Ιn two Vectors V1(x11, x12) και V2(x21,x22) we can compute their inner product as V1 • V2.= (x11* x21 + x12 * x22 ).
I try to compute minimum inner product as (x1ix2j|i-j|, i.j the places of coordinates at V1, V2. Every cooedinate is used once in a sum condition.
I TRIED THIS:
int : vlen;
set of int : LEN = 1..vlen;
set of int : VECS = 1..2;
array[VECS,LEN] of -25..25 : vector;
var -600..700 : sumTotal;
constraint exists(i,j,k,l in LEN where i!=k \/ j!=l)(
exists(v,v2 in VECS)(sumTotal=(vector[v,i] * vector[v2,j] * abs(i-j)+vector[v,k] * vector[v2,l] * abs(k-l)
)));
solve minimize sumTotal;
output ["vector1=["]++[" \(vector[1,j])"|j in LEN]++[" ];\nvector2=["]++[" \(vector[2,j])"|j in LEN]++[" ];\nsumTotal=\(sumTotal);"]
for
vlen = 2;
vector = [|-2,3|-4,5|];
i expect:
vector1 = [-2, 3];
vector2 = [-4, 5];
sumTotal = -22;
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but i take:
vector1=[ -2 3 ];
vector2=[ -4 5 ];
sumTotal=-40;
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I'm afraid I don't understand the meaning of your model, but it does contain some errors in the constraint that should be easy to fix:
VEC, LEN, then the second index should always be part of that set.sum is it's own looping structure; it doesn't need an forall expression.The resulting constraint would be:
constraint sumTotal = sum(i,j in LEN)(
vector[1,i] * vector[2,j] * abs(i,j)
);
This still leaves a rather strange model, so you might want to take a look at the following:
sumTotal is your only variable, but it's defined by parameters. It cannot be optimised as it only has 1 solution.i and j be able to take the same value? If not, then you should use i,j in LEN where i < j.sumTotal?