  Using tools provided by the theory of abstract convexity, we extend
conditions for zero duality gap to the context of nonconvex and nonsmooth
optimization. Mimicking the classical setting, an abstract convex function is
the upper envelope of a family of abstract affine functions (being conventional
vertical translations of the abstract linear functions). We establish new
conditions for zero duality gap under no topological assumptions on the space
of abstract linear functions. In particular, we prove that the zero duality gap
property can be fully characterized in terms of an inclusion involving
(abstract) $\varepsilon-$subdifferentials. This result is new even for the
classical convex setting. Endowing the space of abstract linear functions with
the topology of pointwise convergence, we extend several fundamental facts of
functional/convex analysis. This includes (i) the classical
Banach--Alaoglu--Bourbaki theorem (ii) the subdifferential sum rule, and (iii)
a constraint qualification for zero duality gap which extends a fact
established by Borwein, Burachik and Yao (2014) for the conventional convex
case. As an application, we show with a specific example how our results can be
exploited to show zero duality for a family of nonconvex, non-differentiable
problems.
