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Lagrangean Relaxation. --- Bounding through penalty adjustment. Outline. Brief introduction How to perform Lagrangean relaxation Subgradient techniques Example: Set covering Decomposition techniques and Branch and Bound. Introduction. Lagrangian Relaxation is a technique which has - PowerPoint PPT Presentation
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Lagrangean Relaxation
--- Bounding through penalty adjustment
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Outline
• Brief introduction
• How to perform Lagrangean relaxation
• Subgradient techniques
• Example: Set covering
• Decomposition techniques and Branch and Bound
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Introduction
Lagrangian Relaxation is a technique which has
been known for many years:• Lagrange relaxation is invented by (surprise!) La
grange in 1797 !• This technique has been very useful in conjuncti
on with Branch and Bound methods.• Since 1970 this has been the bounding techniqu
e of choice ...... until the beginning of the 90’ies (branch-and-price)
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Linear Program
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How can we calculate lower bounds ?
We can use heuristics to generate upper bounds, but getting (good) lower bounds is often much harder !
The classical approach is to create a relaxation.
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The classical branch-and-bound algorithm use the
LP relaxation. It has the nice feature of being
general, i.e. applicable to all MIP models.
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Linear Program
Min:
s.t.
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This is called the Lagrangean Lower Bound
Program (LLBP).
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Note:
• For any λ0, the program LLBP provides a lower bound to the original problem.
• To get the tightest lower bound, we should solve
}1,0{
..
)(min
max0
x
dBxts
Axbcx
(This problem is called Lagrangean Dual program.)
• Ideally, the optimal value of Lagrangear Dual program is equal to the optimal value of the original problem. If not, a duality gap exists.
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Lagrangian Relaxation
What can this be used for ?• Primary usage: Bounding ! Because it is a relaxa
tion, the optimal value will bound the optimal value of the real problem !
• Lagrangian heuristics, i.e. generate a “good” solution based on a solution to the relaxed problem.
• Problem reduction, i.e. reduce the original problem based on the solution to the relaxed problem.
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Two Problems
Facing a problem we need to decide:
• Which constraints to relax (strategic choice)
• How to find the best lagrangean multipliers, (tactical choice)
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Which constraints to relax
Which constraints to relax depends on two things:• Computational effort:
– Number of Lagrangian multipliers– Hardness of problem to solve
• Integrality of relaxed problem: If it is integral, we can only do as good as the straightforward LP relaxation !
(Integrality: the solution to relaxed problem is guaranteed to be integral.)
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Multiplier adjustment
Two different types are given:
• Subgradient optimisation• Multiplier adjustment
Of these, subgradient optimisation is the method of choice. This is general method which nearly always works !
Here we will only consider this Method, although more efficient (but much more complicated) adjustment methods have been suggested.
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Lagrangian Relaxation
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Problem reformulation
Note: Each constraint is associated with a multiplier i.iijj ij bxa
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The subgradient
(subgradient of multiplier i)
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Subgradient Optimization
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Example: Set covering
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Relaxed Set coverint
How can we solve this problem to optimality ???
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Optimization Algorithm
The answer is so simple that we are reluctant calling it an optimization algorithm: Choose all x’s with negative coefficients !
What does this tell us about the strength of the relaxation ?
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Rewritten: Relaxed Set covering
where
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Lower bound
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Comments
• This is actually quite interesting: The algorithm is very simple, but a good lower bound is found quickly !
• This relied a lot on the very simple LLBP optimization algorithm.
• Usually the LLBP requires much more work ....
• The subgradient algorithm very often works ...
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So what is the use ?
This is all very nice, but how can we solve our problem ?
• We may be lucky that the lower bound is also a feasible and optimal solution.
• We may reduce the problem, performing Lagrangian problem reduction.
• We may generate heuristic solutions based on the LLBP.
• We may use LLBP in lower bound calculations for a Branch and Bound algorithm.
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Homework
Starting with ZUB=6,=2 and i=0 for i=1,2,3, perform 3 iterations of the subgradient optimization method on the previous example problem. Please show the lower bound and multiplier values of each iteration.
