Perspective cuts for a class of convex 0–1 mixed integer programs
- 763 Downloads
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.
KeywordsMixed-Integer Programs Valid Inequalities Unit Commitment problem Portfolio Optimization
Unable to display preview. Download preview PDF.
- 1.Ahn, S., Escudero, L.F., Guignard-Spielberg, M.: On modeling robust policies for financial trading. In: T.A. Ciriani and R.L. Leachman (eds.) Optimization in Industry 2, Wiley Chichester, 1994 pp 163–184Google Scholar
- 3.Borghetti, A., Frangioni, A., Lacalandra, F., Nucci, C.A.: Lagrangian heuristics based on disaggregated bundle methods for hydrothermal unit commitment. IEEE Transactions on Power Systems 18 (1), 1–10 (2003)Google Scholar
- 6.Frangioni, A., Gentile, C.: Perspective cuts for 0–1 mixed integer programs. Technical report 577, Istituto di Analisi dei Sistemi ed Informatica “Antonio Ruberti” (IASI-CNR), 2002Google Scholar
- 8.Frangioni, A.: About lagrangian methods in integer optimization. Annals of Operations Research, To appear 2005Google Scholar
- 9.Hiriart-Urruty, J.-B., Lemaréchal, C.: Convex analysis and minimization algorithms I–- fundamentals. Grundlehren Math. Wiss. 305 Springer-Verlag, New York 1993Google Scholar
- 10.Jobst, N.J., Horniman, M.D., Lucas, C.A., Mitra, G.: Computational aspects of alternative portfolio selection models in the presence of discrete asset choice constraints. Quantitative Finance 1, Wiley Chichester, 2001, pp 1–13Google Scholar
- 11.Kallrath, J., Wilson, J.M.: Business optimization. Macmillan Press Ltd. Houndmills, 1997Google Scholar