$Q$-subdifferential and $Q$-conjugate for global optimality

Normal cone and subdifferential have been generalized through various continuous functions; in this article, we focus on a non separable $Q$-subdifferential version. Necessary and sufficient optimality conditions for unconstrained nonconvex problems are revisited accordingly. For inequality constrained problems, $Q$-subdifferential and the lagrangian multipliers, enhanced as continuous functions instead of scalars, allow us to derive new necessary and sufficient optimality conditions. In the same way, the Legendre–Fenchel conjugate is generalized into $Q$-conjugate and global optimality conditions are derived by $Q$-conjugate as well, leading to a tighter inequality.