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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.

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