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

We prove sufficient and necessary conditions ensuring zero duality gap forLagrangian duality in some classes of nonconvex optimization problems. To thisaim, we use the $\Phi$-convexity theory and minimax theorems for $\Phi$-convexfunctions. The obtained zero duality results apply to optimization problemsinvolving prox-bounded functions, DC functions, weakly convex functions andparaconvex functions as well as infinite-dimensional linear optimizationproblems, including Kantorovich duality which plays an important role indetermining Wasserstein distance.