Math Is Fun Forum

tt930101

Member

Registered: 2005-09-14

Posts: 2

an optimization problem

Could anyone help me with the following problem? I am not familar with it, this problem is formulated   in my current project, so I am not sure if it is solvable or come up with any reasonable solution. Thank you in advance.

Contraints:
x1 + x2 + x3 = a
y1 + y2 + y3 = b
x1 <= m1, x2<= m2, x3 <= m3
y1 <= n1, y2<=n2, y3<=n3
a < m1+m2+m3
b < n1+n2+n3
x1, x2, x3, y1, y2, y3 positive

Objective function:
f(x1, x2, x3, y1, y2, y3) = n1*m2*m3 + n2*m1*m3 +
n3*m1*m2 - y1*x2*x3 - y2*x1*x3 - y3*x1*x2

Question: Is there any lower bound for the function f?
I.e., does there exist any E>0 such that f >= E for any
xi, yi satisfying the constraints. If yes, what is E?

Last edited by tt930101 (2005-09-14 04:33:49)

winterLover

Guest

Re: an optimization problem

Wow!  This is a hard one.  No wonder it was never answered...  -John E. Franklin

George,Y

Member

Registered: 2006-03-12

Posts: 1,379

Re: an optimization problem

I can tell you this is a typical linear optimization problem and taught in advanced linear algebra.

I haven't learned it and it's too hard for me.

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X'(y-Xβ)=0

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: an optimization problem

Maybe it's easier than it looks.

y1 <= n1
x2 <= m2
x3 <= m3

So n1*m2*m3 >= y1*x2*x3 or rather n1*m2*m3 - y1*x2*x3 >= 0

n2 <= y2
x1 <= m1
x3 <= m3

So n2*m1*m3 >= y2*x1*x3 or rather n2*m1*m3 - y2*x1*x3 >= 0

y3 <= n3
x1 <= m1
x2 <= m2

So n3*m1*m2 >= y3*x1*x2 or rather n3*m1*m2 - y3*x1*x2 >= 0

Combine these three statements, rearrange terms, and you get:

n1*m2*m3 + n2*m1*m3 +
n3*m1*m2 - y1*x2*x3 - y2*x1*x3 - y3*x1*x2 >= 0

So 0 is a lower bound.

Edit:

Why does E have to be greater than 0?  It can't be 0?

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."