  Assuming the best numerical value for the cosmic baryonic density and the
existence of three neutrino flavors, standard big bang nucleosynthesis is a
parameter-free model. It is important to assess if the observed primordial
abundances can be reproduced by simulations. Numerous studies have shown that
the simulations overpredict the primordial $^7$Li abundance by a factor of
$\approx$ $3$ compared to the observations. The discrepancy may be caused by
unknown systematics in $^7$Li observations, poorly understood depletion of
lithium in stars, errors in thermonuclear rates that take part in the lithium
and beryllium synthesis, or physics beyond the standard model. Here, we focus
on the likelihood of a nuclear physics solution. The status of the key nuclear
reaction rates is summarized. Big bang nucleosynthesis simulations are
performed with the most recent reaction rates and the uncertainties of the
predicted abundances are established using a Monte Carlo technique.
Correlations between abundances and reaction rates are investigated based on
the metric of mutual information. The rates of four reactions impact the
primordial $^7$Li abundance: $^3$He($\alpha$,$\gamma$)$^7$Be,
d(p,$\gamma$)$^3$He, $^7$Be(d,p)2$\alpha$, and $^7$Be(n,p)$^7$Li. We employ a
genetic algorithm to search for simultaneous rate changes in these four
reactions that may account for all observed primordial abundances. When the
search is performed for reaction rate ranges that are much wider than recently
reported uncertainties, no acceptable solutions are found. Based on the
currently available evidence, we conclude that it is highly unlikely for the
cosmological lithium problem to have a nuclear physics solution.
