Random Convex Programs: Dealing with the Non-existing Solution Nuisance

Random convex programs (RCPs) are convex optimization problems subject to a finite number of constraints that are extracted at random according to some probability distribution. The optimal objective of an RCP, and its associated optimal solution (when it exists), are random variables: RCP theory is mainly concerned with providing probabilistic assessments on the probability of objective and constraint violation in random convex programs. In a recent contribution, the authors provide a tight upper bound on the constraint violation probability for a restricted class of random convex programs that are assumed to be feasible and attain an optimal solution in every possible problem realization. Here, we remove this restriction and we provide a result holding for general random convex programs.