Stochastic Programming - Index of

BASIC CONCEPTS 65
this case, the reader may consult the wide selection of books devoted to
linear programming in particular. Referring to our former presentation (7.5),
we have, owing to (7.10), that I B (ˆx) = I(ˆx), and, with the basic part
B =(A i | i ∈ I(ˆx)) and the nonbasic part N =(A i | i ∉ I(ˆx)) of the matrix
A, the constraints of (7.1) may be rewritten—using the basic and nonbasic
variables as introduced in (7.4)—as
x {B} = B −1 b − B −1 Nx {NB} ,
x {B} ≥ 0,
x {NB} ≥ 0.
⎫
⎬
⎭
(7.11)
Obviously this system yields our feasible basic solution ˆx iff x {NB} =0,and
then we have, by our assumption (7.10), that x {B} = B −1 b>0. Rearranging
the components of c analogously to (7.4) into the two vectors
= c i , i the kth element of I(ˆx), k=1, ···,m,
l
= c i , i the lth element of {1, ···,n}−I(ˆx), l =1, ···,n− m,
owing to (7.11), the objective may now be expressed as a function of the
nonbasic variables:
c {B}
k
c {NB}
c T x =(c {B} ) T x {B} +(c {NB} ) T x {NB}
=(c {B} ) T B −1 b +[(c {NB} ) T − (c {B} ) T B −1 N] x {NB} .
(7.12)
This representation of the objective connected to the particular feasible basic
solution ˆx implies the optimality condition for linear programming—the socalled
simplex criterion.
Proposition 1.15 Under the assumption (7.10), the feasible basic solution
resulting from (7.11) for x {NB} =0is optimal iff
[(c {NB} ) T − (c {B} ) T B −1 N] T ≥ 0. (7.13)
Proof
By assumption (7.10), the feasible basic solution given by
x {B} = B −1 b − B −1 Nx {NB} ,
x {NB} =0
satisfies x {B} = B −1 b>0. Therefore any nonbasic variable x {NB}
l
may be
increased to some positive amount without violating the constraints x {B} ≥ 0.
Furthermore, increasing the nonbasic variables is the only feasible change
applicable to them, owing to the constraints x {NB} ≥ 0. From the objective
presentation in (7.12), we see immediately that
c Tˆx =(c {B} ) T B −1 b
≤ (c {B} ) T B −1 b +[(c {NB} ) T − (c {B} ) T B −1 N] x {NB} ∀x {NB} ≥ 0