  We investigate the structure of admixed neutron stars with a regular hadronic
component and a fraction of fermionic self-interacting dark matter. Using two
limiting equations of state for the dense baryonic interior, constructed from
piecewise generalised polytropes, and an asymmetric self-interacting fermionic
dark component, we analyse different scenarios of admixed neutron stars
depending on the mass of dark fermions $m_\chi$, interaction mediators
$m_\phi$, and self-interacting strengths $g$. We find that the contribution of
dark matter to the masses and radii of neutron stars leads to tension with mass
estimates of the pulsar J0453+1559, the least massive neutron star, and with
the constraints coming from the GW170817 event. We discuss the possibilities of
constraining dark matter model parameters $g$ and $y \equiv m_\chi/m_\phi$,
using current existing knowledge on neutron star estimations of mass, radius,
and tidal deformability, along with the accepted cosmological dark matter
freeze-out values and self-interaction cross-section to mass ratio,
$\sigma_\mathrm{SI}/m_\chi$, fitted to explain Bullet, Abell, and dwarf galaxy
cluster dynamics. By assuming the most restrictive upper limit,
$\sigma_\mathrm{SI}/m_\chi < 0.1$ cm$^2$/g, along with dark matter freeze-out
range values, the allowed $g$-$y$ region is $0.01 \lesssim g \lesssim 0.1$,
with $0.5 \lesssim y \lesssim 200$. For the first time, the combination of
updated complementary restrictions is used to set constraints on
self-interacting dark matter.
