Local convergence of an inexact proximal algorithm for weakly convex functions

Since introduced by Martinet and Rockafellar, the proximal point algorithm was generalized in many fruitful directions. More recently, in 2002, Pennanen studied the proximal point algorithm without monotonicity. A year later, Iusem and Svaiter joined Pennanen to present inexact variants of the method, again without monotonicity. Building on the foundation laid by these two prior works, we propose a variant of the proximal point algorithm designed specifically for weakly convex functions. Our motivation for introducing this inexact 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 determine the practical feasibility of the proximal algorithm.