A Semismooth Newton-Based Augmented Lagrangian Algorithm for the Generalized Convex Nearly Isotonic Regression Problem

The generalized convex nearly isotonic regression problem addresses a least squares regression model that incorporates both sparsity and monotonicity constraints on the regression coefficients. In this paper, we introduce an efficient semismooth Newton-based augmented Lagrangian ( Ssnal ) algorithm to solve this problem. We demonstrate that, under reasonable assumptions, the Ssnal algorithm achieves global convergence and exhibits a linear convergence rate. Computationally, we derive the generalized Jacobian matrix associated with the proximal mapping of the generalized convex nearly isotonic regression regularizer and leverage the second-order sparsity when applying the semismooth Newton method to the subproblems in the Ssnal algorithm. Numerical experiments conducted on both synthetic and real datasets clearly demonstrate that our algorithm significantly outperforms first-order methods in terms of efficiency and robustness.