  Studying independent sets of maximum size is equivalent to considering the
hard-core model with the fugacity parameter $\lambda$ tending to infinity.
Finding the independence ratio of random $d$-regular graphs for some fixed
degree $d$ has received much attention both in random graph theory and in
statistical physics.
  For $d \geq 20$ the problem is conjectured to exhibit 1-step replica symmetry
breaking (1-RSB). The corresponding 1-RSB formula for the independence ratio
was confirmed for (very) large $d$ in a breakthrough paper by Ding, Sly, and
Sun. Furthermore, the so-called interpolation method shows that this 1-RSB
formula is an upper bound for each $d \geq 3$. For $d \leq 19$ this bound is
not tight and full-RSB is expected.
  In this work we use numerical optimization to find good substituting
parameters for discrete $r$-RSB formulas ($r=2,3,4,5$) to obtain improved
rigorous upper bounds for the independence ratio for each degree $3 \leq d \leq
19$. As $r$ grows, these formulas get increasingly complicated and it becomes
challenging to compute their numerical values efficiently. Also, the functions
to minimize have a large number of local minima, making global optimization a
difficult task.
