A Projected Gradient Method For Nonlinear Inverse Problems With αl₁-βl₂ Regularization

The investigation of the non-convex α‖·‖(?)1-β‖·‖(?)2(α≥β≥0)regularization has attracted attention in the field of sparse recovery over the last five years.As an alternative of the lpnorm with 0≤p<1,the advantage of using the functional α‖·‖(?)1-β‖·‖(?)2(α≥β≥0)lie in the fact that it is a good approximation of the l0-norm and it has a simpler structure than the l0norm from the perspective of computation.In this paper,we investigate the solution of αl1-βl2 regularization of nonlinear ill posed operator equation A(x)=y.One extensively studied way to obtain a minimizer ofα‖·‖(?)1-β‖·‖(?)2(α≥β≥0)regularization is ST-(αl1-βl2)algorithm.Unfortunately,such iteration is similar to the classical iterative soft thresholding algorithm which converges quite slowly.A subtle alternative to the ST-(δl1-βl2)algorithm is the proiected gradient(PG)method.Nevertheless,its current applicability of the PG is limited to liner inverse problems.In this paper,we extend the PG method based on generalized conditional gradient method and surrogate function approach to nonlinear inverse problems with the α‖·‖(?)1-β‖·‖(?)2(α≥β≥0)regularization.Firstly,based on the generalized conditional gradient method,the original nonconvex sparse regularized functional is transformed into a structure of G(x)+Φ(x),and the projected gradient algorithm is used to accelerate the ST-(αl1-βl2)algorithm.In addition,we propose a strategy to determine the radius R of l1-ball constraint by Morozov’s discrepancy principle.Then,we study the second acceleration algorithm in finite dimensional space Rn,namely the PG algorithm based on the surrogate function approach,and prove the convergence and stability of the proposed algorithm.Finally,numerical experiments are given in the context of a nonlinear compressive sensing problem to illustrate the efficiency of the proposed approach.