Chance Constrained Programming with Joint Constraints

Abstract

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.

A tractable approximation of chance constrained stochastic MPC based on affine disturbance feedback

“…In (13) the chance constraint has to be satisfied for each time step i in the future individually, however, since at each time step there are r inequalities that have to be fulfilled, we have a joint chance constraint at each time step. Since individual chance constraints are far easier to handle, the usual procedure is to approximate the joint chance constraints with individual chance constraints and set the probability level by defining α k = α/r, where r is the number of rows [12]. This gives…”

Alternate risk measures for emergency medical service system design

“…Models involving probabilistic constraints were introduced by Charnes (1958), Miller (1965), and Prékopa (1970). Prékopa (1995) discusses in detail the probabilistic optimization theory and the associated numerical techniques.…”

Optimization Under Probabilistic Envelope Constraints

“…Chance constraints date at least as far back as, e.g., Charnes and Cooper 1959, and since then there has been considerable work, e.g., Miller and Wagner (1965), Prékopa (1970), Delage and Mannor (2010), and many others; we refer the reader to the textbook Prékopa (1995) and reference therein for a thorough review. The chance constraint formulation in (1) guarantees that the given constraint will be satisfied with probability p. With the remaining 1 − p probability, the constraint is violated, and no control whatsoever is provided on the degree of violation.…”