Yinyu Ye, MS&E, Stanford. MS&E310 Lecture Note #04. 1. Dual Interpretations
and Duality Applications. Yinyu Ye. Department of Management Science and ...
Yinyu Ye, MS&E, Stanford
MS&E310 Lecture Note #04
Dual Interpretations and Duality Applications
Yinyu Ye Department of Management Science and Engineering Stanford University Stanford, CA 94305, U.S.A.
http://www.stanford.edu/˜yyye (LY, Chapters 4.1-4.2, 6.8)
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Yinyu Ye, MS&E, Stanford
MS&E310 Lecture Note #04
Production Problem I
max pT x s.t. Ax ≤ r, x ≥ 0 where
• p: profit margin vector • A: resources consumption rate matrix • r: available resource vector • x: production level decision vector
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Yinyu Ye, MS&E, Stanford
MS&E310 Lecture Note #04
Production Problem II: Liquidation Pricing
• y: the fair price vector • AT y ≥ p: competitiveness • y ≥ 0: positivity • min rT y: minimize the total liquidation cost
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Yinyu Ye, MS&E, Stanford
MS&E310 Lecture Note #04
maximize
x1
subject to
x1
Primal :
≤1 x2
x1 x1 ,
Dual :
+2x2
minimize
y1
subject to
y1
+y2 y2
y1 , y2 ,
4
+x2 x2
≤1 ≤ 1.5 ≥ 0.
+1.5y3 +y3
≥1
+y3
≥2
y3
≥ 0.
Yinyu Ye, MS&E, Stanford
MS&E310 Lecture Note #04
Optimal Value Function and Shadow Prices
z(b) =
cT x
minimize
Ax = b, x ≥ 0.
subject to Suppose a new right-hand-vector b+ such that
+ = b + δ and b b+ k i = bi , ∀i ̸= k. k
Then, the optimal dual solution y∗ has a property
yk∗ = (z(b+ ) − z(b))/δ as long as y∗ remains the dual optimal solution for b+ , because
z(b+ ) = (b+ )T y∗ = z(b) + δ · yk∗ . Thus, the optimal dual value is the rate of the net change of the optimal objective value over the net change of an entry of the right-hand-vector resources, i.e.,
∇z(b) = y∗ . 5
Yinyu Ye, MS&E, Stanford
MS&E310 Lecture Note #04
Transportation Problem
min s.t.
∑m ∑n i=1
∑n
j=1 cij xij
= si , ∀i = 1, ..., m
j=1 xij ∑m i=1 xij
= dj , ∀j = 1, ..., n ≥ 0, ∀i, j.
xij
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Yinyu Ye, MS&E, Stanford
d1
d2
d3
MS&E310 Lecture Note #04
1
C11, x11
s1
2
s2
2
3 . .
. . dn
1
. . m
n
Supply
Demand 7
sm
Yinyu Ye, MS&E, Stanford
MS&E310 Lecture Note #04
Transportation Dual
max s.t.
∑m
i=1 si ui
+
∑n j=1
ui + vj
ui : supply site unit price vi : demand site unit price ui + vj ≤ cij : competitiveness
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d j vj ≤ cij , ∀i, j.
Yinyu Ye, MS&E, Stanford
MS&E310 Lecture Note #04
Max-Flow and Min-Cut Given a directed graph with nodes 1, ..., m and edges A, where node 1 is called source and node m is called the sink, and each edge (i, j) has a flow rate capacity kij . The Max-Flow problem is to find the largest possible flow rate from source to sink. Let xij be the flow rate from node i to node j . Then the problem can be formulated as maximize subject to
xm1 ∑
∑
xj1 − j:(1,j)∈A x1j + xm1 = 0, ∑ ∑ j:(j,i)∈A xji − j:(i,j)∈A xij = 0, ∀i = 2, ..., m − 1, ∑ ∑ j:(j,m)∈A xjm − j:(m,j)∈A xmj − xm1 = 0, j:(j,1)∈A
0 ≤ xij ≤ kij , ∀(i, j) ∈ A.
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Yinyu Ye, MS&E, Stanford
MS&E310 Lecture Note #04
6
2
2
3 1
3 4
3
3
4 7
Sink
Source 5
3 4
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Yinyu Ye, MS&E, Stanford
MS&E310 Lecture Note #04
The dual of the Max-Flow problem
minimize subject to
∑ (i,j)∈A
kij zij
−yi + yj + zij ≥ 0, ∀(i, j) ∈ A, y1 − ym = 1, zij ≥ 0, ∀(i, j) ∈ A.
yi : node potential value. At an optimal solution has property y1 = 1, ym = 0 and for all other i: 1 if i ∈ S yi = 0 if i ̸∈ S This problem is called the Min-Cut problem.
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Yinyu Ye, MS&E, Stanford
MS&E310 Lecture Note #04
2
1
2
3 3
4 Sink
Source 3
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Yinyu Ye, MS&E, Stanford
MS&E310 Lecture Note #04
Application: Combinatorial Auction Pricing I Given the m different states that are mutually exclusive and exactly one of them will be true at the maturity. A contract on a state is a paper agreement so that on maturity it is worth a notional $1 if it is on the winning state and worth $0 if is not on the winning state. There are n orders betting on one or a combination of states, with a price limit and a quantity limit.
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Yinyu Ye, MS&E, Stanford
MS&E310 Lecture Note #04
Combinatorial Auction Pricing II: an order m ∈ R+ , πj ∈ R+ , qj ∈ R+ ): aj is the combination betting vector where each component is either 1 or 0 a1j a 2j aj = , ... amj
The j th order is given as (aj
where 1 is winning and 0 is non-winning; πj is the price limit for one such a contract, and qj is the maximum number of contracts the better like to buy.
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Yinyu Ye, MS&E, Stanford
MS&E310 Lecture Note #04
World Cup Information Market
Order:
#1
#2
#3
#4
#5
Argentina
1
0
1
1
0
Brazil
1
0
0
1
1
Italy
1
0
1
1
0
Germany
0
1
0
1
1
France
0
0
1
0
0
0.4
0.95
0.75
Bidding Prize:π
0.75 0.35
Quantity limit:q
10
5
10
10
5
Order fill:x
x1
x2
x3
x4
x5
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Yinyu Ye, MS&E, Stanford
MS&E310 Lecture Note #04
Combinatorial Auction Pricing III: Pricing each state Let xj be the number of contracts awarded to the j th order. Then, the j th better will pay the amount
πj · x j and the total collected amount is
n ∑
πj · xj = π T x
j=1
If the ith state is the winning state, then the auction organizer need to pay back
n ∑
aij xj
j=1
The question is, how to decide x
∈ Rn .
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Yinyu Ye, MS&E, Stanford
MS&E310 Lecture Note #04
Combinatorial Auction Pricing IV: LP model
max
πT x − z
s.t.
Ax − e · z
≤ 0,
x
≤ q,
x
≥ 0.
π T x: the optimistic amount can be collected. z : the worst-case amount need to pay back.
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Yinyu Ye, MS&E, Stanford
MS&E310 Lecture Note #04
Combinatorial Auction V: The dual
min
qT y
s.t.
AT p + y
≥ π,
eT p
= 1,
(p, y)
≥ 0.
p represents the state price. What is y? Price information gaps/differentials/slacks where their weighted sum we like to minimize.
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Yinyu Ye, MS&E, Stanford
MS&E310 Lecture Note #04
Combinatorial Auction V: Strict Complementarity
xj > 0
aTj p + yj = πj and yj ≥ 0 so that aTj p ≤ πj
0 < x j < qj
yj = 0 so that aTj p = πj
x j = qj
yj > 0 so that aTj p < πj
xj = 0
aTj p + yj > πj and yj = 0 so that aTj p > πj
The price is Fair:
pT (Ax − e · z) = 0 implies pT Ax = pT e · z = z; that is, the worst case cost equals the worth of total shares. Moreover, if a lower bid wins the auction, so does the higher bid on any same type of bids.
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Yinyu Ye, MS&E, Stanford
MS&E310 Lecture Note #04
World Cup Information Market Result
Order:
#1
#2
#3
#4
#5
State Price
Argentina
1
0
1
1
0
0.2
Brazil
1
0
0
1
1
0.35
Italy
1
0
1
1
0
0.2
Germany
0
1
0
1
1
0.25
France
0
0
1
0
0
0
Bidding Price:π
0.75
0.35
0.4
0.95
0.75
Quantity limit:q
10
5
10
10
5
Order fill:x∗
5
5
5
0
5
Question 1: The uniqueness of dual prices?
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Yinyu Ye, MS&E, Stanford
MS&E310 Lecture Note #04
Combinatorial Auction Pricing VI: convex programming model
max π T x − z + u(s) s.t.
Ax − e · z + s = 0, x
≤ q,
x, s ≥ 0. u(s): a value function for the market organizer on slack shares. If u(·) is a strictly concave function, then the state price vector is unique. Question 2: Online allocation?
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Yinyu Ye, MS&E, Stanford
MS&E310 Lecture Note #04
Online Resource Allocation Linear Programming
maximizex s.t.
∑n j=1 ∑n j=1
πj xj aij xj ≤ bi , ∀ i = 1, 2, ..., m,
0 ≤ xj ≤ 1, ∀ j = 1, ..., n. Approach 1 (SCPM):
maximizexk ,s
πk xk + u(s)
s.t.
aik xk + si = bi −
∑k−1 j=1
aik x ¯k , ∀ i = 1, 2, ..., m,
0 ≤ xk ≤ 1, si ≥ 0, ∀ i = 1, ..., m. Approach 2 (SLPM):
maximizex1 ,...,xk s.t.
∑k j=1 ∑k j=1
πi x i aik xk ≤
k n bi ,
∀ i = 1, 2, ..., m,
0 ≤ xj ≤ 1, ∀ j = 1, ..., k. and use the dual prices of the partial LP for decisions for the next period ... 22
Yinyu Ye, MS&E, Stanford
MS&E310 Lecture Note #04
Online Resource Allocation with Production Costs One may consider more general resource allocation problems with production costs:
maximizex s.t.
∑n ∑ ∑
j=1 (πj xj
k
−
∑
k cijk yijk )
yijk = aij xj ; ∀i, j,
i,j
yijk ≤ ck ; ∀k,
0 ≤ xj ≤ 1, ∀ j = 1, ..., n; where cijk is the cost allocate good/resource i, which is produced by producer k and ck is the production capacity of producer k .
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= 1, ..., K , to bidder j ;
