Mr. Bloomberg did not endorse a presidential candidate in 2008, when Mr. Obama ran against Senator John McCain, Republican of Arizona, and until Thursday, he had steadfastly withheld his support from both presidential candidates this year, largely because he had grown frustrated with the tone and substance of the presidential campaign. He recently derided as “gibberish” the answers that Mr. Obama and Mr. Romney gave during a debate to a question about an assault-weapons ban. He has expressed disappointment with Mr. Obama’s performance over the past few years and concern about what he has described as the shifts in Mr. Romney’s views over time.