Inexact Restoration methods have been proved to be effective to solve constrained optimization problems in which some structure of the feasible set induces a natural way of recovering feasibility from arbitrary infeasible points. Sometimes natural ways of dealing with minimization over tangent approximations of the feasible set are also employed. A recent paper [N. Banihashemi and C. Y. Kaya, Inexact Restoration for Euler discretization of box-constrained optimal control problems, Journal of Optimization Theory and Applications 156, pp. 726--760, 2013] suggests that the Inexact Restoration approach can be competitive with well-established nonlinear programming solvers when applied to certain control problems without any problem-oriented procedure for restoring feasibility. This result motivated us to revisit the idea of designing general-purpose Inexact Restoration methods, especially for large-scale problems. In this paper we introduce an affordable algorithm of Inexact Restoration type for solving arbitrary nonlinear programming problems and we perform the first experiments that aim to assess its reliability.