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

An Applications Oriented Guide to Lagrangian Relaxation

Author: Marshall L. Fisher

Source: Interfaces 15:2 (1985)

Presenter: Lillian Tseng


Outline

Outline

  • Introduction

  • An Example: Maximization

    • Dualizing Constraint

    • Determining u

    • Adjusting Multiplier

  • Comparison with Linear Programming Based Bounds

  • Largrangian Relaxation Algorithm

  • Conclusion


Introduction

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

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 d1

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

Maximization Problem

(1)

(2)

(3)

(4)

(5)

(6)


Dualizing constraint 2

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


Determining u cont d

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 d1

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

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

Adjusting Multiplier (Cont’d)

  • Determining stepsize tk (all tk starts from 1)


Adjusting multiplier cont d1

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

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.


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)


Comparison with linear programming based bounds cont d1

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 d2

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

Largrangian Relaxation Algorithm

  • Generic Largrangian relaxation algorithm.

  • Modified Largrangian relaxation algorithm.


An improved relaxation

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

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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