  The VISTA near-infrared $YJK_\mathrm{s}$ survey of the Magellanic Clouds
System (VMC, PI M.-R. L. Cioni) is collecting deep $K_\mathrm{s}$-band
time-series photometry of the pulsating variable stars hosted in the system
formed by the two Magellanic Clouds and the Bridge connecting them. In this
paper we present for the first time $K_\mathrm{s}$-band light curves for
Anomalous Cepheid (AC) variables. In particular, we have analysed a sample of
48 Large Magellanic Cloud ACs, for which identification and optical magnitudes
were obtained from the OGLE III and IV catalogues. The VMC $K_\mathrm{s}$-band
light curves for ACs are well sampled, with the number of epochs ranging from 8
to 16, and allowing us to obtain very precise mean $K_\mathrm{s}$ magnitudes
with errors on average of the order of 0.01 mag. The $\langle K_\mathrm{s}
\rangle$ values were used to build the first Period-Luminosity and
Period-Wesenheit relations in the near-infrared for fundamental-mode and first
overtone ACs. At the same time we exploited the optical ($V,I$) OGLE data to
build accurate Period-Luminosity, Period-Luminosity-Colour and Period-Wesenheit
relations both for fundamental-mode and first overtone ACs. For the first time
these relations were derived from a sample of pulsators which uniformly cover
the whole AC instability strip. The application of the optical Period-Wesenheit
relation to a sample of dwarf galaxies hosting a significant population of ACs
revealed that this relation is a valuable tool for deriving distances within
the Local Group. Due to its lower dispersion, we expect the $K_\mathrm{s}$
Period-Wesenheit relations first derived in this paper to represent a valuable
tool for measuring accurate distances to galaxies hosting ACs when more data in
near-infrared filters become available.
