We present an estimate of the optical luminosity function (OLF) of LOFAR radio-selected quasars (RSQs) at $1.4<z<5.0$ in the $9.3:textrm{deg}^{2}$ NOAO Deep Wide-field survey (NDWFS) of the Bootes field. The selection was based on optical/mid-ir photometry used to train three different machine learning (ML) algorithms. Objects taken as quasars by the ML algorithms are required to be detected at $5sigma$ significance in deep radio maps to be classified as candidate quasars. The optical imaging came from the SDSS and the PS1 $3pi$ survey; mid-ir photometry was taken from the SDWFS survey; and radio data was obtained from deep LOFAR imaging of the NDWFS-Bootes field. The requirement of a $5sigma$ LOFAR detection allowed us to reduce the stellar contamination in our sample by two orders of magnitude. The sample comprises 134 objects, including both photometrically selected candidate quasars (47) and spectroscopically confirmed quasars (83). The depth of our LOFAR observations allowed us to detect the radio-emission of quasars that would be otherwise classified as radio-quiet. Around $65%$ of the quasars in the sample are fainter than $M_{textrm{1450}}=-24.0$, a regime where the OLF of quasars selected through their radio emission, has not been investigated in detail. It has been demonstrated that in cases where mid-ir wedge-based AGN selection is not possible due to a lack of appropriate data, the selection of quasars using ML algorithms trained with optical/mid-ir photometry in combination with LOFAR data provides an excellent approach for obtaining samples of quasars. We demonstrate that RSQs show an evolution similar to that exhibited by faint quasars $(M_{textrm{1450}}leq-22.0)$. Finally, we find that RSQs may compose up to $sim20%$ of the whole faint quasar population (radio-detected plus radio-undetected).
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