This presentation starts with a brief introduction on topology optimization as a mathematical tool for optimal design of mechanical components. Although now routinely used in the industry, software for topology optimization suffers from limitations, in particular when used for complex three-dimensional structures. Several ways will be presented on how to substantially improve efficiency of topology optimization software using modern tools of numerical linear algebra and numerical optimization. These are based on domain decomposition and multigrid techniques and, for the more involved problems, on decomposition of large-scale matrix inequalities using recent results of graph theory.