Maximise the function

$f(x_1,x_2)=837\mathrm{.}96-x_{1}sin\sqrt[\phantom{2}]{x_1}-x_{2}sin\sqrt[\phantom{2}]{x_2}$

by performing $15$ simulated annealing iterations, starting out from the initial estimate $x^0={\left[\begin{array}{cc}100& 400\end{array}\right]}^T.$ Discretise the search space according to a square lattice structure in which each cell has sides length $lon=20.$ Work accurately to three decimal places and adopt the stochastic neighbourhood perturbation operator described in Figure $1.$ Adopt both a geometric reheating schedule and a geometric cooling schedule with heating factor $B^{\text{'}}=1\mathrm{.}15$ and cooling factor $B=0\mathrm{.}85.$ Take the initial temperature as $T^0=1,$ and terminate a search epoch either if $a_{\text{m}\text{a}\text{x}}=3$ solutions have been accepted $($in which case cooling takes place$)$ of if $r_{\text{m}\text{a}\text{x}}=2$ solutions have been rejected $($in which case reheating occurs$).$ Produce a table of algorithmic output in the tabular format shown below. Use as many numbers from the following list as are necessary to resolve stochastic decisions: $,$

$0\mathrm{.}68,0\mathrm{.}95,0\mathrm{.}38,0\mathrm{.}11,0\mathrm{.}54,0\mathrm{.}03,0\mathrm{.}84,0\mathrm{.}33,0\mathrm{.}25,0\mathrm{.}68,0\mathrm{.}51,0\mathrm{.}83,0\mathrm{.}47.$