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/spl isin/Q then -q/spl isin/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 /spl Fscr/. 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 /spl epsi/ for all possible distributions f/spl isin//spl Fscr/. It is proven that the PRLP is also a convex program for the class /spl Fscr/.