Characterizations and applications of beyond convex functions in Banach spaces

We investigate a variant of the class of prox-convex functions introduced in Grad and Lara [An extension of the proximal point algorithm beyond convexity. J Glob Optim. 2022;82:313–329] to the context of Banach spaces, namely prox-convexity and generalized prox-convexity, respectively. The proximity operators of such a function is single-valued, generalized firmly nonexpansive and weakly continuous. The Moreau envelope of such a function is directionly differentiable. We show that the classical proximal point algorithm remains convergent when the convexity of the proper lower semicontinuous function to be minimized is relaxed to prox-convexity or generalized prox-convexity. We also proved the convergence of the function values; the asymptotic rate n−1 reaches its optimal value.