This paper deals with particular families of DC optimization problems involving suprema of convex functions. We show that the specific structure of this type of function allows us to cover a variety of problems in nonconvex programming. Necessary and sufficient optimality conditions for these families of DC optimization problems are established, where some of these structural features are conveniently exploited. More precisely, we derive necessary and sufficient conditions for (global and

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... optimality in DC semi-infinite programming and DC cone-constrained optimization, under natural constraint qualifications. Finally, a penalty approach to DC abstract programming problems is developed in the last section.