# Duality theory

1. Oct 11, 2005

### pinki82

Obtain an upper bound for the optimal value in the following problem;
Max (4x_1 + x_2 + 2x_3 + 3x_4 )
2x_1 - x_2 + x_3 - 2x_4 <= 2
7x_1 + x_2 + 5x_3 + 10x_4 <= 4
2x_1 + 3x_2 - x_3 - x_4 <= 2
x_i >= 0 , i= 1,2,3,4

any hint.help. please.
thanks

note: >= means > or equal to
<= means < or equal to

WORK DONE :

I understnad how to find the optimal value...but how do i find the
upper bound?
i dont really understand what is meant by the Upper Bound??

thanks.

2. Oct 13, 2005

### Maxos

Definition of upper bound:

Given an ordered set $$X and A \subset X , s \in X is its "upper bound" ( sup\{A\}) \leftrightharpoons \forall x \in A, x \preceq s et \forall y \prec s \exists a \in A : a \prec y$$

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