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The Huber norm Huber (1973) is an alternative to Iteratively Reweighted Least Square
programs for solving the hybrid *l*^{2}-*l*^{1} problem. In this note, I detail a method for
minimizing the Huber norm.
Because the Huber norm gives rise to a non-linear problem with non-twice continuously differentiable
objective functions, its use is quite challenging. Claerbout (1996) implemented
a Huber regression based on conjugate-gradient descents. However, the final results were not satisfying.
Here I propose to solve the Huber problem using a quasi-Newton update of the solution with
the computation of an approximated Hessian (second derivative of the objective function).
This strategy is innovative in seismic processing and merits some explanation.
In this paper I first provide general definitions plus sufficient conditions to solve
the optimization problem. Then, I present the quasi-Newton method and the complete algorithm used to
solve the Huber problem.

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Stanford Exploration Project

4/27/2000