FindMinimum challenge

The problem is the following:Given a symmetric square matrix
a
ij
(1≤i,j≤n) with integer entries and zeros along the main diagonal find
x
kl
(1≤k≤n, 1≤l≤d) such that the following expression is minimized​​
n
∑
i,j=1
(
2

d
∑
l=1
2
(
x
il
-
x
jl
)
-
a
ij

An example: n=16, d=2
mat=
0
1
4
3
1
2
3
2
2
3
4
3
1
2
3
2
1
0
3
4
2
1
2
3
3
2
3
4
2
1
2
3
4
3
0
1
3
2
1
2
4
3
2
3
3
2
1
2
3
4
1
0
2
3
2
1
3
4
3
2
2
3
2
1
1
2
3
2
0
3
4
1
1
2
3
2
2
3
4
3
2
1
2
3
3
0
1
4
2
1
2
3
3
2
3
4
3
2
1
2
4
1
0
3
3
2
1
2
4
3
2
3
2
3
2
1
1
4
3
0
2
3
2
1
3
4
3
2
2
3
4
3
1
2
3
2
0
1
4
3
1
2
3
2
3
2
3
4
2
1
2
3
1
0
3
4
2
1
2
3
4
3
2
3
3
2
1
2
4
3
0
1
3
2
1
2
3
4
3
2
2
3
2
1
3
4
1
0
2
3
2
1
1
2
3
2
2
3
4
3
1
2
3
2
0
3
4
1
2
1
2
3
3
2
3
4
2
1
2
3
3
0
1
4
3
2
1
2
4
3
2
3
3
2
1
2
4
1
0
3
2
3
2
1
3
4
3
2
2
3
2
1
1
4
3
0
;
mat==Transpose[mat]
True
TrialMatrix[n_,d_]:=Table[Sum[
2
(x[i,k]-x[j,k])
,{k,d}],{i,n},{j,n}]
toMinimize=Apply[Plus,Flatten[TrialMatrix[16,2]-mat^2]^2]/.​​ {x[1,1]->0,x[1,2]->0};
vars=Union[Cases[toMinimize,x[_,_],∞]];
FindMinimumEvaluate[toMinimize],​​ Evaluate[Sequence@@({#,Random[]}&/@vars)][[1]]//Timing
{5.22Second,1811.02}
The following crashes the kernel from June 19 (as it
crashed on May 19 when I sent this already):
FindMinimumEvaluate[toMinimize],​​ Evaluate[Sequence@@({#,Random[]}&/@vars)],​​Method->Newton[[1]]//Timing
The following too crashes the kernel from June 19 (like
it did on May 19):
FindMinimumEvaluate[toMinimize],​​ Evaluate[Sequence@@({#,Random[]}&/@vars)],​​Method->QuasiNewton[[1]]//Timing
FindMinimumEvaluate[toMinimize],​​ Evaluate[Sequence@@({#,Random[]}&/@vars)],​​Method->LevenbergMarquardt[[1]]//Timing
{4.84Second,1811.02}