All Questions
Tagged with nonlinear-optimization ds.dynamical-systems
15 questions
1
vote
0
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62
views
Under which condition, such that all second-order critical points satisfy $\sum_j\cos(\theta_i-\theta_j)>0$ for all $i\in[n]$?
Consider the following non-convex function
$$E(\theta):=-\sum_{i,j}A_{ij}\cos(\theta_i-\theta_j)$$
where $A$ is a symmetric, diagonal-free matrix whose non-diagonal element are $\pm 1$. In other words,...
- 49
0
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0
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27
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How to control the angles of Kuramoto model by controlling its order parameter?
Consider Homogenous Kuramoto model in this paper. In theorem 3.1, the author derive condition on $A$ such that all second-order critical points of $E(\theta)$ are in two opposite quadrants, by saying ...
- 49
4
votes
0
answers
121
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$f(n) = \frac{n^2 + n + 4}{2}$, $g(f(n)) = f(g(n))$ such that $g(n)$ is an integer
Let $n$ be a strict positive integer and let's define an integer sequence $f(n)$ :
$$f(n) = \frac{n^2 + n + 4}{2}$$
so
$$
\begin{split}
f (\Bbb N)& \triangleq {3,5,8,12,17,23,30,38,47,\ldots}\\
f(...
- 763
5
votes
1
answer
265
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Equivalence of bounded gradient flow solutions and uniformly bounded gradient descent trajectories for definable functions
I am reading paper [1] by C. Josz regarding the global convergence of the gradient method. The main result is the following:
$\textbf{Theorem}$: For a definable differentiable function $f : \mathbb{R}^...
- 171
1
vote
0
answers
53
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Unique solution to nonlinear optimization through gradient descent
I am trying to estimate the path of a random walk described by the following SSM
$$
\begin{align}
x_{t+1} &= x_{t} + q_{t+1} \newline
y_{t+1} &= h(x_{t+1}) + r_{t+1}
\end{align}
$...
- 11
4
votes
2
answers
272
views
The mower's challenge
Weeds have taken over the paths (two squares). If mowed, they don't grow back, but unmowed weeds spread at speed $1$ along the road. What's the minimum speed of the mower to get rid of all the weeds? ...
- 2,619
1
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0
answers
56
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Minimising risk in dynamical systems
I have been reading the paper of Goerner and Ulancowicz - "Quantifying economic sustainability" in which it is suggested that there is a tradeoff between sustainability and efficiency. ...
user478214
0
votes
0
answers
222
views
Convergence of ODE solutions almost everywhere to a stable equilibrium point
Theorem: Suppose ${\bf g} :\mathbb{R}^n \mapsto \mathbb{R}^n$ is continuously differentiable, there exists a set $\mathcal{A} \subset \mathbb{R}^n$ such that $\bf g$ is uniformly Lipschitz on $\...
- 1
8
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0
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278
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The busy Star Guardian
On an infinite plane, the Prime Star has disintegrated into four constituent stars, the North Star, the South Star, the East Star and the West Star, each traveling at a constant speed of $1$ in their ...
- 2,619
4
votes
1
answer
90
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Do Pareto critical points of a multicriteria optimization problem form an attractor of the dynamical system induced by a descent algorithm?
Let $d\in\mathbb N$, $k\in\mathbb N$ and $f:\mathbb R^d\to\mathbb R^k$ be differentiable. Say that $v\in\mathbb R^d$ is a descent direction at $x\in\mathbb R^d$ if ${\rm D}f(x)v<0$ (component-wise) ...
- 167
4
votes
1
answer
242
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Nonlinear system of integral equations
I have encountered a system of nonlinear integral equations in my work. They take the form
$$\int_{0}^{1} \frac{1}{g(y)}e^{f(x)/g(y)}(x+f(x)/g(y)-f(x))dy=0$$
$$\int_{0}^{1}\frac{f(x)}{g(y)^2} e^{f(x)/...
- 143
3
votes
0
answers
68
views
Convergence of iteration of a convex program
Let $\mathbf{V} \in \mathbb{R}_{+}^{n \times m}, \ \ \mathbf{E} \in \mathbb{R}_{+}^{n \times m}$, with $\mathbf{V} \mathbf{1}_{m} = \mathbf{1}_{n}$ and $\mathbf{E}^{T} \mathbf{1}_{n} = \mathbf{1}_{m}$...
- 53
2
votes
0
answers
100
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Fixed point of dynamic system
Let $F(\cdot, \cdot)\colon \mathbb{R}^n \times \mathbb{R}^n\rightarrow \mathbb{R}$ be a bivariate and nonnegative function. Suppose $F(x,y)$ is not convex with repect to $x$ or $y$. Moreover, asusme $...
- 1,127
7
votes
1
answer
483
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Generalized Rayleigh-quotient gradient flow on Grassmannian
The following theorem appears without proof in :
Helmke, Uwe, and John B. Moore. Optimization and dynamical systems. Springer Science & Business Media, 2012.
Let $A$ be a symmetric $n\times n$ ...
- 318
2
votes
1
answer
412
views
Non-linear 1st order difference equation
I have been trying to solve the following difference equation for some time now : $$u^3(n+1) = a - b\cdot u^2(n) + u^3(n), \qquad a \ne 0 \ne b$$
I have tried various substitutions, simplifications ...
- 173