The proposed feedback control approach, originally designed for smooth and strongly convex optimization problems, can be extended to handle non-smooth or non-convex optimization problems by incorporating techniques that address the challenges posed by these types of functions. One potential method is to utilize subgradient methods, which are effective for non-smooth optimization. In this context, the gradient in the feedback control dynamics can be replaced with a subgradient, allowing the algorithm to adapt to the non-smooth nature of the objective function.
Additionally, for non-convex problems, the algorithm could be modified to include mechanisms that escape local minima, such as introducing stochastic elements or perturbations in the control dynamics. This could involve randomizing the updates to the primal and dual variables, thereby enabling the algorithm to explore the solution space more broadly.
Moreover, the convergence analysis would need to be adjusted to account for the lack of strong convexity, potentially relying on concepts from nonsmooth analysis and generalized convergence criteria. This would involve redefining the Lyapunov function to ensure stability and convergence in the presence of non-smoothness or non-convexity.