Local convergence of a proximal type method for weakly convex minimization

The main purpose of this paper is to develop and analyze a proximal-type method for the local minimization of weakly convex functions. Our motivation for introducing this algorithm is to increase its versatility and applicability in a broader range of scenarios in optimization and introduce a more adaptable version of the method for typical generalizations. Our study relies heavily on the Moreau envelope, a well-known mathematical tool used to analyze the behavior of the proximal operator. By leveraging the properties of the Moreau envelope, we are able to prove that the proximal algorithm converges in local contexts. Moreover, we present a complexity result to assess the practical feasibility of the proximal algorithm and provide a numerical example to illustrate the practical behavior of the algorithm.