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Say I discover an “interesting topic” (at least from my perspective), and I do all sort of work necessary to formulate and tackle the problem (it is an optimization problem), nearly a month of work, and now I suddenly come to realize that the close-form analytical solution to this problem is not possible (or that the solution is too complicated).

What should I do? Should I continue to solve the problem using a numerical method (using computer simulation tools from Matlab), or should I give up my topic?

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closed as too broad by David Ketcheson, Nate Eldredge, scaaahu, Fomite, silvado Jul 3 '14 at 9:05

There are either too many possible answers, or good answers would be too long for this format. Please add details to narrow the answer set or to isolate an issue that can be answered in a few paragraphs.If this question can be reworded to fit the rules in the help center, please edit the question.

Have you talked to your advisor about your problem? – Mad Jack Jul 1 '14 at 3:43
This question appears to be off-topic because it is about mathematical research. It's also stated in terms that are much too vague for anyone to give a reasonable answer. – David Ketcheson Jul 1 '14 at 4:12
@David why is mathematical research off topic? This is not a question on how to solve a particular mathematical problem, it's a question on the research process. – ff524 Jul 1 '14 at 4:33
This question is lacking some information about the context; what is this research being done for? I can see some reason for concern if this is, for example, for a thesis with a fixed time limit of either 3 or 6 months, but if, on the other hand, you are doing PhD research and that was an attempt at trying something that might become a paper, "nearly a month of works" is not a long time for working on a particular idea - and possible dead-end - at all. – O. R. Mapper Jul 1 '14 at 7:19
To answer the title question: Then the problem becomes interesting! – JeffE Jul 1 '14 at 20:13

Most real world problems don't have closed-form solutions. Somehow, we manage.

You write:

the close-form analytical solution to this problem is not possible

If the impossibility of a closed-form analytic solution is in itself a new result, then that may well be a publishable result. If you can also find an efficient method to get a numerical (approximate) solution, that's either a heavier-weight first paper (combined with the proof of the non-existence of closed-form solution), or it's a second paper in its own right.

You then wrote:

OR the solution is too complicated.

If there may be a closed-form analytical solution, but it's just too complicated for you to find, then that's an entirely different matter. In which case, you've got nothing to publish. Just a very hard problem that you can either persevere with, or you can stop working on it for a while, and go do something more promising. There's no harm in stopping working on it for a while. Just keep the problem in the back of your mind, along with a handful of other unsolved problems: and every time you learn a new heuristic, algorithm, or solution pattern (or an enhancement to an old one), then try to apply it to the unsolved problems you've been storing up (kudos to the late Richard Feynman for this).

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+1 for the distinction between not possible and too complicated. Though for some fields, "Difficult + Compelling simulation results" is publishable. – Fomite Jul 2 '14 at 4:35

This depends entirely on the standards in the particular field, previous work on this topic, and the specific topic and problem domain itself.

A closed form solution is not always necessary in order to make a research contribution. Sometimes formulating the problem is itself significant; sometimes numerical methods offer useful insight; sometimes saying "we tried applying technique X to problem Y and found that it cannot work" is a contribution.

Another possibility in some cases is to simplify or modify your problem formulation to something that lends itself more easily to a closed form solution.

Reading the literature surrounding the problem is a good way to become familiar with what kinds of contributions are considered useful in this area. Then you can decide whether to proceed with numerical methods or do something else. You should also try and find a faculty mentor (if you don't already have one) to advise you on what your next steps should be.

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To support the "depends on the field", closed form solutions may not even be particularly meaningful, if a particular set of numerical results is what's going to be carried forward. – Fomite Jul 2 '14 at 4:35

My topic is numerical (computational) mathematics. In this area we develop algorithms and apply them to (hopefully) important problems, which typically do not have a pen and paper solution. I am doing this for more than 15 years and see no reason to give up.

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Never, never, never give up. – Pete L. Clark Jul 1 '14 at 5:41
Never giving up is overrated. Sometimes you need to sit back and change tracks (but not until you know you have done everything you could). – user13107 Jul 2 '14 at 2:08

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