Characterizing Existence of Minimizers and Optimality to Nonconvex Quadratic Integrals

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

“…We have the following result. The quadratic case in the Euclidean space has been studied in [9]. Proof.…”

“…On the one hand, the classical existence result, due to Tonelli, requires some convexity and superlinear growth assumptions on the function f 0 (t, \cdot ), which imply the weak lower semicontinuity of the integral functional \int T f 0 (t, \cdot )d\mu and the weak compactness of its sublevel sets restricted to the feasible set (see, for instance, Theorem 16.2 in [6]). Even more, there are situations (when Z = \BbbR n ) in which the feasible set is bounded but not compact with respect to any (locally convex) topology in L 1 ; see [9] for details. To be more precise, the case when f 0 (t, \cdot ) is a quadratic function on a finite-dimensional space is analyzed in [9] by establishing a complete description about the (non)existence of solutions.…”

“…Even more, there are situations (when Z = \BbbR n ) in which the feasible set is bounded but not compact with respect to any (locally convex) topology in L 1 ; see [9] for details. To be more precise, the case when f 0 (t, \cdot ) is a quadratic function on a finite-dimensional space is analyzed in [9] by establishing a complete description about the (non)existence of solutions. This is carried out via arguments based on the existence of measurable selections.…”

“…In this section, Y is a finitedimensional Banach space, Z is a normed real space, \BbbC \subsete Y is a nonempty closed convex set, and (T, \scrE , \mu ) is a measure space with a complete, nonnegative, \sigma -finite, and nonatomic measure \mu . In this setting, Pettis and Bochner integrals of Y -valued measurable functions coincide, and the value function ( 1) is written as (9) \psi \scrL (a) . = inf \bigl\{ \int T f 0 (t, z(t))d\mu : z \in \scrD \subset \scrL (T ; Z),…”