On the sample complexity of uncertain linear and bilinear matrix inequalities

In this paper, we consider uncertain linear and bilinear matrix inequalities which depend in a possibly nonlinear way on a vector of uncertain parameters. Motivated by recent results in statistical learning, we show that probabilistic guaranteed solutions can be obtained by means of randomized algorithms. In particular, we show that the Vapnik-Chevonenkis dimension (VC-dimension) of the two problems is finite, and we compute upper bounds on it. In turn, these bounds allow us to derive explicitly the sample complexity of the problems. Using these bounds, in the second part of the paper, we derive a sequential scheme, based on a sequence of optimization and validation steps. The algorithm is on the same lines of recent schemes proposed for similar problems, but improves both in terms of complexity and generality.