A variable-frequency light ray (VF ray) is a new concept in optics; it is a ray that is monochromatic at every point of the path (no frequency spread) but its frequency changes along the path due to interactions occurring along that path. For example, the frequency of the light ray can be affected by gravitational time dilation, reflection from a moving mirror, or coherent Raman scattering. Since the Fermat principle of least time (PLT) implicitly assumes that the frequency of the light ray is an irrelevant constant, the PLT-based geometrical optics is insufficient to handle propagation of VF rays. In this paper, we derive a new extremum principle (NEP) that is sensitive to changes of frequency along the path of the light ray and constitutes a basis for geometrical optics of VF rays. The NEP is derived directly from the principle of least action applied to the quantized electromagnetic field associated with the light ray propagating in a vacuum or in a transparent material medium in the presence of the gravitational field. The relationship between the NEP, the classical version of the Fermat principle, and the general relativistic generalization of the Fermat principle (the extremum principle for null geodesics) is discussed in detail. Applications of the NEP are suggested in the nonrelativistic, special relativistic, and general relativistic regimes. PACS Nos.: 42.15-i, 12.20-m, 04.20-q