Perspective cuts for a class of convex 0-1 mixed integer programs

We show that the convex envelope of the objective function of Mixed-Integer Programming problems with a specific structure is the perspective function of the continuous part of the objective function. Using a characterization of the subdifferential of the perspective function, we derive ``perspective cuts'', a family of valid inequalities for the problem. Perspective cuts can be shown to belong to the general family of disjunctive cuts, but they do not require the solution of a potentially costly nonlinear programming problem to be separated. Using perspective cuts substantially improves the performance of Branch \& Cut approaches for at least two models that, either ``naturally'' or after a proper reformulation, have the required structure: the Unit Commitment problem in electrical power production and the Mean-Variance problem in portfolio optimization.