We introduce and investigate a new generalized convexity notion for functions called prox-convexity. The proximity operator of such a function is single-valued and firmly nonexpansive. We provide examples of (strongly) quasiconvex, weakly convex, and DC (difference of convex) functions that are prox-convex, however none of these classes fully contains the one of prox-convex functions or is included into it. 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.

中文翻译：

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超越凸性的近点算法的扩展

我们引入并研究了一种新的广义凸性概念，用于称为 prox-convexity 的函数。这种函数的邻近算子是单值的并且是非扩展的。我们提供了近似凸函数的（强）拟凸函数、弱凸函数和 DC（凸差）函数的示例，但是这些类中没有一个类完全包含或包含在其中。我们表明，当要最小化的适当下半连续函数的凸性放松到近似凸性时，经典的近点算法保持收敛。