A Probabilistic Approach to Optimal Estimation - Part II: Algorithms and Applications

In this paper, we develop randomized and deterministic algorithms for computing the probabilistic radius of information associated to an identification problem, and the corresponding optimal probabilistic estimate. To compute this estimate, in the companion paper [11] the concept of optimal violation function is introduced. Moreover, for the case of uniform distributions, it is shown how its computation is related to the solution of a (quasi) concave optimization problem, based on to the maximization of the volume of a specially constructed polytope. In this second paper, we move a step further and develop specific algorithms for addressing this problem. In particular, since the problem is NP-hard, we propose both randomized relaxations (based on a probabilistic volume oracle and stochastic optimization algorithms), and deterministic ones (based on semi-definite programming). Finally, we present a numerical example illustrating the performance of the proposed algorithms.