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

Authors:
A. Frangioni;C. Gentile
Affiliations:
Department of Computer Science, University of Pisa, Largo B. Pontecorvo 3, 56124, Pisa, Italy;Istituto di Analisi dei Sistemi ed Informatica “Antonio Ruberti”, C.N.R., Viale Manzoni 30, 00185, Rome, Italy
Venue:
Mathematical Programming: Series A and B
Year:
2006

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Abstract

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.