I don't know what the answer is but I know what it isn't as my ex husband hasn't seen my children (now adults) for 9 years. They absolutely hate getting unwarranted messages from him or his wife as they are full of self pitying explanations and tales of ' I thought about you every day'. This just really angers my DC. I would think that they would react better if he just said hi and that he would like to keep in contact etc and just start slowly.
I agree with eliza's post.
I had no contact with my father for 20 years. If he had sent me a Facebook friend request I would have pulled this face 
and deleted it. It's an incredibly frivolous way to initiate contact.
As it happens, contact was eventually initiated by me, after DS1 was born. It was purely a matter of courtesy on my part and I had no real stake in whether or not he replied. As it happens he did, and we now have a cordial, if extremely distant relationship. He sends presents to the kids, and we visit - in a neutral setting - once or twice a year. He has changed slightly with age, but not enough to make me want a closer relationship.
Sometimes I get intensely irritated with him. Cutesy, over-familiar 'daughter' cards are one thing. Faintly accusing voicemail messages about not having heard for a while and being worried - usually followed up by another one 5 minutes later, since he clearly hasn't figured out that voicemail messages aren't transmitted directly into the recipient's head the second they are recorded - are another. Perhaps these are unfair reactions on my part, but it is what it is. He didn't bother for 20 years and I was the one who made the initial approach, and yet he can't wait 5 minutes for a response.
I would approach with extreme caution if you don't want your children to react in a similar way. I would not have reacted well to a casual friend request on Facebook. I would not have reacted well to a self-flagellating 'woe is me' type approach. I would have reacted extremely badly to any attempt to justify or blame others for his long absence.
If you are serious about building bridges then I would suggest a formal, matter-of-fact approach. A letter would be best. Just provide your contact details, acknowledge that a great deal of time has passed and things ended badly for the family relationship, and ask them to make contact if they wish to do so. Make it clear that you will not push the issue if they choose not to pursue a relationship but that you would be pleased if they did, and that any contact will be on their terms. Nothing that could be taken as undue pressure or emotional blackmail. Not too much about how sad it makes you feel. Keep it simple and honest.
And you need to be prepared for the fact that, even if they make contact, you may never have anything more than a cordial relationship. Sometimes that ship has just sailed. Children can be far more resilient than we might expect. I was. I was sad for a while, and wondered why my father didn't love me as much as my friends were loved by their fathers. And then I got over it and moved on.
Sorry if this isn't the soft-focus, lovey-dovey sort of personal story you were hoping for, but you need to be realistic about the fact that there are several ways that this could go - and not all of them involve happy endings. Your children are the wronged parties in all of this, and now they are adults and they have the right to choose the shape of any relationship they have with you. From the tone of your posts I do have some concerns about how you might approach the situation, and how they might be made to feel as a result.
Tread very carefully.