Local Minimizers of Nonconvex Functions in Banach Spaces via Moreau Envelopes

Abstract: This paper investigates the preservation of local minimizers and strong minimizers of extended-real-valued lower semicontinuous functions under taking their Moreau envelopes. We address a general setting of Banach spaces, while all the obtained results are new even for functions in finite dimensions. Our main motivation came from applications to numerical methods in nonsmooth optimization dealing with broad classes of nondifferentiable and nonconvex functions. The paper also formulates and discusses some open questions stemming from this study.