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 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

    • Adjusting Multiplier

  • 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.


Maximization Problem

(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


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


Adjusting Multiplier

  • 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∞ ).


Adjusting Multiplier (Cont’d)

  • Determining stepsize tk (all tk starts from 1)


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 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

Comparison with Linear Programming Based Bounds (Cont’d)

(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.


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