Random Convex Programs

Random convex programs (RCPs) are convex optimization problems subject to a finite number N of random constraints. The optimal objective value J∗ of an RCP is thus a random variable. We study the probability with which J∗ is no longer optimal if a further random constraint is added to the problem (violation probability, V ∗). It turns out that this probability rapidly concentrates near zero as N increases. We first develop a theory for RCPs, leading to explicit bounds on the upper tail probability of V ∗. Then we extend the setup to the case of RCPs with r a posteriori violated constraints (RCPVs): a paradigm that permits us to improve the optimal objective value while maintaining the violation probability under control. Explicit and nonasymptotic bounds are derived also in this case: the upper tail probability of V ∗ is upper bounded by a multiple of a beta distribution, irrespective of the distribution on the random constraints. All results are derived under no feasibility assumptions on the problem. Further, the relation between RCPVs and chanceconstrained problems (CCP) is explored, showing that the optimal objective J∗ of an RCPV with the generic constraint removal rule provides, with arbitrarily high probability, an upper bound on the optimal objective of a corresponding CCP. Moreover, whenever an optimal constraint removal rule is used in the RCPVs, then appropriate choices of N and r exist such that J∗ approximates arbitrarily well the objective of the CCP.