ADMM and Linearized ADMM for Weakly Convex Minimization
We study a class of weakly convex optimization problems in which the objective is the sum of a smooth convex term and a weakly convex term that may be nonsmooth. To exploit this structure, we develop a splitting technique based on the alternating direction method of multipliers (ADMM), which decouples the minimization of the two components into tractable subproblems. Because the update associated with the smooth term may require an inner iterative solver, we further linearize this term, yielding a linearized ADMM (LADMM) scheme with an inexpensive one-step update. Under mild conditions, we establish the subsequence convergence of both ADMM and LADMM methods to directional stationary solutions, which are equivalent to critical points and Clarke stationary solutions for our weakly convex problem. Numerical experiments on two low-dimensional test functions and a high-dimensional logarithmic regularized logistic regression model demonstrate that the proposed approaches are computationally efficient and produce solutions of comparable quality to baseline methods.