The problem of automatic extraction of sentiment expressions from informal text, as in microblogs such as tweets is a recent area of investigation. Compared to formal text, such as in product reviews or news articles, one of the key challenges lies in the wide diversity and informal nature of sentiment expressions that cannot be trivially enumerated or captured using predefined lexical patterns. In this work, we present an optimization-based approach to automatically extract sentiment expressions for a given target (e.g., movie, or person) from a corpus of unlabeled tweets. Specifically, we make three contributions: (i) we recognize a diverse and richer set of sentiment-bearing expressions in tweets, including formal and slang words/phrases, not limited to pre-specified syntactic patterns; (ii) instead of associating sentiment with an entire tweet, we assess the target-dependent polarity of each sentiment expression. The polarity of sentiment expression is determined by the nature of its target; (iii) we provide a novel formulation of assigning polarity to a sentiment expression as a constrained optimization problem over the tweet corpus. Experiments conducted on two domains, tweets mentioning movie and person entities, show that our approach improves accuracy in comparison with several baseline methods, and that the improvement becomes more prominent with increasing corpus sizes.