<p>We propose a numerical algorithm for solving axisymmetric problems of the theory of elasticity on the indentation of a punch into a half space under the conditions of wear caused by the action of friction forces. The contact problem is formulated in the form of the Signorini integral equations/inequalities, which are transformed into a system of linear algebraic equations/inequalities. The solution of this system is reduced to finding the minimum of a quadratic form under linear constraints imposed on unknown contact stresses. The wear process is studied by the method of steps along the time coordinate in each of which the contact zone and the stress distribution in this zone are determined numerically. By using the developed algorithm, we also consider an axisymmetric problem of interaction of an elastic half space with a punch containing local inhomogeneities.</p>