  LSP(n), the largest small polygon with n vertices, is the polygon of unit
diameter that has maximal area A(n). It is known that for all odd values $n
\geq 3$, LSP(n) is the regular n-polygon; however, this statement is not valid
for even values of n. Finding the polygon LSP(n) and A(n) for even values $n
\geq 6$ has been a long-standing challenge. In this work, we develop
high-precision numerical solution estimates of A(n) for even values $n \geq 4$,
using the Mathematica model development environment and the IPOPT local
nonlinear optimization solver engine. First, we present a revised (tightened)
LSP model that greatly assists the efficient solution of the model-class
considered. This is followed by numerical results for an illustrative sequence
of even values of n, up to $n \leq 1000$. Our results are in close agreement
with, or surpass, the best results reported in all earlier studies. Most of
these earlier works addressed special cases up to $n \leq 20$, while others
obtained numerical optimization results for a range of values from $6 \leq n
\leq 100$. For completeness, we also calculate numerically optimized results
for a selection of odd values of n, up to $n \leq 999$: these results can be
compared to the corresponding theoretical (exact) values. The results obtained
are used to provide regression model-based estimates of the optimal area
sequence {A(n)}, for all even and odd values n of interest, thereby essentially
solving the entire LSP model-class numerically, with demonstrably high
precision.
