0
$\begingroup$

Consider a $C^\infty$ smooth nonconvex function $f$, and ODE $$ \begin{cases} \dot{x}=-x\circ\nabla f(x),\\ x(0)\in\mathbb{R}^d_{++}. \end{cases} $$ Here $\circ$ is elementwise product, $\mathbb{R}^d_{++}$ the set of strict positive vectors. Then, can you prove that the solution has finite length under mild condition, e.g. $f$ satisfies the Kurdyka-Łojasiewicz inequality?

We shold be careful that $x(t)$ can approach the boundary of $\mathbb{R}^d_{++}$. I believe it should be correct, as I have tried many numerical test and cannot find any counterexamples.

Improve this question
$\endgroup$
5
  • $\begingroup$ What is the meaning of $\circ$, given that it is not composition? What is $\mathbb R_{++}^d$? $\endgroup$
    – Pietro Majer
    Commented Oct 9 at 22:08
  • $\begingroup$ Hi, the $\circ$ is elementwise product, $\mathbb{R}^d_{++}$ the set of strict positive vectors. I have updated the question. $\endgroup$
    – dkyopt
    Commented Oct 10 at 9:18
  • $\begingroup$ Thank you -maybe then $\cdot$ is preferable $\endgroup$
    – Pietro Majer
    Commented Oct 10 at 9:39
  • $\begingroup$ It would also be useful to write the inequality in your case. This paper seems to be related to your problem mathreader.github.io/files/… $\endgroup$
    – Pietro Majer
    Commented Oct 10 at 9:58
  • $\begingroup$ Thanks. I am a mathematical optimization researcher. Indeed, the dynamical system comes from an optimization problem. I have tried the KL assumption, but it is not enough. $\endgroup$
    – dkyopt
    Commented Oct 11 at 8:10

0

Reset to default

You must log in to answer this question.

Browse other questions tagged .