Near equality and almost convexity of functions with applications to optimization and error bounds

In this paper, we investigate near equality and almost convexity of extended real valued functions defined on finite-dimensional Euclidean spaces. The main result states that an almost convex function (respectively, its domain, lower level set) is nearly equal to (respectively, the domain, lower level set of) its closure, convex hull and closed convex hull. It is proved that almost convexity of an extended real valued function is equivalent to near equality of itself and another almost convex function. Moreover, it is shown that the operations given by sum, scalar multiple, pointwise supremum, epi-sum and epi-multiple of almost convex functions preserve almost convexity, the formulation of the subdifferential of sum and scalar multiple of almost convex functions on the relative interior of their domain is analogous to that of convex functions under suitable additional assumptions and the proximal average of almost convex functions enjoys analogous properties of lower semi-continuous and convex functions. The episum of almost convex functions is proved to be convex under additional assumptions and the Moreau envelope of an almost convex function is shown to be convex and continuously differentiable with the gradient given by the one related to its closure. Applications to almost convex optimization problems are provided, in particular, under suitable assumptions, the solution set is proved to be nearly convex and the solution sets of two almost convex optimization problems are shown to be nearly equal if the related objective functions are nearly equal. Another application shows that the classical Hoffman' error bound holds for almost convex inequalities under a generalized Slater condition. Several examples are given to illustrate these results.