Nonlinear Elastic-Net Regularization and Its Iterative Soft Thresholding Algorithm

Abstract Elastic-net regularization, as a variational method, demonstrates enhanced stability compared to classical <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi mathvariant="normal">ℓ</m:mi> <m:mn>1</m:mn> </m:msub> </m:math> {\ell_{1}} sparsity regularization, making it suitable for addressing ill-conditioned problems. However, conventional elastic-net regularization is typically limited to linear equations. In this paper, we extend the elastic-net regularization method to nonlinear problems. We investigate the well-posedness of this regularization and demonstrate that it serves as a sparsity regularization approach. The iterative soft thresholding algorithm, commonly used for classical <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi mathvariant="normal">ℓ</m:mi> <m:mn>1</m:mn> </m:msub> </m:math> {\ell_{1}} sparsity regularization, features a straightforward structure and is easy to implement. We show that, under widely accepted conditions regarding the nonlinearity of the function F , this algorithm is effective in solving the elastic-net regularization for nonlinear ill-conditioned equations. Our numerical results highlight the efficiency of the proposed method.