send large amount of emails to user's first degree connections (OPTED IN, NOT SPAM).
That's a bit hard to read. I will take it to mean that the "first degree connections" have themselves opted in, not that the "users" have opted in the "first degree connections" without asking them. The former would be fine, the latter could be debated, depending on the specifics of the situation.
Is there any paid service provider/API for managing such a large amount of outgoing mails?
Yep. You can start off with this thread over at Stack Overflow.
You'll need to decide whether you want a pure outgoing mail server (i.e. your own application handles all logic), or whether you want a newsletter provider (the provider handles un-subscription, has some templating logic, etc). Search around on Stack Overflow, there are multiple variations on this question already, and good answers over there.
If you want pure outgoing "SMTP-server-as-an-API", then I think SendGrid is the largest provider. For the newsletter providers, there are many good services -- MailChimp, Campaign Monitor, AWeber, etc. Most of the newsletter service providers also have an API for just sending a mail from your app.
Can we do this by installing mail servers like postfix in our hosting (RackSpaceCloud)
You could if Rackspace would allow you. It's probably cheaper than using a 3rd party provider. But generally I'd advise against it. Getting emails delivered without being marked as spam is real work; and it's a constantly evolving field. See my answer here for a introduction, and then go read more on Server Fault and Stack Overflow.
One last thing: If you're cheap, then Google App Engine can be used to send emails. But watch out for the quota limits, and in general, research this before settling on it.