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

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-and-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

A computational comparison of reformulations of the perspective relaxation: SOCP vs. cutting planes

“…Semi-continuous variables are very often found in models of real-world problems such as production planning problems [17,5,7,8], financial trading and planning problems [10,6], and many others [4,1,9]. These are variables that are constrained to either assume the value 0, or to lie in some given convex compact set P ⊆ R m ; in our applications P will always be a polyhedron.…”

Disjunctive Inequalities: Applications And Extensions

“…Ç ezik and Iyengar [18] and lately Drewes [23] proposed, for MICP where all disjunctions are generated from binary variables, an application of the lifting procedure to the conic case, whereby disjunctive inequalities are obtained by solving a continuous conic optimization problem. Analogously to Frangioni and Gentile [26], restricting to a special type of convex constraint (second order or semidefinite cone) allows to obtain more specialized and thus efficient procedures for obtaining a disjunctive inequality.…”

On interval-subgradient and no-good cuts

“…This is still nonlinear and nonconvex, but we can linearize it using (6)- (11). Some considerations are needed in order to compute the tightest possible d i and d i for each index i.…”

“…In [18], lifting techniques are discussed in the framework of NLP; [20] discusses an extension of the RLT to convex Mixed-Integer Programming (MIP). A certain attention has been devoted to conic MIP [8,2]; in part, this is due to the fact that Lift&Project techniques (see, e.g., [3]) to compute valid inequalities for the union of two convex sets can easily be extended to the nonlinear setting [9], and this may produce strong conical reformulations of MIPs [22,12] out of which effective cuts may be obtained [11].…”