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Quadratic Programming
G. Cornelis van Kooten
REPA Group University of Victoria
October 18, 2012 2
Problem specification in matrix notation:
Max F(x) = c x + x′ Ω x
s.t. A x b
x ≥ 0
where x′Ωx is the quadratic form.
For a maximum, the objective function must be concave; for a minimum it must be convex
Concave → Ω is negative definite or negative semi-definite
Convex → Ω is positive definite or positive semi-definite
0220223
024262
02441
)22(22264L
definite negative032 1
1 2,02,
2 1
1 2
2,2 1,,6 4
0,
22s.t.
22264max
2
1
21
22
22
yxyxL
yxy
L
yxx
L
yxyxyxyxyx
bAy
xxc
yx
yx
yxyxyx
Example
solution. feasible basic theofout driven be first to the
are that variablesartificial are and where
223'
62422'
4241'
:follows as 31 resultsConvert
:methodsimplex theusing
solved becan that one toproblem isconvert th We
0,,,,6
0,05
0)22(4
21
22
11
21
21
AA
Syx
Ayx
Ayx
yx
yx
yx
S is a slack variable, as usual; (4) and (5) are
complementary slackness conditions.
any time.at solution in becan pair each of oneonly
or ,or nonbasic, is or either :last three handle To
0 and
0
0
0
and
2 2
6 242
4 24s.t.
36610max
:implies This
)(maxmin
:asproblem theRestate
21
21
2
1
22
11
21
2121
yxS
μ,μλ,y,x,
y
x
S
Syx
Ayx
Ayx
yxZ
AAAA
October 18, 2012 6
QP Conclusions
If we maximize Z = – A1 – A2 , we have solved the original QP.
Why?
The new problem takes into account the optimization as the 1st-
order conditions are met already, plus we have shown the
problem to be a maximum as Ω was negative definite.
The advantage of QP is that a QP problem can be re-specified
as an LP. Hence, QP problems are treated as separate
options/solvers in Matlab and GAMS. In Excel, the problem
needs to be set up as an LP as shown above (i.e., solving for the
1st-order conditions).
Price Endogenous Models
Let Pd = α – β Qd (demand function)
Ps = a + b Qs (supply function)
In equilibrium:
Pd = Ps or [α – β Qd] = [a + b Qs]
and Qd = Qs
It is important to recognize that quantity supplied must be equal to or greater than demand Qs ≥ Qd, but if Qs>Qd, then P* = 0, where P* is equilibrium price.
October 18, 2012 7
Thus: (–Qs + Qd )P* = 0
which is a Kuhn-Tucker condition.
October 18, 2012 8
Price
0
quantity
demand
supply
Case where supply exceeds demand and price is zero.
Price Endogenous Model (cont)
To solve for the equilibrium quantity and price, the objective is to maximize the area under the demand curve minus the area under the supply function. Thus, we get the following QP problem:
Max α Qd – ½ β Qd2 – a Qs– ½ b Qs
2
s.t. Qd – Qs 0
Qd, Qs ≥ 0
P* is the dual variable associated with the 1st constraint.
October 18, 2012 9
Spatial Price Equilibrium (SPE) or
Trade Model
• Production and/or consumption occur in spatially separated markets, each with its own supply and demand. Trade occurs if prices between regions differ by the amount of the transportation cost plus tariffs/taxes
• Developed by Takayama & Judge (Spatial and Temporal Price and Allocation Models 1971) and Judge & Takayama (Studies in Economic Planning over Space and Time 1973) (both North-Holland)
October 18, 2012 10
Graphic representation of SPE models follows.
October 18, 2012 11
(a) Canada (b) International Market (c) United States
Lumber quantity
Price
Pc
0 q* qcS
a
b
Sc
ES+ transportation costs
PU
Sus
Dus
0 0 q*
Dc
Canadian surplus = a + b
U.S. surplus = α + β
Canada-U.S. Trade in Softwood Lumber
October 18, 2012 12
ES = excess supply
ED = excess demand
(a) Canada (b) International Market (c) United States
Lumber quantity
Price
Pc
0 q* qcS
a
b
Sc
EDCanada
PU
Sus
Dus
0 0 q*
Dc
ESCanada
Canada-U.S. Trade in Softwood Lumber (cont)
October 18, 2012 13
ES = excess supply
ED = excess demand
(a) Canada (b) International Market (c) United States
Lumber quantity
Price
Pc
0 q* qcS
a
b
Sc
PU
Sus
Dus
0 0 q*
Dc
ESCanada
EDUS
P trade
Q trade
Canada-U.S. Trade in Softwood Lumber (cont)
October 18, 2012 14
U.S. and Canadian prices differ by the transportation cost =
PUtrade – PC
trade
(a) Canada (b) International Market (c) United States
Lumber quantity
Price
Pc
0 q* qcS
a
b
Sc
PU
Sus
Dus
0 0 q*
Dc
ES
EDUS
PU trade
Q trade
PC trade
ES+ transportation costs
Canada-U.S. Trade in Softwood Lumber (cont)
(a) Canada (b) International Market (c) United States
Lumber quantity
Price
PcT
Pc
0 qcD q* qc
S
a
b c
d e
g
Sc
ES+ transportation costs
A
B C
D
E
G
QT
ES
ED
PU
PTU
Sus
Dus
0 0 qUs q* qU
d
Dc
Cdn surplus = a+b+c+g+e+d
U.S. surplus = α+β+ ϕ+δ+γ
Cdn gain = g = B+E
U.S. gain = ϕ+δ = A
Complete Canada-U.S. Lumber Trade Model
SPE Model: Mathematical Formulation
s.t. iqij ≤ qj, j (region cannot export more than supply)
jqij qi, i (regional demands satisfied)
qi, qj, qij 0 (non-negativity)
qij = sales by region j to region i,
tij = unit transport cost from region j to region i,
X selling regions; M buying regions (X≠M, i may equal j)
jiqtqbqa
qqZqqq
M
i
X
j
ijij
X
j
jjjj
M
i
iiii
ijji
,,)5.0(
)5.0(,,
Max
1 11
2
1
2
Solution Exists IF:
1. Each region’s demand is downward sloping
2. Each region’s supply is upward sloping
3. Linear demand and supply → quadratic
program
4. Z is strictly concave in qi and qj, concave in
qij, and bounded from above.
October 18, 2012 17
Solution exists and is unique in terms of qi
and qj, but not necessarily for qij
(see Takayama & Judge p.142)
Example:
Trade between Europe, Japan & U.S. (Ch 13, McCarl & Spreen)
Supplies: Ps,U = 25 + Qs,U (Only U.S. & Europe Ps,E = 35 + Qs,E supply commodity)
Demands: Pd,U = 150 – Qd,U
Pd,E = 155 – Qd,E
Pd,J = 160 – Qd,J
Transport costs: U.S.–Europe = 3 (both directions)
U.S.–Japan = 4
Europe–Japan = 5
October 18, 2012 18
GAMS file available here
October 18, 2012 23
quantity shadow price
US .demand 46.400 103.600
US .supply 78.600 103.600
EUROPE.demand 50.400 104.600
EUROPE.supply 69.600 104.600
JAPAN .demand 51.400 108.600
Sales to → US EUROPE JAPAN
from
US 46.40 0 32.20
EUROPE 0 50.40 19.20
Objective value = 9193.60
Reduced cost: US to Europe = -4
Europe to US = -2
All others = 0
Undistorted No-Trade Quota Tax/Subsidy
Objective 9193.6 7506.3 8761.6 9178.6
U.S. Demand 45.4 62.5 61.5 46.4
U.S. Supply 9.6 62.5 63.5 78.6
U.S. Price 104.6 87.5 88.5 103.6
Europe Demand 51.4 60 40.7 50.4
Europe Supply 68.6 60 79.3 69.6
Europe Price 103.6 95 114.3 104.6
Japan Demand 51.4 0 40.7 51.4
Japan Price 108.6 160 119.3 108.6
Scenario Analysis