Moreau–Rockafellar Formulae and Closedness-Type Regularity Conditions

Throughout this chapter, we assume that all topological dual spaces of the separated locally convex spaces considered are endowed with the corresponding weak* topologies. In this section, we give generalized Moreau–Rockafellar formulae expressed via the perturbation function Φ considered in Section 1 as well as closedness-type regularity conditions for the general optimization problem (PG). These will be particularized in the following sections to the different classes of convex functions and corresponding convex optimization problems, respectively, introduced in the previous chapter (see also [27]). Let X and Y be separated locally convex spaces and X ∗ and Y ∗ be their topological dual spaces, respectively. Given a function $$\Phi : X \times Y \rightarrow \overline{\mathbb{R}}$$ , we have that Φ∗ is convex and, consequently, the infimal value function of Φ∗, $${\rm inf}_{{y^{\ast}\in Y^{\ast}}} \phi^{\ast} (., y^{\ast}) : X^{\ast} \rightarrow \overline{\mathbb{R}}$$ is also convex. For a subset of X ∗ and a function defined on X ∗ the closure and the lower semicontinuous hull, respectively, in the weak∗ topology are denoted by $${\rm cl}_{\omega^{\ast}}$$ , while $${\rm cl}_{\omega^{\ast} \times \mathcal{R}}$$ denotes the closure of a subset of $$(X^{\ast}, \omega(X^{\ast}, X)) \times \mathbb{R}$$ . Here, we denote by $$\mathcal{R}$$ the natural topology on $$\mathbb{R}$$ . The following theorem can be obtained from [110] and plays a determinant role in the investigations we make in this chapter.