On Some Extensions and Performance of Fast-Lipschitz Optimization

Abstract: A huge range of problems in applied sciences such as engineering and economics can be formulated as mathematical optimization prob- lems. In general, these must be solved by iterative methods whose con- vergence properties to a large extent determines what is achievable. In decentralized applications such as wireless sensor networks, informa- tion exchange can be expensive and optimization presents additional challenges. A theory for efficient network and distributed optimiza- tion is still in its infancy. Fast-Lipschitz optimization is a recently proposed class of optimization problems, where unique solutions fully determined by a system of equations allow for effective algorithms to solve the problem. This master thesis further investigates F-Lipschitz theory and its possible benefits. In particular, goals were to extend the class, and to compare convergence of the algorithms for the com- putation of the optimal solution to traditional Lagrangian methods. By carefully proving the main properties of the theory, new results are achieved both through relaxation of existing qualifying conditions and introduction of a new one. New forms, different from the original problem, are also studied. They are shown to belong in the class under mild assumptions. The convergence of a F-Lipschitz method is com- pared to a gradient method, for problems under the assumption of real eigenvalues. Novel conditions for guaranteed faster performance of the F-Lipschitz method are established. Several possibilities of expanding the theory still remain, and some suggestions are given throughout the thesis.