Abstract A bundle method for minimizing the difference of convex (DC) and possibly nonsmooth functions is developed. The method may be viewed as an inexact version of the DC algorithm, where each subproblem is solved only approximately by a bundle method. We always terminate the bundle method after the first serious step. This yields a descent direction for the original objective function, and it is shown that a stepsize of at least one is accepted in this way. Using a line search, even larger stepsizes are possible. The overall method is shown to be globally convergent to critical points of DC programs. The new algorithm is tested and compared to some other solution methods on several examples and realistic applications.

Complementarity solvers are continually being challenged by modelers demanding improved reliability and scalability. Building upon a strong theoretical background, the semismooth algorithm has the potential to meet both of these requirements. We discuss relevant theory associated with the algorithm and then describe a sophisticated implementation in detail. Particular emphasis is given to the use of preconditioned iterative methods to solve the (nonsymmetric) systems of linear equations generated at each iteration and robust methods for dealing with singularity. Results on the MCPLIB test suite indicate that the code is reliable and efficient and scales well to very large problems.