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This paper investigates the preservation of local minimizers and strongminimizers of extended-real-valued lower semicontinuous functions under takingtheir Moreau envelopes. We address a general setting of Banach spaces, whileall the obtained results are new even for functions in finite dimensions. Ourmain motivation came from applications to numerical methods in nonsmoothoptimization dealing with broad classes of nondifferentiable and nonconvexfunctions. The paper also formulates and discusses some open questions stemmingfrom this study.

Local Minimizers of Nonconvex Functions in Banach Spaces via Moreau Envelopes