# An Applications Oriented Guide to Lagrangian Relaxation - PowerPoint PPT Presentation

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An Applications Oriented Guide to Lagrangian Relaxation. Author: Marshall L. Fisher Source: Interfaces 15:2 (1985) Presenter: Lillian Tseng. Outline. Introduction An Example: Maximization Dualizing Constraint Determining u Adjusting Multiplier

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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

### Outline

• Introduction

• An Example: Maximization

• Dualizing Constraint

• Determining u

• Comparison with Linear Programming Based Bounds

• Largrangian Relaxation Algorithm

• Conclusion

### 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.

### 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?

### Introduction (Cont’d)

• (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)

### Dualizing Constraint (2)

(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.

### 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.

### 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

• 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)

• 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.

### 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.

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)

### 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.

### Comparison with Linear Programming Based Bounds (Cont’d)

• 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.

### Largrangian Relaxation Algorithm

• Generic Largrangian relaxation algorithm.

• Modified Largrangian relaxation algorithm.

### 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)

### 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.