Dual formulation of the utility maximization problem: the case of   nonsmooth utility
  We study the dual formulation of the utility maximization problem in incomplete markets when the utility function is finitely valued on the whole real line. We extend the existing results in this literature in two directions.   First, we allow for nonsmooth utility functions, so as to include the shortfall minimization problems in our framework. Second, we allow for the presence of some given liability or a random endowment. In particular, these results provide a dual formulation of the utility indifference valuation rule. 
B. Bouchard, N. Touzi and A. Zeghal