Fast-Lipschitz optimization has been recently proposed as a new framework with numerous computational advantages for both centralized and decentralized convex and non-convex optimization problems. Such a framework generalizes the interference function optimization, which plays an essential role distributed radio power optimization over wireless networks. The characteristics of Fast-Lipschitz methods are low computational and coordination complexity compared to Lagrangian methods, with substantial benefits particularly for distributed optimization. These special properties of Fast-Lipschitz optimization can be ensured through qualifying conditions, which allow the Lagrange multipliers to be bound away from zero. In this paper, the Fast-Lipschitz optimization is substantially extended by establishing new qualifying conditions. The results are a generalization of the old qualifying conditions and a relaxation of the assumptions on problem structure so that the optimization framework can be applied to many more problems than previously possible. The new results are illustrated by a non-convex optimization problem, and by a radio power optimization problem which cannot be handled by the existing Fast-Lipschitz theory.