Lagrangian duality for nonconvex optimization problems with abstract convex functions

We investigate Lagrangian duality for nonconvex optimization problems. To this aim we use the $\mathrm{\Phi }$-convexity theory and minimax theorem for $\mathrm{\Phi }$-convex functions. We provide conditions for zero duality gap and strong duality. Among the classes of functions, to which our duality results can be applied, are prox-bounded functions, DC functions, weakly convex functions and paraconvex functions.