Before revision:  In order to compute market equilibria, we introduce (infinite-dimensional) convex programs over Banach spaces, thereby generalizing the Eisenberg-Gale convex program and its dual . Regarding the new convex programs, we establish existence of optimal solutions, the existence of KKT-type conditions , as well as strong duality .
After revision:  In order to compute market equilibria, we introduce (infinite-dimensional) convex programs over Banach spaces, thereby generalizing the Eisenberg-Gale convex program and its dual . Regarding the new convex programs, we establish existence of optimal solutions, the existence of KKT-type conditions , as well as necessity and sufficiency of KKT-type conditions .