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