A Lagrangian-Based Method with "False Penalty'' for Linearly Constrained Nonconvex Composite Optimization

Abstract: We introduce a primal-dual framework for solving linearly constrained nonconvex composite optimization problems. Our approach is based on a newly developed Lagrangian, which incorporates \emph{false penalty} and dual smoothing terms. This new Lagrangian enables us to develop a simple first-order algorithm that converges to a stationary solution under standard assumptions. We further establish global convergence, provided that the objective function satisfies the Kurdyka-{\L}ojasiewicz property. Our method provides several advantages: it simplifies the treatment of constraints by effectively bounding the multipliers without boundedness assumptions on the dual iterates; it guarantees global convergence without requiring the surjectivity assumption on the linear operator; and it is a single-loop algorithm that does not involve solving penalty subproblems, achieving an iteration complexity of $\mathcal{O}(1/\epsilon^2)$ to find an $\epsilon$-stationary solution. Preliminary experiments on test problems demonstrate the practical efficiency and robustness of our method.