Manager dating subordinate
Given that potential scenario the companies discourage dating in the workplace to the point of making it a cause for firing an employee. By falling in love / having romantic relationship (sounds cheap, I'm sorry if it hurts you) may end up losing your prestige, dignity, respect.
The dating of employees within the company is usually allowed by a special dispensation from Human Resources after a consultation with lawyers. Because if you are in love and want to date and get married that will be great.
The subordinate may reasonably believe that rejecting the invitation will have adverse consequences at work.
Relationships between colleagues may be OK, but could still cause problems if one party has a higher position in the company, due to the influence the more senior person may have with the junior's supervisor. There's at least a 90% chance it wont work out and a 30% chance it will turn ugly.
How can you be 100% certain that your subordinate shares your feelings?
Because if they don't, and you approach them, the subordinate may rightly think that saying no could harm them professionally.
Otherwise try to tackle the whole not dating your subordinates lifestyle first, to at least look like you don't need to leverage your ownership/managerial position for dating.
I have worked several places where the boss was dating one of the employees and in two out of three cases, it was a cancer in the workplace.
They do want to see you happy in your dating life, but they don't want to see their firm's reputation suffer because they have families/wives they need to provide for.