WordWeb Online

1. In an unpleasantly offensive manner
   "he smelled offensively unwashed"
    
2. In an obnoxious manner
   "he said so in one of his more offensively intellectually arrogant sentences";
   - objectionably, obnoxiously
    
3. Aggressively
   "'In this crisis, we must act offensively,' the President said"; "the admiral intends to act offensively in the Mediterranean"

Antonym: defensively, inoffensively