Summary: The treatment of contact problems with friction by mathematical programming (MP) is considered. Since MP is concerned with finite dimensional problems, a formal finite element discretization of the continuous problem is performed. The finite dimensional model obtained is investigated with respect to two physical problems. Firstly, we consider the problem of determining the evolution of contact tractions and displacements for a prescribed loading history. This problem is shown to result in a new type of parametric linear complementarity problem (LCP), which can be solved numerically by a sequence of principle pivots. Secondly, we consider the problem of “shakedown”, i.e. the determination of the cyclic load histories for which cyclic slip may cease in a possible steady state. The numerical solution of this problem is obtained by linear programming (LP). Finally, numerical solutions are given for a mechanical model that represents a shrink-fitted gearwheel.