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.
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