I will talk about expected utility maximization under ratchet and drawdown constraints on consumption in incomplete semimartingale markets. The drawdown constraint on consumption means that the consumption rate process does not fall below a fraction \lambda\in[0,1] of its current running maximum; ratchet constraint, a special case corresponding to \lambda=1, means that consumption rate is non-decreasing. For each \lambda\in[0,1], the optimization is considered via convex duality methods and with respect to two parameters: the initial wealth and the essential lower bound on consumption process. In order to state the problem and define the primal domains in this general semimartingale market setting, I introduce a natural extension of the notion of running maximum to arbitrary non-negative optional consumption rate processes. The dual domains for optimization are then characterized in terms of the closed solid hull of the set of equivalent martingale deflators with respect to a certain ordering on the set of non-negative optional processes. Finally, I will show that the abstract duality result for incomplete markets can be used in order to derive a more detailed characterization of solutions in the complete market case.