Many times, people who suffer the absence of abusive partners do so for one or more of three reasons that variably existed as a result of, or as a foundation for the relationship.
Validation: Tulip mentioned this earlier when she stated that it was sometimes better to have someone recognize you and treat you badly, than not recognize you at all. Even negative attention can be a source of validation that we exist - that we affect association from and between ourself and another person. Wanting to be a part of something greater than ourselves is a human need. However, combining that need with low self worth, or desperation for affection can sometimes be a powerful attractant to the wrong kind of partner.
Identity: This is just about the most common reason for asociating with, and mourning the absence of an abusive partner. Low self esteem creates a desire to rely upon the strength of others as a means of giving identity to oneself. Abused adults will acquiesce to the strength of their overbearing partners, even in cases of physical abuse because they've been conditioned (probably since childhood) to measure their own self worth through the eyes of others. This creates a constant attempt to please the abusive partner, even though the actions result in a continuation of the same low self image.
Awakening: Terorizer touched on this point in his post. There is something initially exciting about someone who can or will do things that make you feel energized; alive, or stimulated in a way that you may not have had the courage or experience to do for yourself. Perhaps it's an initial feeling of protection, of being guided through an amazing adventure, or just being in the presence of someone who demonstrates themselves to be so much stronger than you. In a healthy relationship, both partners grow as a result of the union. In an abusive relationship, it is the weaker partner who becomes the inevitable target as soon as the newness has worn out.