Sampled convex programs are studied to deal with convex optimization
problems including uncertainty. A deterministic approach called robust
optimization is commonly applied to solve these problems.
On the other hand, sampled convex programs are randomized approach
based on constraint sampling. Calafiore and Campi have proposed sufficient
number of samples such that only small portion of original constraints
are violated at randomized solution. Our main concern is not only the
probability of violation, but also the degree of violation, that is,
the worst-case violation.
We derive an upper bound of the worst-case violation for sampled convex
programs under general uncertainty set, and provide the relation between
the probability of violation and the worst-case violation. As well as
the probability of violation, the degree of violation is also assured to
be depressed to small value with sufficiently large number of random samples.