  Warm dark matter (WDM) can potentially explain small-scale observations that
currently challenge the cold dark matter (CDM) model, as warm particles
suppress structure formation due to free-streaming effects. Observing
small-scale matter distribution provides a valuable way to distinguish between
CDM and WDM. In this work, we use observations from the Dark Energy Survey and
PanSTARRS1, which observe 270 Milky-Way satellites after completeness
corrections. We test WDM models by comparing the number of satellites in the
Milky Way with predictions derived from the Semi-Analytical SubHalo Inference
ModelIng (SASHIMI) code, which we develop based on the extended Press-Schechter
formalism and subhalos' tidal evolution prescription. We robustly rule out WDM
with masses lighter than 4.4 keV at 95% confidence level for the Milky-Way halo
mass of $10^{12} M_\odot$. The limits are a weak function of the (yet
uncertain) Milky-Way halo mass, and vary as $m_{\rm WDM}>3.6$-$5.1$ keV for
$(0.6$-$2.0) \times 10^{12} M_\odot$. For the sterile neutrinos that form a
subclass of WDM, we obtain the constraints of $m_{\nu_s}>11.6$ keV for the
Milky-Way halo mass of $10^{12} M_{\odot}$. These results based on SASHIMI do
not rely on any assumptions of galaxy formation physics or are not limited by
numerical resolution. The models, therefore, offer a robust and fast way to
constrain the WDM models. By applying a satellite forming condition, however,
we can rule out the WDM mass lighter than 9.0 keV for the Milky-Way halo mass
of $10^{12} M_\odot$.
