Weak Convexity and Approximate Subdifferentials

We explore and construct an enlarged subdifferential for weakly convex functions. The resulting object turns out to be continuous with respect to both the function argument and the enlargement parameter. We carefully analyze connections with other constructs in the literature and particularize to the weakly convex setting well-known variational principles. By resorting to the new enlarged subdifferential, we provide an algorithmic pattern of descent for weakly convex minimization. Under minimal assumptions, we show subsequential convergence to a critical point. Links with difference-of-convex functions algorithms and criticality conditions are also discussed.