An Applications Oriented Guide to Lagrangian Relaxation

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

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.

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?

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.

6. Maximization Problem (1) (2) (3) (4) (5) (6)

7. 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.

8. Determining u

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.

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

11. 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∞ ).

12. Adjusting Multiplier (Cont’d) • Determining stepsize tk (all tk starts from 1)

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 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.

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.

15. 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)

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.

17. 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.

18. Largrangian Relaxation Algorithm • Generic Largrangian relaxation algorithm. • Modified Largrangian relaxation algorithm.

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)

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.