Extensions and Applications of Fast-Lipschitz Optimization

Abstract: Fast-Lipschitz optimization is a framework for optimization problems having a special structure in which the optimal solution is given by a set of equations, easily solvable through fixed point iterations. This solution method is simple to implement and particularly well suited for distributed optimization applications, such as those arising in wireless sensor networks.This thesis provides several theoretical contributions to the Fast-Lipschitz framework. In particular, it further develops the qualifying conditions under which a problem is guaranteed to be Fast-Lipschitz.Known qualifying conditions are unified and extended by a new set of conditions. This is done through a newapproach to the analysis of existing conditions, based on the Karush--Kuhn--Tucker (KKT) conditions.The Fast-Lipschitz framework is further extended by examining problem structures that are not treatable by the existing framework, such as problems with more or fewer constraints than variables. Finally, this thesis provides a comparison between the convergence properties of Fast-Lipschitz optimization and those of a traditional method based on gradient descent.The thesis then applies the theory of Fast-Lipschitz optimization to the area of radio power control over wireless networks.Conditions are given under which well known distributed power control algorithms,such as those involving standard types of interference functions, fall in the Fast-Lipschitz framework.This gives a direct connection between these algorithms and a family of optimization problems, and the fixed points thereby assume a meaning of optimality.Finally, the thesis gives illustrative examples of the new theory and examples of applications such as in a general non-convex optimal control problem and a non-monotonic power control problem.