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By applying the perturbation function approach, we propose the Lagrangian andthe conjugate duals for minimization problems of the sum of two, generallynonconvex, functions. The main tools are the $\Phi$-convexity theory andminimax theorems for $\Phi$-convex functions. We provide conditions ensuringzero duality gap and introduce $\Phi$-Karush-Kuhn-Tucker conditions thatcharacterize solutions to primal and dual problems. We also discuss therelationship between the dual problems introduced in the present investigationand some conjugate-type duals existing in the literature.

On duality for nonconvex minimization problems within the framework of abstract convexity