An Applications Oriented Guide to Lagrangian Relaxation

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An Applications Oriented Guide to Lagrangian Relaxation

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An Applications Oriented Guide to Lagrangian Relaxation

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An Applications Oriented Guide to Lagrangian Relaxation

Author: Marshall L. Fisher

Source: Interfaces 15:2 (1985)

Presenter: Lillian Tseng

- Introduction
- An Example: Maximization
- Dualizing Constraint
- Determining u
- Adjusting Multiplier

- Comparison with Linear Programming Based Bounds
- Largrangian Relaxation Algorithm
- Conclusion

- 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.

- 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?

- (P) (LRu)
- ZD(u) is an upper bound of the original problem Z, Z≤ZD(u), and u is non-negative.

(1)

(2)

(3)

(4)

(5)

(6)

(3)

(4)

(5)

(6)

- Considering constraint (2) as a resource constraint with supply (right) and demand (left), and u is the “price” charged for the resource.

- 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.

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

- Using subgradient method
- tkis 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)ZDif (tk 0 & Σki = 0ti∞ as k∞ ).

- Determining stepsize tk (all tk starts from 1)

- 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 λkis 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.

- 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.

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

(LP)

(dual LP)

- 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.

- Geoffrion(1974)
- ZD ≤ZLP for any Largrangian relaxation.

- Proof
- ZD=ZLPonly 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.

- Generic Largrangian relaxation algorithm.
- Modified Largrangian relaxation algorithm.

- 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)

- 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.