Developing Lagrangian-based Methods for Nonsmooth Nonconvex Optimization

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In this paper, we consider the minimization of a nonsmooth nonconvex objective function $f(x)$ over a closed convex subset $\mathcal{X}$ of $\mathbb{R}^n$, with additional nonsmooth nonconvex constraints $c(x) = 0$.

• We develop a unified framework for developing Lagrangian-based methods, which takes a single-step update to the primal variables by some subgradient methods in each iteration.
• These subgradient methods are ``embedded'' into our framework, in the sense that they are incorporated as black-box updates to the primal variables.
• We prove that our proposed framework inherits the global convergence guarantees from these embedded subgradient methods under mild conditions.