Strong and Total Conjugate Duality

In this chapter, we are interested in formulating regularity conditions of closednesstype, which do not necessarily guarantee stable strong duality, but are sufficient for having strong duality. First, we do this for the primal–dual pair (PG)–(DG) and after that we particularize the general result to the different classes of problems investigated in the previous chapters. Assume that X and Y are separated locally convex spaces, with X ∗ and Y ∗ their topological dual spaces, respectively, and $$\Phi : X \times Y \rightarrow \overline{\mathbb{R}}$$ is a proper and convex function such that $$0 \in {\rm Pr}Y ({\rm dom} \Phi)$$ . Throughout this chapter we assume that the dual spaces are endowed with the weak∗ topologies. As proved by Theorem 5.1, if Φ is lower semicontinuous, then for all $$x^{\ast} \in X^{\ast}$$ one has $$(\Phi(\cdot,0))^{\ast}(x^{\ast}) = {\rm cl}_{\omega\ast}\ ({\rm inf}_{y\ast\in Y\ast}\Phi^{\ast}(\cdot, y^{\ast}))(x^{\ast})$$ . Starting from this fact, one can formulate the following regularity condition for (PG) and its conjugate dual (DG $$(Rc_{5}^{\Phi} \left|\begin{array}{rcl}&&\Phi\ {\rm is\ lower\ semicontinuous\ and\ inf}_{y^{\ast} \in Y^{\ast}} \Phi^{\ast}(\cdot,y^{\ast})\ {\rm is\ lower}\\ &&{\rm semicontinuous\ and\ exact\ at\ 0}.\end{array}\right.$$ We say that $${\inf}_{y^{\ast}\in Y^{\ast}} \Phi^{\ast} (\cdot, y^{\ast})$$ is exact at $$\overline{x}^{\ast} \in X^{\ast}$$ if there exists $$\overline{y}^{\ast} \in Y^{\ast}$$ such that $${\rm inf}_{y^{\ast} \in Y^{\ast}} \Phi^{\ast}(\overline{x}^{\ast}, y^{\ast}) = \Phi^{\ast}(\overline{x}^{\ast}, \overline{y}^{\ast})$$ . We have the following general result.