  By applying the perturbation function approach, we propose the Lagrangian and
the conjugate duals for minimization problems of the sum of two, generally
nonconvex, functions. The main tools are the $\Phi$-convexity theory and
minimax theorems for $\Phi$-convex functions. We provide conditions ensuring
zero duality gap and introduce $\Phi$-Karush-Kuhn-Tucker conditions that
characterize solutions to primal and dual problems. We also discuss the
relationship between the dual problems introduced in the present investigation
and some conjugate-type duals existing in the literature.
