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Unformatted text preview: try to upper bound the optimal value by a linear combination
of the constraints. Precisely, for all i, let us multiply the ith constraint by yi ≥ 0 and then
sum the resulting constraints. In the previous two examples, we had (y1 , y2 , y3 ) = (0, 5 , 0) and
3
(y1 , y2 , y3 ) = (0, 1, 1). More generally, we obtain the following inequality:
y1 (x1 − x2 − x3 + 3x4 ) + y2 (5x1 + x2 + 3x3 + 8x4 ) + y3 (−x1 + 2x2 + 3x3 − 5x4 )
= (y1 − 5y2 − y3 )x1 + (−y1 + y2 + 2y3 )x2 + (−y1 + 3y2 + 3y3 )x3 + (3y1 + 8y2 − 5y3 )x4
≤
y1 + 55y2 + 3y3
For this inequality to provide an upper bound of 4x1 + x2 + 5x3 + 3x4 , we need to ensure that,
for all x1 , x2 , x3 , x4 ≥ 0,
4x1 + x2 + 5x3 + 3x4
≤ (y1 − 5y2 − y3 )x1 + (−y1 + y2 + 2y3 )x2 + (−y1 + 3y2 + 3y3 )x3 + (3y1 + 8y2 − 5y3 )x4 .
That is, y1 − 5y2 − y3 ≥ 4, −y1 + y2 + 2y3 ≥ 1, −y1 + 3y2 + 3y3 ≥ 5, and 3y1 + 8y2 − 5y3 ≥ 3.
Combining all inequalities, we obtain the following minimization linear programme:
Minimize y1 + 55y2 + 3y3
Subject to:
y1 − 5y2 − y3
−y1 + y2 + 2y3
−y1 + 3y2 + 3y3
3y1 + 8y2 − 5y3
y1 , y2 , y3 ≥
≥
≥
≥
≥ 4
1
5
3
0 This problem is called the dual of the initial maximization problem. 140 9.3.2 CHAPTER 9. LINEAR PROGRAMMING Dual problem We generalize the example given in Subsection 9.3.1. Consider the following general maximization linear programme:
Problem 9.5. Maximize
∑n=1 c j x j
j
Subject to: ∑n=1 ai j x j ≤ bi
j
xj ≥ 0 for all 1 ≤ i ≤ m
for all 1 ≤ j ≤ n Problem 9.5 is called the primal. The matricial formulation of this problem is
Maximize
cT x
Subject to: Ax ≤ b
x≥0 where xT = [x1 , . . . , xn ] and cT = [c1 , . . . , cn ] are vectors in Rn , and bT = [b1 , . . . , bm ] ∈ Rm ,
and A = [ai j ] is a matrix in Rm×n .
To ﬁnd an upper bound on cT x, we aim at ﬁnding a vector yT = [y1 , . . . , ym ] ≥ 0 such that,
for all feasible solutions x ≥ 0 of the initial problem, cT x ≤ yT Ax ≤ yT b = bT y, that is:
Minimize
bT y
Subject to: AT y ≥ c
y≥0 In other words, the dual of Problem 9.5 is deﬁned by:
Problem 9.6. Minimize
∑m 1 bi yi
i=
Subject to: ∑m 1 ai j yi ≥ c j
i=
yi ≥ 0 for all 1 ≤ j ≤ n
for all 1 ≤ i ≤ m Notice that the dual of a maximization problem is a minimization problem. Moreover, there
is a onetoone correspondence between the m constraints of the primal ∑ j=1...n ai j x j ≤ bi and
the m variables yi of the dual. Similarly, the n constraints ∑m 1 ai j yi ≥ c j of the dual correspond
i=
onetoone to the n variables xi of the primal.
Problem 9.6, which is the dual of Problem 9.5, can be equivalently formulated under the
standard form as follows.
Maximize
∑m 1 (−bi )yi
i=
Subject to: ∑m (−ai j )yi ≤ −c j
i=
yi ≥ 0 for all 1 ≤ j ≤ n
for all 1 ≤ i ≤ m (9.22) Minimize
∑n=1 (−c j )x j
j
Subject to: ∑n=1 (−ai j )x j ≥ −bi
j
xj ≥ 0 for all 1 ≤ i ≤ m
for all 1 ≤ j ≤ n (9.23) Then, the dual of Problem 9.22 has the following formulation which is equivalent to Problem 9.5. 9.3. DUALITY OF LINEAR PROGRAMMING 141 We deduce the following lemma.
Lemma 9.7. If D is t...
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 Fall '13
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