Chance-constrained optimization via randomization: feasibility and optimality

In this paper we study the link between a semi-infinite chance-constrained optimization problem and its randomized version, i.e. the problem obtained by sampling a finite number of its constraints. Extending previous results on the feasibility of randomized convex programs, we establish here the feasibility of the solution obtained after the elimination of a portion of the sampled constraints. Constraints removal allows one to improve the cost function at the price of a decreased feasibility. The cost improvement can be inspected directly from the optimization result, while the theory here developed permits to keep control on the other side of the coin, the feasibility of the obtained solution. In this way, trading feasibility for performance through a sampling-and-discarding approach is put on solid mathematical grounds by the results of this paper. The feasibility result obtained in this paper applies to all chance-constrained optimization problems with convex constraints, and has the distinctive feature that it holds true irrespective of the algorithm used for constraints removal. One can thus e.g. use a greedy algorithm – which is computationally low-demanding – and the corresponding feasibility remains guaranteed. We further prove in this paper that if constraints removal is optimally done – i.e. one deletes those constraints leading to the largest possible cost improvement – a precise optimality link to the original semi-infinite chanceconstrained problem in addition holds.