Missing Person/Abduction Related Question

Hypothetically, if a women, early 20s, physically fit & able to take care of herself, were to be possibly "randomly" abducted, what would be the proper course of action by law enforcement to take on this case if the abductor were to of eventually been profiled to be a possible male, high functioning sociopath, caucasian, mid 20s, early 30s, with possible low self estime but the ability to travel great distances without being noticed, potentially with the ability to suede his victim into doing what he wants.
I just want to know, hypothetically, what would the response by law enforcement be to find this missing women, either an all out media barrage to get the word out to the public or to keep the investigation quiet, in hopes of not alarming the abductor... Or if there are any other options? What would they be?
Serious answers please! Preferably from anyone with Crim or law enforcement background! And the sooner then better!
Thanks!

Oh I hope this is for a book you're writing, or a paper you're studying for and NOT because you're a high functioning sociopath