On the convexity of probabilistically constrained linear programs

The focal point of this paper is the so-called Probabilistically Constrained Linear Program (PCLP), also known as Chance Constrained Linear Program. We prove that PCLP is a convex program when the uncertain parameters are uniformly distributed over a convex symmetric set Q. By symmetric we mean that if q∈Q then -q∈Q. Furthermore, we provide a deterministic equivalent of the PCLP which, in some cases, leads to an implementation that does not require the use of stochastic programming; only commonly available optimization tools are required. Finally, the concept of Probabilistically Robust Linear Program (PRLP) is introduced. The PRLP is a generalization of the PCLP for the case when the distribution of the uncertain coefficients is not known. The only assumption is that their distribution belongs to a known class ℱ. Since the true distribution of the uncertain parameters is not available, in the PRLP it is required that the probability of satisfaction of the constraints is greater than a given risk level ε for all possible distributions f∈ℱ. It is proven that the PRLP is also a convex program for the class ℱ