Chance Constrained Programming with Joint Constraints - Miller, Bruce L.; Wagner, Harvey M. (Operations Research, 1965)
This paper considers the mathematical properties of chance constrained programming problems where the restriction is on the joint probability of a multivariate random event. One model that is considered arises when the right-handside constants of the linear constraints are random. Another model treated here occurs when the coefficients of the linear programming variables are described by a multinormal distribution. It is shown that under certain restrictions both situations can be viewed as a deterministic nonlinear programming problem. Since most computational methods for solving nonlinear programming models require the constraints be concave, this paper explores whether the resultant problem meets the concavity assumption. For many probability laws of practical importance, the constraint in the first type of model is shown to violate concavity. However, a simple logarithmic transformation does produce a concave restriction for an important class of problems. The paper also surveys the “generalized linear programming” method for solving such problems when the logarithmic transformation is justified. For the second type model, the constraint is demonstrated to be nonconcave.