In this paper new results are established in multiobjective DC programming with infinite convex constraints ($MOPIC$ for abbr.) that are defined on Banach space (finite or infinite) with objectives given as the difference of convex functions subject to infinite convex constraints. This problem can also be called multiobjective DC semi-infinite and infinite programming, where decision variables run over finite-dimensional and infinite-dimensional spaces, respectively. This problem has not been studied yet at present. Necessary and sufficient optimality conditions for weak Pareto-optimality are introduced. Also, we seek a connection between multiobjective linear infinite programming and $MOPIC$. The Wolfe and Mond-Weir dual problems are presented and the weak, strong and strict converse duality theorems are also derived. The results obtained are new in both semi-infinite and infinite frameworks.

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