Conjugate duality for constrained optimization via image space analysis and abstract convexity (Q2414117)

Let \(X\), \(Z\) be two Banach spaces equipped, respectively, with order relations \(\succeq_{X}\) and \(\succeq_{Z}\) defined by pointed closed cones with nonempty interior. Given two functions \(f:X\rightarrow \mathbb{R}\) and \(g:X\rightarrow Z\), the paper considers the following constrained optimization problem: \[ \min \left \{ f(x):g(x)\succeq_{Z}0_{Z}\right \} . \] Using the methods of abstract convexity, a conjugate duality scheme is proposed. It is shown that in several instances, this scheme achieves zero duality gap, while the classical conjugate duality may give nonzero duality gap.