An algorithm for the minimization of mixed l1 and l2 norms with application to Bayesian estimation

The regularizing functional approach is widely used in many estimation problems. In practice, the solution is defined as one minimum point of a suitable functional, the main part of which accounts for the underlying physical model, whereas the regularizing part represents some prior information about the unknowns. In the Bayesian interpretation, one has a maximum a posteriori (MAP) estimator in which the main and regularizing parts are represented, respectively, by likelihood and prior distributions. When either the prior or likelihood is a Laplace distribution and the other is a Gaussian distribution, one is led to consider functionals that include both absolute and square norms. The authors present a characterization of the minimum points of such functionals, together with a descent-type algorithm for numerical computations. The results of Monte-Carlo simulations are also reported