Try a hidden spot first to try this out, but I'd give polishing compound a try first. An orbital polisher and some good automotive products might be just the ticket.
The reason I say this is because it works great on "cultured marble" sinks. Makes them shine like new.
I use rubbing compound (scratch remover) on those though, and I do it by hand.
And while I'm at it, Griot's 3 in 1 has become my new favorite accessory in the kitchen!
Countertops, stovetops, and even stainless steel sink stays clean longer and shines pretty well.