线性规划对偶理论

• Likewise, at least $20 must be paid for the combination of
resources used to produce one chair. Thus the y
variables must satisfy: y1 + 1.5y2 + 0.5y3 ≥ 20
IE5001
2
LP Duality
Primal LP :
max s.t.
Dual LP :
z c1x1 c2 x2 ... cn xn a11x1 a12 x2 ... a1n xn b1 a21x1 a22 x2 ... a2n xn b2 am1x1 am2 x2 ... amn xn bm x1, x2,..., xn 0
y1 ≥ 0, y2 ≥ 0, y3 ≥ 0, y4 ≥ 0
IE5001
4
LP Duality: Dakota Example
• The first dual constraint is associated with desks, the second dual constraint with tables, and the third dual constraint with chairs
• For this reason, dual variables are often referred to as resource shadow prices
IE5001
8
LP Duality
For every primal LP, we have an associated dual LP
Dual
• The decision variable y1 is associated with lumber, y2 with finishing hours, y3 with carpentry hours and y4 with table demands
• Suppose an entrepreneur wants to purchase all of Dakota’s resources. The entrepreneur must determine the price he or she is willing to pay for a unit of each of Dakota’s resources
2x1 + 1.5x2 + 0.5x3 ≤ 8
x2
≤5
x1 ≥ 0, x2 ≥ 0, x3 ≥ 0
• The dual problem is
(lumber constraint) (finishing constraint) (carpentry constraint) (table demand constraint)
Minimize w
Variable yi
↔
yi ≥ 0
↔
yi unrestricted in sign
↔
yi 0
Constraint j
↔
≥ form
↔
= form
↔
form
IE5001
12
LP Duality: Example
• Consider the following LP
y1, y2,..., ym 0
3
LP Duality: Dakota Example
Consider the Dakota problem
• The primal problem is
max z = 60x1 + 30x2 + 20x3
s.t.
8x1 + 6x2 + x3 ≤ 48
4x1 + 2x2 + 1.5x3 ≤ 20
min
w b1 y1 b2 y2 ... bm ym
s.t.
a11 y1 a21 y2 ... am1 ym c1
a12 y1 a22 y2 ... am2 ym c2
a1n y1 a2n y2 ... amn ym cn
IE5001
combination of resources that includes 8 board feet of
lumber, 4 finishing hours, and 2 carpentry hours because
Dakota could, if it wished, use the resources to produce a
Linear Programming Duality
LP Duality
• For each LP, we can define another LP directly or systematically such that for this pair of LPs, the optimal solution of one LP automatically yields the optimal solution of the other LP
• Thus, the solution to the Dakota dual will yield optimal
prices for lumber, finishing hours, and carpentry hours
IE5001
7
LP Duality: Shadow Prices
• In summary, when the primal is a normal maximization problem, the dual variables are related to the value of resources available to the decision maker
min w = 48y1 + 20y2 + 8y3 + 5y4
s.t.
8y1 + 4y2 + 2y3 ≥ 60 (desk constraint)
6y1 + 2y2 + 1.5y3 + y4 ≥ 30 (table constraint)
y1 + 1.5y2 + 0.5y3 ≥ 20 (chair constraint)
• In setting the resource prices, the prices must be high enough to induce Dakota to sell. They must be nonnegative
IE5001
6
LP Duality: Dakota Example
• The entrepreneur must offer Dakota at least $60 for a
• For this pair of LPs, the original LP is called the primal problem, while the other LP is called the dual problem
• Knowledge of the dual problem also provides interesting economic insights
• The total price that should be paid for these resources is 48y1 + 20y2 + 8y3 + 5y4. Since the cost of purchasing the resources is to be minimized,
min w = 48y1 + 20y2 + 8y3 + 5y4 is the objective function for the Dakota dual
• Then
• The dual LP is a minimization problem
• A dual variable is defined for each of the m primal constraints and is required to be nonnegative
• A dual constraint is defined for each of the n primal variables and is of “≥” type
• Objective coefficients of the dual LP are given by RHS of primal constraints
IE5001
百度文库
10
LP Duality
Let x = (x1, x2,…, xn)T, y = (y1, y2,…, ym)T, c = (c1, c2,…, cn)T, b = (b1, b2,…, bm)T ,
8y1 + 4y2 + 2y3 ≥ 60
• Similar reasoning shows that at least $30 must be paid
for the resources used to produce a table. Thus the y
variables must satisfy: 6y1 + 2y2 + 1.5y3 + y4 ≥ 30
Primal Variables
Right-hand
Variables
x1
…
xj
…
xn
y1
a11
…
a1j
…
a1n
y2
a21
…
a2j
…
a2n
…
……………
Side
b1 b2 …
…
……………
…
ym
am1
…
amj
…
amn
bm
c1
…
cj
…
cn
1st Dual
jth Dual
nth Dual
Dual
Constraint
a11 a12 ... a1n
A


a21
a22
...
a2n


am1
am2
...
amn

Primal LP :
Dual LP :
Max z cT x
Min w bT y
s.t. Ax b
s.t. yT A cT
IE5001
x0
y0 11
LP Duality
Constraint
Constraint Objective
IE5001
9
LP Duality
• Assume that the primal LP is a maximization problem with all the constraints being of “” type and all decision variables are assumed to be nonnegative
IE5001
5
LP Duality: Dakota Example
• To determine these prices, define:
• y1 = price paid for 1 board ft of lumber • y2 = price paid for 1 finishing hour • y3 = price paid for 1 carpentry hour • y4 = price paid for 1 table demand
desk that could be sold for $60. Since the entrepreneur is
offering 8y1 + 4y2 + 2y3 for the resources used to produce a desk, he or she must chose the y variables to satisfy:
• LHS coefficients of the dual constraint are given by constraint (column) coefficients of the associated primal variable, while RHS of the dual constraint is given by the objective coefficient of the same primal variable
Primal Problem (or Dual Problem) Maximize z Constraint i form = form ≥ form Variable xj xj ≥ 0 xj unrestricted in sign xj 0
Dual Problem (or Primal Problem)