We present an extended conjugate duality for a generalized semi-infinite programming problem (P). The extended duality is defined in the context of the absence of convexity of problem (P), by means of a decomposition into a family of convex subproblems and a conjugate dualization of the subproblems. Under appropriate assumptions, we establish strong extended duality and provide necessary and sufficient optimality conditions for problem (P). These extended conjugate duality and optimality conditions are new in the literature of generalized semi-infinite programming .