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 2018/11/19

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. 2018/11/19

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? 2018/11/19

Introduction (Cont’d) (P) (LRu) ZD(u) is an upper bound of the original problem Z, Z≤ZD(u), and u is non-negative. 2018/11/19

Maximization Problem (1) (2) (3) (4) (5) (6) 2018/11/19

Dualizing Constraint (2) (3) (4) (5) (6) talk about the relationship between supply, demand and price Considering constraint (2) as a resource constraint with supply (right) and demand (left), and u is the “price” charged for the resource. 2018/11/19

Determining u u x1 x2 x3 x4 ZD(u) Z* 1 20 6 60 3 34 10 2 26 18 16 14 ½ 1 20 6 60 3 34 10 2 26 18 16 14 ½ 19 ¾ 18.5 so far, the best u is 1, and Z* is 16, minimum UB is 18 2018/11/19

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. 2018/11/19

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 2018/11/19

Adjusting Multiplier Using subgradient method tk is 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)  ZD if (tk  0 & Σki = 0 ti  ∞ as k  ∞ ). 2018/11/19

Adjusting Multiplier (Cont’d) Determining stepsize tk (all tk starts from 1) rules of tk tk=1 tk+1 =tk /2 tk+1 =tk /3 u0 u1 2 u2 u3 ½ 2/9 u4 ¾ 0.296 u5 7/8 0.321 u6 15/16 0.329 u7 31/32 0.332 So the best stepsize is tk=tk/2 2018/11/19

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 λk is 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. 2018/11/19

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. compare LR with UB by relaxing the integrality requirement and solving the resulting problem 2018/11/19

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) 2018/11/19

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. 2018/11/19

Comparison with Linear Programming Based Bounds (Cont’d) Geoffrion(1974) ZD ≤ZLP for any Largrangian relaxation. Proof ZD=ZLP only 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. just like this example, the integrity doesn’t matter Zd is the UB of Largrangian relaxation, Zlp is the UB of LP relaxation In BIP, if the integrity will affect the answer, Zd must less than Zlp, so LR can improve the UB better. 2018/11/19

Largrangian Relaxation Algorithm Generic Largrangian relaxation algorithm. Modified Largrangian relaxation algorithm. 2018/11/19

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) 2018/11/19

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. Picking up checks at branch banks The US Navy, both make use of the skills and the job satisfaction Assembly system by choosing from people, single purpose machine, different skilled robots Model: network flow, shortest route, min spanning tree, knapsack 2018/11/19