  The union-closed sets conjecture states that in any nonempty union-closed
family $\mathcal{F}$ of subsets of a finite set, there exists an element
contained in at least a proportion $1/2$ of the sets of $\mathcal{F}$. Using
the information-theoretic method, Gilmer \cite{gilmer2022constant} recently
showed that there exists an element contained in at least a proportion $0.01$
of the sets of such $\mathcal{F}$. He conjectured that his technique can be
pushed to the constant $\frac{3-\sqrt{5}}{2}$ which was subsequently confirmed
by several researchers
\cite{sawin2022improved,chase2022approximate,alweiss2022improved,pebody2022extension}.
Furthermore, Sawin \cite{sawin2022improved} showed that Gilmer's technique can
be improved to obtain a bound better than $\frac{3-\sqrt{5}}{2}$, but this new
bound is not explicitly given by Sawin. This paper further improves Gilmer's
technique to derive new bounds in the optimization form for the union-closed
sets conjecture. These bounds include Sawin's improvement as a special case. By
providing cardinality bounds on auxiliary random variables, we make Sawin's
improvement computable, and then evaluate it numerically which yields a bound
around $0.38234$, slightly better than $\frac{3-\sqrt{5}}{2}\approx0.38197$. }
