Well, it seems that your problem is not very serious, it is simply some cosmetic problem. Anyway, the oil bumps is a sign of your bad skin conditions. I mean, the balance of water and oil does not exist any more. So you must take some steps to tackle this problem. My advice for you is to keep a healthier diet, more vegetables and fruits,less oil and meat. Also, try to get some mild face wash to help. Meantime, it would be nice for you to keep a good rest on a regular basis.
I think there are four ways to get rid of it. First, you should figure out that if this oil bumps under eyes are caused by your eye cream. If so, you should switch to another brand of eye cream which is suitable for you. Second, try to make sure that whether you overdoes on eye cream when you use it or not. If so, you should cut down the quality once. Third, try to use eye gel instead of eye care product or cream because care product and cream are more oily than gel. Fourth, apply a wet and hot washcloth on your eyes until it is cool and then heat it up and repeat doing this twice. If you insist on this over night, it will gone within a week.