Constraint qualifications for convex inequality systems with applications in constrained optimization

For an inequality system defined by an infinite family of proper convex functions, we introduce some new notions of constraint qualifications in terms of the epigraphs of the conjugates of these functions and study relationships between these new constraint qualifications and other well-known constraint qualifications including the basic constraint qualification studied by Hiriart-Urrutty and Lemarechal and by Li, Nahak, and Singer. Extensions of known results to more general settings are presented, and applications to particular important problems, such as conic programming and approximation theory, are also studied.