70
CHAPTER 5. NONLINEAR PROGRAMMING
with the components 1 − π
i
ts
i
tm
i
t of r we can see that 5.37 is equivalent to the set of N T problems:
Maximize − f
i
π
i
t, s
i
t, m
i
t − p
∗ i
t1 − π
i
ts
i
tm
i
t 0 ≤ π
i
t ≤ 1 , i = 1, . . . , N ; t = 1, . . . , T .
5.38 Thus,
p
∗ i
t is optimum tax per mg of BOD in area i during period t. Before we leave this example let us note that the optimum dual variable or shadow price
λ
∗
plays an important role in a larger framework. We noted earlier that the quality standard q, ¯
z was somewhat arbitrary. Now suppose it is proposed to change the standard in the
ith area during period
t to q + ∆q
i
t and ¯ z + ∆z
i
t. If the corresponding components of λ
∗
are λ
q∗ i
t and λ
z∗ i
t, then the change in the minimum cost necessary to achieve the new standard will be approximately
λ
q∗ i
t∆q
i
t + λ
z∗ i
t∆z
i
t. This estimate can now serve as a basis in making a benefitscost analysis of the proposed new standard.
5.3 Quadratic Programming
An important special case of NP is the quadratic programming QP problem: Maximize
c
′
x −
1 2
x
′
P x subject to
Ax ≤ b, x ≥ 0 , 5.39
where x ∈ R
n
is the decision variable , c ∈ R
n
, b ∈ R
m
are fixed, A is a fixed m × n matrix and
P = P
′
is a fixed positive semi-definite matrix. Theorem 1:
A vector x
∗
∈ R
n
is optimal for 5.39 iff there exist λ
∗
∈ R
m
, µ
∗
∈ R
n
, such that Ax
∗
≤ b, x
∗
≥ 0 c − P x
∗
= A
′
λ
∗
− µ
∗
, λ
∗
≥ 0, µ
∗
≥ 0 , λ
∗ ′
Ax
∗
− b = 0 , µ
∗ ′
x
∗
= 0 . 5.40
Proof: By Lemma 3 of 1.3, CQ is satisfied, hence the necessity of these conditions follows from
Theorem 2 of 1.2. On the other hand, since P is positive semi-definite it follows from Exercise 6
of Section 1.2 that f
: x 7→ c
′
x − 12 x
′
P x is a concave function, so that the sufficiency of these conditions follows from Theorem 4 of 1.2.
♦ From 5.40 we can see that
x
∗
is optimal for 5.39 iff there is a solution x
∗
, y
∗
, λ
∗
, µ
∗
to 5.41, 5.42, and 5.43:
Ax + I
m
Y = b −P x − A
′
λ + I
n
µ = −c , 5.41
x ≥ 0 y ≥ 0, λ ≥ 0, µ ≥ 0 , 5.42
µ
′
x = 0 , λ
′
y = 0 . 5.43
Suppose we try to solve 5.41 and 5.42 by Phase I of the Simplex algorithm see 4.3.2. Then we must apply Phase II to the LP:
Maximize −
m
X
i=1
z
i
−
n
X
j=1
ξ
j
5.4.
COMPUTATIONAL METHOD 71
subject to Ax + I
m
y + z
= b −P x
− A
′
λ + I
n
µ + ξ = −c
x ≥ 0, y ≥ 0, λ ≥ 0, µ ≥ 0, z ≥ 0, ξ ≥ 0, 5.44
starting with a basic feasible solution z = b, ξ = −c. We have assumed, without loss of generality,
that b ≥ 0 and −c ≥ 0. If 5.41 and 5.42 have a solution then the maximum value in 5.44 is 0.
We have the following result. Lemma 1:
If 5.41, 5.42, and 5.43 have a solution, then there is an optimal basic feasible solution of 5.44 which is also a solution f 5.41, 5.42, and 5.43.
Proof: Let
ˆ x, ˆ
y, ˆ λ, ˆ
µ be a solution of 5.41, 5.42, and 5.43. Then ˆ x, ˆ
y, ˆ λ, ˆ
µ, ˆ z = 0, ˆ
ξ = 0 is an optimal solution of 5.44. Furthermore, from 5.42 and 5.43 we see that at most
n + m components of
ˆ x, ˆ
y, ˆ λ, ˆ
µ are non-zero. But then a repetition of the proof of Lemma 1 of 4.3.1 will also prove this lemma.
♦ This lemma suggests that we can apply the Simplex algorithm of 4.3.2 to solve 5.44, starting
with the basic feasible solution z = b, ξ = −c, in order to obtain a solution of 5.41, 5.42, and
5.43. However, Step 2 of the Simplex algorithm must be modified as follows to satisfy 5.43: If a variable
x
j
is currently in the basis, do not consider µ
j
as a candidate for entry into the basis; if a variable
y
i
is currently in the basis, do not consider λ
i
as a candidate for entry into the basis. If it not possible to remove the
z
i
and ξ
j
from the basis, stop. The above algorithm is due to Wolfe [1959]. The behavior of the algorithm is summarized below.
Theorem 2: Suppose
P is positive definite. The algorithm will stop in a finite number of steps at an optimal basic feasible solution
ˆ x, ˆ
y, ˆ λ, ˆ
µ, ˆ z, ˆ
ξ of 5.44. If ˆ z = 0 and ˆ
ξ = 0 then ˆ x, ˆ
y, ˆ λ, ˆ
µ solve 5.41, 5.42, and 5.43 and
ˆ x is an optimal solution of 5.39. If ˆ
z 6= 0 or ˆ ξ 6= 0, then there is no
solution to 5.41, 5.42, 5.43, and there is no feasible solution of 5.39. For a proof of this result as well as for a generalization of the algorithm which permits positive
semi -definite
P see Cannon, Cullum, and Polak [1970], p. 159 ff.
5.4 Computational Method
