Convexity and Closure in Optimal Allocations Determined by Decomposable Measures

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

“…Introduction. This paper discusses and analyzes some convex representatives of the value function \psi : Y \rightar \BbbR : (1) \psi (a) . = inf \bigl\{ \int T f 0 (t, z(t))d\mu : z \in \scrD \subset \scrL (T ; Z), \int T g 0 (t, z(t))d\mu \in a -\BbbC \bigr\} , which are convex functions lying between the function \psi and its closed hull.…”

“…In particular, the function \psi and its representatives have the same closed hull, and one derives the convexity of the closure of the value function. The data of the underlying optimization problem in (1) are the (possibly incomplete) normed spaces Y and Z, the closed convex set \BbbC \subset Y , the extended real-valued function f 0 : T \times Z \rightar \BbbR \infty , and the vector-valued function g 0 : X \rightar Y (see section 2 for details). The set \scrL (T ; Z) stands for either the space of Bochner or the space of Pettis integrable functions, and \scrD is a given decomposable subset of \scrL (T ; Z).…”

“…There are several issues which make standard approaches less useful to be employed directly to the optimization problem in (1). 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]).…”

“…The following example briefly explains the connection between a particular problem of type (1) and the convexity of the Aumann integral of an associated multifunction (we refer the reader to section 2 for the definitions and notation used).…”

“…1 , \cdot \cdot \cdot , T k ) of T such that the vector (\q, \z, \w, \p) \in L 1 (T ) \times \scrL (T ; Z) \times \scrS (T ; Y ) \times L 1 (T ) which in turn leads us to \psi \varepsi \scrL (a) \leq \varphi \varepsi (a) as \beta \downarr \varphi \varepsi (a). Therefore the functions \psi \varepsi \scrL and \varphi \varepsi coincide, and \psi \varepsi \scrL as well as the upper envelope sup \varepsi >0 \psi \varepsi \scrL are convex.…”