Additively separable duality theory

Abstract

In duality theory, there is a trade-off between generality and tractability. Thus, the generality of the Tind-Wolsey framework comes at the expense of an infinite-dimensional dual solution space, even if the primal solution space is finite dimensional. Therefore, the challenge is to impose additional structure on the dual solution space and to identify conditions on the primal program, such that the properties that are typically associated with duality, like weak and strong duality, are preserved.

In this paper, we consider real-valuedness, continuity, and additive separability as such additional structures. The virtue of the latter property is that it restores the one-to-one correspondence between primal constraints and dual variables as it exists in Lagrangian duality. The main result of this paper is that, roughly speaking, the existence of realvalued, continuous, and additively separable dual solutions that preserve strong duality is guaranteed, once the primal program satisfies a certain stability condition. The latter condition is ensured by the well-known regularity conditions that imply constraint qualification in Karush-Kuhn-Tucker points. On the other hand, if instead of additive separability, a mild tractability condition is imposed on the dual solution space, then stability turns out to be a necessary condition for strong duality in a well-defined sense. This result, combined with the observation that applicability of some well-known augmented Lagrangian methods to constrained optimization.