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Efficient Algorithms for Optimization and Fixed Points

In the first part of this dissertation we present algorithms for the fixed point problem over a class of Lipschitz continuous (constant 1) functions with respect to the infinity norm. The algorithms compute approximations satisfying the residual error criterion. The first algorithm handles bivariate functions and has what we conjecture to be optimal worst-case complexity. The second algorithm handles functions of all dimensions and has better than exponential worst-case complexity in terms of the dimension and error tolerance. For the absolute error criterion and Lipschitz constant greater than 1, the problem has infinite worst-case complexity when information consists of function evaluations. We include C implementations of the two algorithms in the Appendix. 
In the second part we report implementation of a massively-parallel linear-scaling algorithm for the ab initio tight-binding FIREBALL method. The use of local-orbitals yields a very sparse Hamiltonian matrix, which facilitates using a linear-scaling algorithm to obtain the electronic band-structure energy. The general functional form of Kim, Mauri, and Galli has been optimized utilizing the conjugate gradient method. The results of this approach are reported here. The linear-scaling optimization procedures are parallelized using the standard MPI message passing interface mixed with an OpenMP strategy.