# Question about theorems of the form "Any limit point of the sequence is a minimum of the problem"

In Optimization, we have problems of the form \begin{align*} &\min f(x)\\ &\text{s.t. }\hspace{0.2cm} x \in S \end{align*} and many theorems are of the form "If the problem satisfies certain conditions and $$x^k$$ is a certain sequence, then every limit point of $$x^k$$ is a minimum of the problem."

My question is: Let's say you run the algorithm and produce $$x^1, x^2, ..., x^{1000}$$. Do you write a little problem to pick arbitrary subsequences and see if they look like they might converge?

Also, how do you tell if a sequence looks like it might converge? Do you literarily look at it and see if it seems to stabilize around a point?

• Are you interested to use composition method to solve the problem? Dec 3 '20 at 9:03
• @A.Omidi Hmm no I haven't learned composition method. I do not have a specific problem to solve. I am just reading some textbooks and seeing these theorems, and I am curious how one might apply them to real life problems.
– Ovi
Dec 3 '20 at 14:25

I can't say that I've ever done this -- my experience with iterative methods has largely been confined to cases where the master sequence converges to a single limit point -- but one possibility would be to run the algorithm long enough to generate a decent size sequence, sort the sequence (using lexicographic order if $$x$$ has dimension greater than 1), and then scan the sorted sequence for clusters of observations that are close together (perhaps by looking at L1 difference in consecutive observations). If you find such a cluster, one of those limit points would be nearby.