Characterisation of zero duality gap for optimization problems in spaces without linear structure

We prove sufficient and necessary conditions ensuring zero duality gap for Lagrangian duality in some classes of nonconvex optimization problems. To this aim, we use the $\mathrm{\Phi }$-convexity theory and minimax theorems for $\mathrm{\Phi }$-convex functions. The obtained zero duality results apply to optimization problems involving prox-bounded functions, DC functions, weakly convex functions and paraconvex functions as well as infinite-dimensional linear optimization problems, including Kantorovich duality which plays an important role in determining Wasserstein distance.