Convex Representatives of the Value Function and Aumann Integrals in Normed Spaces

Abstract

Convex representatives are proposed for the value function of an infinite-dimensional constrained nonconvex variational problem. All the involved variables in this problem take their values in (possibly of infinite dimension, not necessarily separable or complete) normed spaces, while the associated measure can be any $\sigma$-finite, nonnegative, and nonatomic complete measure. This in particular shows that the closure hull of the (possibly nonconvex) value function is always convex, as long as the sense of the integral within the cone-valued functional constraint is given and the type of the closure is appropriately determined. Correspondingly, similar convexity properties for the Aumann integral in general normed spaces of infinite dimension are established. Applications are given in a fairly general positively homogeneous framework.