Several optimal algorithms have been proposed for the complementation of nondeterministic Büchi word automata. Due to the intricacy of the problem and the exponential blow-up that complementation involves, these algorithms have never been used in practice, even though an effective complementation construction would be of significant practical value. Recently, Kupferman and Vardi described a complementation algorithm that goes through weak alternating automata and that seems simpler than previous algorithms. We combine their algorithm with known and new minimization techniques. Our approach is based on optimizations of both the intermediate weak alternating automaton and the final nondeterministic automaton, and involves techniques of rank and height reductions, as well as direct and fair simulation.