Linearized ADMM for Nonsmooth Nonconvex Optimization with Nonlinear Equality Constraints

This paper proposes a new approach for solving a structured nonsmooth nonconvex optimization problem with nonlinear equality constraints, where both the objective function and constraints are 2-blocks separable. Our method is based on a 2-block linearized ADMM, where we linearize the smooth part of the cost function and the nonlinear term of the functional constraints in the augmented Lagrangian at each outer iteration. This results in simple subproblems, whose solutions are used to update the iterates of the 2 blocks variables. We prove global convergence for the sequence generated by our method to a stationary point of the original problem. To demonstrate its effectiveness, we apply our proposed algorithm as a solver for the nonlinear model predictive control problem of an inverted pendulum on a cart.