distributed lagrangean relaxation protocol for the generalized mutual assignment problem
DESCRIPTION
Distributed Lagrangean Relaxation Protocol for the Generalized Mutual Assignment Problem. Katsutoshi Hirayama (平山 勝敏). Faculty of Maritime Sciences (海事科学部) Kobe University (神戸大学) [email protected]. Summary. - PowerPoint PPT PresentationTRANSCRIPT
Distributed Lagrangean Relaxation Protocol for the Generalized Mutual Assignment Problem
Katsutoshi Hirayama (平山 勝敏)
Faculty of Maritime Sciences (海事科学部)Kobe University (神戸大学)[email protected]
Summary
This work is on the distributed combinatorial optimization rather than the distributed constraint satisfaction.
I present the Generalized Mutual Assignment Problem (a distributed fo
rmulation of the Generalized Assignment Problem) a distributed lagrangean relaxation protocol for the GMAP a “noise” strategy that makes the agents (in the protocol) qu
ickly agree on a feasible solution with reasonably good quality
Outline
Motivation distributed task assignment
Problem Generalized Assignment Problem Generalized Mutual Assignment Problem Lagrangean Relaxation Problem
Solution protocol Overview Primal/Dual Problem Convergence to Feasible Solution
Experiments Conclusion
Motivation: distributed task assignment
Example 1: transportation domain A set of companies, each having its own
transportation jobs. Each is deliberating whether to perform a job by
myself or outsource it to another company. Seek for an optimal assignment that satisfies their
individual resource constraints (#s of trucks).
Kobe
Kyoto
Tokyojob1
job2job3
Company1 has {job1} and 4 trucks
Company2 has {job2,job3} and 3 trucks
profit trucks
Co.1job1 5 2
job2 6 2
job3 5 1
Co.2job1 4 2
job2 2 2
job3 2 2
Motivation: distributed task assignment
Example 2: info gathering domain A set of research divisions, each having its own
interests in journal subscription. Each is deliberating whether to subscribe a journal by
myself or outsource it to another division. Seek for an optimal subscription that does not exceed
their individual budgets. Example 3: review assignment domain
A set of PCs, each having its own review assignment Each is deliberating whether to review a paper by
myself or outsource it to another PC/colleague. Seek for an optimal assignment that does not exceed
their individual maximally-acceptable numbers of papers.
Problem: generalized assignment problem (GAP)
These problems can be formulated as the GAP in a centralized context.
job1 job2 job3
Company1(agent1)
Company2(agent2)
(5,2)
(4,2)
(6,2)
(2,2)
(5,1)
(2,2)
43
(profit, resource requirement)
Assignment constraint:each job is assigned to exactlyone agent.
Knapsack constraint:the total resource requirementof each agent does not exceedits available resource capacity.
01 constraint:each job is assigned or notassign to an agent.
Problem: generalized assignment problem (GAP)
232221131211 224565 xxxxxx
}3,2,1{ },2,1{ },1,0{
3222
422
1
1
1
232221
131211
2313
2212
2111
jix
xxx
xxx
xx
xx
xx
ij
max.
s. t.
The GAP instance can be described as the integer program.
GAP: (as the integer program)
However, the problem must be dealt by the super-coordinator.
xij takes 1 if agent i is to perform job j; 0 otherwise.
assignmentconstraints
knapsackconstraints
Problem: generalized assignment problem (GAP)
Drawbacks of the centralized formulation Cause the security/privacy issue
Ex. the strategic information of a company would be revealed.
Need to maintain the super-coordinator (computational server)
Distributed formulation of the GAP: generalized mutual assignment problem (GMAP)
Problem: generalized mutual assignment problem (GMAP)
The agents (not the supper-coordinator) solve the problem while communicating with each other.
job1 job2 job3
Company1 (agent1) Company2 (agent2)
43
Problem: generalized mutual assignment problem (GMAP)
Assumption: The recipient agent has the right to decide whether it will undertake a job or not.
job1
43
job2 job3 job1 job2 job3
Sharing theassignmentconstraints
(profit, resource requirement)
Company1 (agent1) Company2 (agent2)
(5,2) (6,2) (5,1) (4,2) (2,2) (2,2)
Problem: generalized mutual assignment problem (GMAP)
The GMAP can also be described as a set of integer programs
232221 224 xxx
}3,2,1{ },1,0{
3222
1
1
1
2
232221
2313
2212
2111
jx
xxx
xx
xx
xx
j
max.
s. t.
131211 565 xxx
}3,2,1{ },1,0{
422
1
1
1
1
131211
2313
2212
2111
jx
xxx
xx
xx
xx
j
max.
s. t.
Agent1 decides x11, x12, x13 Agent2 decides x21, x22, x23
Sharing theassignmentconstraints
GMP1 GMP2
: variables of others
Problem: lagrangean relaxation problem
By dualizing the assignment constraints, the followings are obtained.
)1(2
)1(2
)1(2
224
23133
22122
21111
232221
xx
xx
xxxxx
}3,2,1{ },1,0{
3222
2
232221
jx
xxx
j
max.
s. t.
)1(2
)1(2
)1(2
565
23133
22122
21111
131211
xx
xx
xxxxx
}3,2,1{ },1,0{
422
1
131211
jx
xxx
j
max.
s. t.
Agent1 decides x11, x12, x13 Agent2 decide x21, x22, x23
LGMP1(μ) LGMP2(μ)
: variables of others),,( 321 : lagrangean multiplier vector
Problem: lagrangean relaxation problem
Two important features: The sum of the optimal values of {LGMPk(μ) | k in all of the a
gents} provides an upper bound for the optimal value of the GAP.
If all of the optimal solutions to {LGMPk(μ) | k in all of the agents} satisfy the assignment constraints for some values of μ, then these optimal solutions constitute an optimal solution to the GAP.
LGMP1(μ) LGMP2(μ)
Opt.Value1 Opt.Value2
Opt.Sol1 Opt.Sol2
solve solve
GAP
+ Opt.Value
Opt.Sol (if Opt.Sol1 and Opt.Sol2 satisfy the assignment constraints)
=
Solution protocol: overview
The agents alternate the following in parallel while performing P2P communication until all of the assignment constraints are satisfied. Each agent k solves LGMPk(μ), the primal problem, using a k
napsack solution algorithm. The agents exchange solutions with each other. Each agent k finds appropriate values for μ (solves the (lagr
angean) dual problem) using the subgradient optimization method.
Agent1 Agent2 Agent3sharing sharing
Solve dual & primal prlms Solve dual & primal prlms
Solve dual & primal prlms Solve dual & primal prlms Solve dual & primal prlms
Solve dual & primal prlms
exchange
time
Solution protocol: primal problem
Primal problem: LGMPk(μ) Knapsack problem Solved by an exact method (i.e., an optimal solution is nee
ded)
)2,2
5( 1
)1(2
)1(2
)1(2
565
23133
22122
21111
131211
xx
xx
xxxxx
}3,2,1{ },1,0{
422
1
131211
jx
xxx
j
max.
s. t.
LGMP1(μ)
job1
agent1
4
job2 job3
)2,2
6( 2 )1,
25( 3
(profit, resource requirement)
Solution protocol: dual problem
Dual problem The problem of finding appropriate values for μ Solved by the subgradient optimization method
Subgradient Gj for the assignment constraint on job j
Updating rule for μj
agents
1i
ijj xG
jtjj Gl
tl : step length at time t
Solution protocol: example
When )0,0,0(),,( 321
)2,2
5( 1
job1
agent1
4
job2 job3
)2,2
6( 2 )1,
25( 3
3
job1 job2 job3
agent2
)2,2
4( 1 )2,
22( 2 )1,
22( 3
Select {job1,job2} Select {job1}
)1,0,1(),,( 321 GGG
1tland
Therefore, in the next,
)1,0,1(),,( 321
Note: the agents involved in job j must assign μj to a common value.
Solution protocol: convergence to feasible solution
A common value to μj ensures the optimality when the protocol stops. However, there is no guarantee that the protocol will eventually stop.
You could force the protocol to terminate at some point to get a satisfactory solution, but no feasible solution had been found. In a centralized case, lagrangean heuristics are usually devis
ed to transform the “best” infeasible solution into a feasible solution.
In a distributed case, such the “best” infeasible solution is inaccessible, since it belongs to global information.
I introduce a simple strategy to make the agents quickly agree on a feasible solution with reasonably good quality.
Noise strategy: let agents assign slightly different values to μj
Solution protocol: convergence to feasible solution
Noise strategy The updating rule for μj is replaced by
jtjj GlN )1(
N : random variable whose value is uniformly distributed over ],[
This rule diversifies agents’ views on the value of μj, and being able to break an oscillation in which agents repeat “clustering and dispersing” around some job.
For δ≠0, the optimality when the protocol stops does not hold.
For δ=0, the optimality when the protocol stops does hold.
Solution protocol: rough image
optimal
feasible region
value of theobject functionof the GAP
• Controlled by multiple agents• No window, no altimeter, but a touchdown can be detected.
Experiments
Objective Clarify the effect of the noise strategy
Settings Problem instances (20 in total)
feasible instances #agents ∈ {3,5,7}; #jobs = 5×#agents profit and resource requirement of each job: an integer randomly
selected from [1,10] capacity of each agent = 20 Assignment topology: chain/ring/complete/random
Protocol Implemented in Java using TCP/IP socket comm. step length lt=1.0 δ∈{0.0, 0.3, 0.5, 1.0} 20 runs of the protocol with each value of δ for each instance; c
utoff a run at (100×#jobs) rounds
Experiments
Measure the followings for each instance Opt.Ratio: the ratio of the runs where optimal solutions were f
ound Fes.Ratio: the ratio of the runs where feasible solutions were f
ound Avg/Bst.Quality: the average/best value of the solution qualiti
es Avg.Cost: the average value of the numbers of rounds at which
feasible solutions were found
optimal
feasible
value ofobject function
o
f o
falitySolutionQu
Experiments
Observations The protocol with δ= 0.0 failed to find an optimal solution fo
r almost all of the instances. In the protocol with δ ≠ 0.0, Opt.Ratio, Fes.Ratio, and Avg.C
ost were obviously improved while Avg/Bst.Quality was kept at a “reasonable” level (average > 86%, best > 92%).
In 3 out of 6 complete-topology instances, an optimal solution was never found at any value of δ.
For many instances, increasing the value of δ may generally have an effect to rush the agents into reaching a compromise.
Conclusion
I have presented Generalized mutual assignment problem Distributed lagrangean relaxation protocol Noise strategy that makes the agents quickly agree on a fea
sible solution with reasonably good quality Future work
More sophisticated techniques to update μ The method that would realize distributed calculation of an u
pper bound of the optimal value.