I'm interested in a simple property of the gradient flow $$x'(t) = - \nabla f(x)$$: under what conditions on $$f$$ does the gradient flow converge to a stationary point?
In particular, I'm interested in the simple case of $$f : \mathbb{R}^n \rightarrow \mathbb{R}$$, where we know that $$f$$ obtains a global minimum somewhere. No assumption of convexity, we allow for arbitrarily many local minima and/or multiple global minima, saddle points, etc. It is fine if part of the conditions on the function require an assumption on the nature of these critical points (e.g. that $$f$$ has no degenerate saddle points, or something like that). I know that gradient flows are quite general, and PDE's can in many cases be cast as gradient flows on functionals, but I'm not interested in this level of abstraction at the moment - I'm thinking more about nonlinear optimization.
• If there is an $x_{\infty} \in \mathbb{R}^n$ with $\displaystyle \lim_{t \rightarrow \infty} x(t) = x_{\infty}$, then we must have $\nabla f(x_{\infty}) = 0$, i. e. $x_{\infty}$ must be a stationary point of $f$. Isn't that right? This doesn't answer the question, but just to clarify. Jan 24, 2019 at 5:47
• I would also say that convergence/divergence depends a lot on the initial value of $x$, not just on $f$. Jan 24, 2019 at 6:24
• If $f$ is real-analytic, then it satisfies a Lojasiewicz inequality: this is often used to argue single limit points of gradient flow trajectories. Aug 10 at 12:10