  The peaceable queens problem asks to determine the maximum number $a(n)$ such
that there is a placement of $a(n)$ white queens and $a(n)$ black queens on an
$n \times n$ chessboard so that no queen can capture any queen of the opposite
color. In this paper, we consider the peaceable queens problem and its variant
on the toroidal board. For the regular board, we show that $a(n) \leq
0.1716n^2$, for all sufficiently large $n$. This improves on the bound $a(n)
\leq 0.25n^2$ of van Bommel and MacEachern.
  For the toroidal board, we provide new upper and lower bounds. Somewhat
surprisingly, our bounds show that there is a sharp contrast in behaviour
between the odd torus and the even torus. Our lower bounds are given by
explicit constructions. For the upper bounds, we formulate the problem as a
non-linear optimization problem with at most $100$ variables, regardless of the
size of the board. We solve our non-linear program exactly using modern
optimization software.
  We also provide a local search algorithm and a software implementation which
converges very rapidly to solutions which appear optimal. Our algorithm is
sufficiently robust that it works on both the regular and toroidal boards. For
example, for the regular board, the algorithm quickly finds the so-called
Ainley construction. Thus, our work provides some further evidence that the
Ainley construction is indeed optimal.
