SSNCVX: A primal-dual semismooth Newton method for convex composite optimization problem

Abstract: In this paper, we propose a uniform semismooth Newton-based algorithmic framework called SSNCVX for solving a broad class of convex composite optimization problems. By exploiting the augmented Lagrangian duality, we reformulate the original problem into a saddle point problem and characterize the optimality conditions via a semismooth system of nonlinear equations. The nonsmooth structure is handled internally without requiring problem specific transformation or introducing auxiliary variables. This design allows easy modifications to the model structure, such as adding linear, quadratic, or shift terms through simple interface-level updates. The proposed method features a single loop structure that simultaneously updates the primal and dual variables via a semismooth Newton step. Extensive numerical experiments on benchmark datasets show that SSNCVX outperforms state-of-the-art solvers in both robustness and efficiency across a wide range of problems.