Stochastic Programming - Index of

54 STOCHASTIC PROGRAMMING
transformed to assume the form (7.1). If, for instance, we have the problem
min c T x
s.t. Ax ≥ b
x ≥ 0,
then, by introducing a vector y ∈ IR m +
of slack variables, we get the problem
min c T x
s.t. Ax − y = b
x ≥ 0
y ≥ 0,
which is of the form (7.1). This LP is equivalent to (7.1) in the sense that
the x part of its solution set and the solution set of (7.1) as well as the two
optimal values obviously coincide. Instead, we may have the problem
min c T x
s.t. Ax ≥ b,
where the decision variables are not required to be nonnegative—so-called free
variables. In this case we may introduce a vector y ∈ IR m + of slack variables
and—observing that any real number may be presented as the difference of two
nonnegative numbers—replace the original decision vector x by the difference
z + − z − of the new decision vectors z + ,z − ∈ IR n + yielding the problem
min{c T z + − c T z − }
s.t. Az + − Az − − y = b,
z + ≥ 0,
z − ≥ 0,
y ≥ 0,
which is again of the form (7.1). Furthermore, it is easily seen that this
transformed LP and its original formulation are equivalent in the sense that
• given any solution (ẑ + , ẑ − , ŷ) of the transformed LP, ˆx := ẑ + − ẑ − is a
solution of the original version,
• given any solution ˇx of the original LP, the vectors ˇy := Aˇx − b and
ž + , ž − ∈ IR n + , chosen such that ž+ − ž − =ˇx, solve the transformed version,
and the optimal values of both versions of the LP coincide.
1.7.1 The Feasible Set and Solvability
From linear algebra, we know that the system Ax = b of linear equations
in (7.1) is solvable if and only if the rank condition
rk(A, b) =rk(A) (7.2)