singular value optimization problems arise in various applications in control theory. for instance the $h_{\infty}$ norm of the transfer function of a linear dynamical system, and the distance problems such as complex (or real) stability and controllability radii have singular value optimization characterizations. these problems are non-convex and non-smooth. the existing commonly employed algorithms for these problems are derivative-free, but do not exploit the lipschitz nature of singular values in a systematic manner. here we solve these problems largely depending on a lipschitz optimization algorithm due to piyavskii and shubert, that never got attention in the context of optimization of eigenvalues or singular values. the piyavskii-shubert based algorithm outperforms the commonly employed algorithms for medium to large scale problems when a few digit accuracy is sought.