1. An Applications Oriented Guide to Lagrangian Relaxation
Author: Marshall L. Fisher
Source: Interfaces 15:2 (1985)
Presenter: Lillian Tseng

2. Outline Introduction An Example: Maximization
Dualizing Constraint
Determining u
Adjusting Multiplier
Comparison with Linear Programming Based Bounds
Largrangian Relaxation Algorithm
Conclusion
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3. Introduction The lack of a “how to do it” exposition.
LR can be used to provide bounds in a branch and bound algorithm.
Using a “maximization” problem as example.
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4. Introduction (Cont’d)
Three major questions in designing a Largrangian-relaxation-based system:
How to compute good multipliers u?
How to deduce a good, feasible solution to the original problem?
Which constraints should be relaxed?
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5. Introduction (Cont’d)
(P) (LRu)
ZD(u) is an upper bound of the original problem Z, Z≤ZD(u), and u is non-negative.
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6. Maximization Problem
(1)
(2)
(3)
(4)
(5)
(6)
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7. Dualizing Constraint (2)
(3)
(4)
(5)
(6)
talk about the relationship between supply, demand and price
Considering constraint (2) as a resource constraint with supply (right) and demand (left), and u is the “price” charged for the resource.
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8. Determining u u x1 x2 x3 x4 ZD(u) Z* 1 20 6 60 3 34 10 2 26 18 16 14 ½1 20 6 60 3 34 10 2 26 18 16 14 19
18.5
so far, the best u is 1, and Z* is 16, minimum UB is 18
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9. Determining u (Cont’d)
The ZD(u) function is given by the upper envelope of the family of linear equations.
The ZD(u) function is convex and differentiable except at points where the Lagrangian problem has multiple optimal solutions.
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10. Determining u (Cont’d)
x1=x2=x3=x4=0
u=0
x2=1
x1=x3=x4=0
10+8u
x2=x4=1
x1=x3=0
14+4u
x1=1
x2=x3=x4=0
16+2u
x1=x4=1
x2=x3=0
20-2u
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11. Adjusting Multiplier Using subgradient method
tk is a scalar stepsize (positive).
xk is an optimal solution to (LRuk), the Lagrangian problem with dual variables set to uk.
In this example, b-Axk
10-(8x1+2x2+x3+4x4).
ZD(uk)  ZD if (tk  0 & Σki = 0 ti  ∞ as k  ∞ ).
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12. Adjusting Multiplier (Cont’d)
Determining stepsize tk (all tk starts from 1)
rules of tk
tk=1
tk+1 =tk /2
tk+1 =tk /3 u0 u1 2 u2 u3
2/9 u4
0.296 u5
7/8
0.321 u6
15/16
0.329 u7
31/32
0.332
So the best stepsize is tk=tk/2
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13. Adjusting Multiplier (Cont’d)
Step size adjustment
Z* is the objective value of the best known feasible solution to (P), and is usually set to 0 initially.
λk is a scalar between 0 and 2.
The sequence λk is determined by setting λk =2 and reducing λk by a factor of two whenever ZD(uk) has failed to decrease in some fixed number of iterations.
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14. Comparison with Linear Programming Based Bounds
The minimum UB known so far is 18.
Let (LP) denote (P) with integrality on x relaxed, and u, vi, wj dual variables on constraints.
compare LR with UB by relaxing the integrality requirement and solving the resulting problem
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15. Comparison with Linear Programming Based Bounds (Cont’d)
the optimal solution: ZLP=18
primal LP
x1=1, x2=0, x3=0,x4=1/2
dual LP
u=1, v1=8, v2=w1=w2=w3=w4=0
Comparison with Linear Programming Based Bounds (Cont’d)
(LP)
(dual LP)
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16. Comparison with Linear Programming Based Bounds (Cont’d)
The result shows that:
ZLP=18, the same UB in LR.
the value of u=1 in dual LP is exactly the value that gave the minimum UB of 18 on the Largrangian problem.
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17. Comparison with Linear Programming Based Bounds (Cont’d)
Geoffrion(1974)
ZD ≤ZLP for any Largrangian relaxation.
Proof
ZD=ZLP only if the Largrangian problem is unaffected by the integrality requirement on x.
UB can be improved by using a Lagrangian relaxation in which the variables are not naturally integral.
just like this example, the integrity doesn’t matter
Zd is the UB of Largrangian relaxation, Zlp is the UB of LP relaxation
In BIP, if the integrity will affect the answer, Zd must less than Zlp, so LR can improve the UB better.
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18. Largrangian Relaxation Algorithm
Generic Largrangian relaxation algorithm.
Modified Largrangian relaxation algorithm.
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19. An Improved Relaxation
Dualizing constraints (3) & (4):
A knapsack problem
It can be solved by subgradient method with λk=1, and Z*=ZD(v1,v2)=16.
The choice of which constraints to dualize is to some extent an art, much like formulation itself.
(2)
(5)
(6)
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20. Some Applications & Conclusions
Past applications
Vehicle routing.
Manpower planning problem.
Resource allocation
Finding the embedded well-known model.
The ability to exploit special problem structure can be applied to real problems.
Picking up checks at branch banks
The US Navy, both make use of the skills and the job satisfaction
Assembly system by choosing from people, single purpose machine, different skilled robots
Model: network flow, shortest route, min spanning tree, knapsack
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