Normal cones to sublevel sets of suprema of convex and quasiconvex functions

We provide sharp and explicit characterizations of the normal cone to sublevel sets of suprema of arbitrary functions, expressed exclusively in terms of subdifferentials of the data functions. In the convex case, the resulting formulas involve the approximate and exact subdifferentials of the individual data functions at the nominal and nearby points. In contrast, the quasiconvex framework requires the use of the Fréchet subdifferential of these data functions but evaluated at nearby points. These results are applied to derive optimality conditions for infinite convex and quasiconvex optimization problems.