The paper focuses on the problem of normal surface waves in an open metal-dielectric inhomogeneous waveguide of arbitrary cross-section with losses. The medium filling the waveguide is characterized by a isotropic inhomogeneous complex permittivity. The setting is reduced to a boundary eigenvalue problem for longitudinal components of the electromagnetic field in Sobolev spaces. Variational formulation of the problem is considered in terms of the analysis of operator-functions. The discreteness of the set of the sought-for eigenvalues is proved.