Consistency of the Scenario Approach
本文证明了在凸约束下，情景方法（一种随机优化方法）的解随着样本量增大几乎必然收敛到某个本质鲁棒问题的最优解，为该方法提供了理论基础。
This paper is meant to prove the consistency of the scenario approach à la Calafiore, Campi, and Garatti with convex constraints. Scenario convex problems are usually stated in two equivalent forms: first, as the minimum of a linear function over the intersection of a finite random sample of independent and identically distributed convex sets, or second, as the min-max of a finite random sample of independent and identically distributed convex functions. The paper shows that, under fairly general assumptions, as the size of the sample increases the minimum attained by the solution of a problem of the first kind converges almost surely to the minimum attained by a suitably defined “essential” robust problem (or diverges if such a robust problem is infeasible), and that the minimum attained by the solution of a scenario problem of the second kind converges almost surely to the minimum of the pointwise essential supremum taken over all the possible convex functions (or diverges if such essential supremum takes the only value $+\infty$). In both cases, if the solution of the essential problem exists and is unique, the solution of the scenario problem converges to it almost surely.