Soft inequality constraints in gradient method and fast gradient method for quadratic programming

A quadratic program (QP) with soft inequality constraints with both linear and quadratic costs on constraint violation can be solved with the dual gradient method (GM) or the dual fast gradient method (FGM). The treatment of the constraint violation influences the efficiency and usefulness of the algorithm. We improve on the classical way of extending the QP: our novel contribution is that we obtain the solution to the soft-constrained QP without explicitly introducing slack variables. This approach is more efficient than solving the extended QP with GM or FGM and results in a similar algorithm than if the soft constraints were replaced with hard ones. The approach is intended for applications in model predictive control with fast system dynamics, where QPs of this type are solved at every sampling time in the millisecond range.