This paper explores mechanical systems undergoing unilateral frictionless contact conditions in the framework of nonlinear modal analysis. The nonlinear eigenproblem is formulated in the frequency domain through the minimization of a Rayleigh quotient subject to non-penetration inequality constraints. An additional equality constraint is introduced for normalization purposes. The resulting constrained minimization problem is then solved using an augmented Lagrangian strategy. Two applications are proposed: a thin longitudinal rod in unilateral contact with a rigid obstacle and a turbomachinery compressor blade in contact with a rigid casing. The first application illustrates the complexity of the nonlinear modal characterization of a system experiencing unilateral contact conditions while the second demonstrates the applicability of the proposed approach to large-scale mechanical systems involving non-smooth nonlinear terms.