  The need of fast distributed solvers for optimization problems in networked
systems has motivated the recent development of the Fast-Lipschitz optimization
framework. In such an optimization, problems satisfying certain qualifying
conditions, such as monotonicity of the objective function and contractivity of
the constraints, have a unique optimal solution obtained via fast distributed
algorithms that compute the fixed point of the constraints. This paper extends
the set of problems for which the Fast-Lipschitz framework applies. Existing
assumptions on the problem form are relaxed and new and generalized qualifying
conditions are established by novel results based on Lagrangian duality. It is
shown for which cases of more constraints than decision variables, and less
constraints than decision variables Fast-Lipschitz optimization applies. New
results are obtained by imposing non strict monotonicity of the objective
functions. The extended Fast-Lipschitz framework is illustrated by a number of
examples, including a non-linear optimal control problem.
