A new insight into Lagrange duality on DC optimization
M. D. Fajardo, J. Vidal-Nunez
TL;DR
This paper advances duality theory for DC optimization under the additivity condition by introducing a novel Lagrange dual based on $c$-conjugation and infimal convolution. It first relaxes previous regularity requirements, showing that a generalized ECC-type condition (ECC and its variants) suffices to guarantee strong duality between the DC primal and the new dual, while also analyzing weak duality. A key contribution is the equivalence between ECC, a conjugate-equality, and an epsilon-$c$-subdifferential characterization, which together yield a practical criterion for strong duality and attainment. The results extend the duality toolkit beyond strictly $e$-convex constraints, connecting evenly convex concepts to traditional Lagrange duality in DC settings and providing a framework for future exploration of Fenchel-Lagrange duality links.
Abstract
In this paper we present a new Lagrange dual problem associated to a primal DC optimization problem under the additivity condition (AC). As usual for DC programming, even weak duality is not guaranteed for free and, due to this issue, we investigate conditions not only for weak, but also for strong duality between this dual and the primal DC problem. In addition, we also analyze conditions for strong duality between the primal and its standard Lagrange dual problem utilizing a revisited regularity condition which is guaranteed by a more general class of functions than the one used in a prior work (see [14]).