Is it good to publish a research paper with many partially answered questions with sub-optimal results? I work in abstract mathematics and I'm currently in the mid stage of my PhD. I try to get bounds on computational aspects of mathematics. I have been working on one problem for almost half a year and I have got four/five really challenging questions in this way. I am trying to solve these new questions but some of them seem to be very tough. I have been able to solve some of these new questions partially, but did not obtain the optimal results.

Question:  Is it good to publish a research paper with many partially answered questions with sub-optimal results?
This depends on many things. Most important is the significance of the questions themselves and whether the partial solutions add anything beyond standard techniques. The questions can be "challenging" without being significant, of course. Do they lead to other things, for example. 

If your solution techniques are standard, then they won't add much either, but if you have developed a new way of looking at such problems, then it could be very important. 

"Almost half a year" doesn't sound very long, actually, for the stage you are at. 

Your advisor should give you good advice here. He or she is more familiar with your problem and your attack and degree of attainment. 

However, I come down a bit on the side of waiting to publish. For one thing, if you publish partial results it may lead others to join the search for solutions in competition with you, perhaps denying you a more significant publication (or even a dissertation) in the future. 

If it makes sense to separate them, having four or five papers, each with a complete solution would be much better for you than rushing now. The publishing venue is likely to be more prestigious, and also more likely to accept complete results. 

Carry on.