
Linear Optimization under Uncertainty 317
where e
i
is the vector of n components with a 1 in the ith position and 0
elsewhere.
Now suppose that B = {ζ| kζk
∞
≤ 1}, then the uncertain constraint
becomes, by Theorem 8.7,
−t ≥ −1.05y
1
−
n
P
i=2
(µ
i
− σ
i
)y
i
.
The robust counterpart can now be written as
minimize
t,y
{−t | −t ≥ −1.05y
n
−
n−1
P
i=1
(µ
i
− σ
i
)y
i
,
n
P
i=1
y
i
= 1, y
i
≥ 0 ∀i }.
which is a linear program.
Solving the cases when n = 10, 25, 50, 100, 150, and 200 each results in a
optimal solution that gives a guaranteed return of 5%. The optimal non-zero
decision variables are t = 1.05 and y
n
= 1 over all of the instances. This should
not be surprising as the robust counterpart ...