On Integrated Chance Constraints

Abstract

In Chapter 3 we introduced integrated chance constraints (ICCs) as a modeling tool for here-and-now stochastic programming problems; see (3.28). In the present chapter we analyze the mathematical properties of this new concept in some detail. Let us review its rationale. As indicated in Section 3.2, if in the constraints of a linear programming problem random coefficients occur with unknown realizations, then in order to have a unequivocal meaning of “feasibility” one has to make additional specifications. There are two well-known modeling techniques for this: in chance-constrained programming (CCP) the probability of infeasibility is restricted, and in stochastic programming with recourse (SPR) the effects of infeasibility are penalized. For convenience, we here consider “penalty cost” models as recourse models, see Remark 3.6. Several authors [12,10,11,3,2] established certain equivalences between CCP and SPR. Their results are not completely convincing, however; for example, CCP problems may be nonconvex whereas SPR problems are always convex [5 page 90]. Even if mathematical equivalence can be established there still are differences between CCP and SPR models which might be important for the model builder.