We consider the generalized minimax programming problem (P) in which functions are locally Lipschitz (G,β)-invex. Not onlyG-sufficient but alsoG-necessary optimality conditions are established for problem (P). WithG-necessary optimality conditions and (G,β)-invexity on hand, we construct dual problem (DI) for the primal one (P) and prove duality results between problems (P) and (DI). These results extend several known results to a wider class of programs.