVert}_1 \leq \sqrt{q} {\lVert u \rVert}_2$ where $\textbf{Card}(u)$ is the number of non-zero element (so the L0-norm). Why does the condition ${\lVert u \rVert}_1 \leq…
user26767
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Calculus of variation with inequality constraints

Find the function $y$ which maximizes the functional $$J[y] = \int_0^1 g(x) y(x) dx$$ subject to $0 \leq y(x) \leq 1$ for all $x\in [0,1]$ and $$\int_0^1 y(x) dx = k$$ where $g$ is a strictly increasing function. I know that I can take care of the…
sami
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Place points on circle with constraints

Context Consider the following circle where each point on the circle is associated with a real in $[0,1)$: Problem I am looking for a mathematical function $f(x)$ defined on at least $\mathbb{N}↦\mathbb{Q}+$ and range $[0,+\infty)↦[0,1)$ that can…
Flo
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Different Approaches for Proving Kantorovich Inequality

Here is a statement of the famous Kantorovich inequality. Thoerem (Kantorovich). Let $A$ be a $n\times n$ symmetric and positive matrix. Furthermore, assume that its eigenvalues are $0 < \lambda_1 \leq \dots \leq \lambda_n$. Then, the following…
Hosein Rahnama
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Maximizing $\int_0^\infty (1+xy')^2e^y dx$ subject to $\int_0^\infty e^ydx = 1$

I'm trying to solve a calculus of variations-type problem, which requires finding the extrema of: $$\int_0^\infty (1+xy')^2e^y dx, $$ subject to the constraint that $\int_0^\infty e^ydx = 1$. Intuitively, $e^y$ is the probability density function of…
Fred Li
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What is "box-constrained mathematical optimization problem"?

I came across the term "box-constrained mathematical optimization problem" while reading a research paper. Can someone please explain what 'box-constrained' and "box-constrained mathematical optimization problem" mean ?
ahmedshahriar
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Involving indicator function as a constraint in a LP problem

I am trying to solve following LP problem \begin{align} &\min_x –c^\top x \\ \text{s.t.} & \sum_{i=1}^M I(-a_i x\leq b) \geq m \\ & \sum_{i=1}^N x_i =1 \\ & x_i\geq 0 \end{align} where $m
Stephen Ge
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