As

L(a,b,c,d,u1,u2,u3,u4,u5,u6,u7,u8) = (a-b)(a-c)(a-d)(b-c)(b-d)(c-d) - u1(1-a) - u2(1-b) - u3(1-c) - u4(1-d) - u5(1+a) - u6(1+b) - u7(1+c) - u8(1+d)

and then try to identify which a,b,c,d,u1,u2,u3,u4,u5,u6,u7,u8 that satifies dL/da = dL/db = dL/dc = dL/dd = dL/du1 = ... = 0.

I guess that I've made a misstake when I set up the inequality constraint in L as

dL/du1 = a-1 = 0 => a = 1
dL/du5 = -a-1 = 0 => a = -1

so dL/du1 and dL/du5 cannot be equal to zero at the same time.

Help from someone?

I don't know if this method is acceptable but I suppose we could argue like this.

First, we take

so all the factors in the product will be positive. And as we want the product to be as big as possible, we take

to be as big as possible and

to be as small as possible; thus we have

As the expression is now antisymmetrical in

and

, we can let

(so

is as large as possible); thus we have

Now you can maximize

using normal calculus methods.

Last edited by scientia (2012-11-13 22:51:01)