Stable Matching-Based Selection
in Evolutionary Multiobjective
Optimization
Sam Kwong
Department of Computer Science
City University of Hong Kong
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Contents
◦ Introduction
◦ Stable Matching-Based selection
◦ Two-Level Stable Matching-Based selection
◦ Adaptive Two-Level Stable Matching-Based selection
◦ Conclusion
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Contents
◦ Introduction
◦
◦
◦
◦
Multiobjective optimization problems
Pareto dominance
Selection in EMO
MOEA/D
◦ Stable Matching-Based selection
◦ Two-Level Stable Matching-Based selection
◦ Adaptive Two-Level Stable Matching-Based selection
◦ Conclusion
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Introduction
◦ Multiobjective optimization problem (MOP)
◦ 𝑚𝑖𝑛𝑖𝑚𝑖𝑧𝑒/𝑚𝑎𝑥𝑖𝑚𝑖𝑧𝑒 𝐹 𝑥 = [𝑓1 𝑥 , 𝑓2 𝑥 , … , 𝑓𝑚 𝑥 ]
◦ 𝑠𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜 𝑥 ∈ Ω
◦ Most often, the objectives cannot be optimized at the same time
4
Real world MOPs
◦ Multiobjective traveling salesman problem (TSP)
A salesman need to travel from the origin city to a couple of cities to
meet customers and then return back to the origin city. Each city need
to be visited exactly once. There are different ways to travel from each
city to another, each of which takes different travelling times and
different costs. The sequence to visit each city could be arbitrary.
◦ The goal of the planning is to minimize the travel time and travel cost
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Real world MOPs
◦ Multiobjective traveling salesman problem
◦ Obviously, different visiting sequence result in significance difference in traveling
time and cost
◦ Even if the visiting sequence is fixed, different transportation may result in
different objectives
Low cost
Long traveling time
Short traveling time
High cost
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Real world MOPs
◦ Antenna design
There is growing interest for small antennas that concurrently have
higher functionality and operability. Designers are interested in
antennas with higher gain and larger bandwidth.
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Real world MOPs
◦ Antenna design
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Pareto Dominance
◦ Pareto Dominance
◦ 𝑚𝑖𝑛𝑖𝑚𝑖𝑧𝑒 𝐹 𝑥 = [𝑓1 𝑥 , 𝑓2 𝑥 ]
Solution 1 dominates solution 2
2
1
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Pareto Dominance
◦ Pareto Dominance
◦ 𝑚𝑖𝑛𝑖𝑚𝑖𝑧𝑒 𝐹 𝑥 = [𝑓1 𝑥 , 𝑓2 𝑥 ]
Solution 1 and solution 2 cannot
be compared
They are nondominated with each
other
1
2
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Pareto Dominance
◦ Pareto Dominance
◦ 𝑚𝑖𝑛𝑖𝑚𝑖𝑧𝑒 𝐹 𝑥 = [𝑓1 𝑥 , 𝑓2 𝑥 ]
Objective space
Pareto font (PF) consists of
the objective vectors of all
Pareto optimal solutions
Pareto optimal solution
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Selection
◦ EA framework
Start
Population initialization
Offspring reproduction
Selection
A set of solutions are randomly
sampled in the search space
Produce a set of offspring solutions
using evolutionary operators
Promising solutions are selected and
survived in the next iteration
Terminate?
End
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Selection in MOEA
◦ Two main requirements
◦ Convergence
◦ Approaching the true Pareto front (PF)
◦ Diversity
◦ Widely distributed along the direction of the PF
Convergence
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Selection in MOEA
◦ Two main requirements
◦ Convergence
◦ Approaching the true Pareto front (PF)
◦ Diversity
◦ Widely distributed along the direction of the PF
Diversity
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MOEA/D
◦ Multiobjective evolutionary algorithm based on decomposition
[1]
◦ Decomposes an MOP into a number of single objective optimization
problems (SOPs) and optimizes them simultaneously
◦ Each single objective optimization problem is defined by a scalarizing function
using a weight vector. In MOEA/D, there are several scalarizing approaches such
as the weighted Tchebycheff, the weighted sum, and the PBI (penalty-based
boundary intersection).
◦ E.g. weighted Tchebychev approach
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MOEA/D
◦ Selection in MOEA/D
◦ Select a solution for each subproblem to minimize the aggregated
single objective
◦ Minimizing the single objective
Convergence
◦ Subproblems are constructed using a set of evenly distributed weight vectors
Diversity
Problem:
When one solution minimizes more
than one subproblems. MOEA/D does
not has a effective mechanism to
select diverse solutions.
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Contents
◦ Introduction
◦ Stable Matching-Based selection
◦ Two-Level Stable Matching-Based selection
◦ Adaptive Two-Level Stable Matching-Based selection
◦ Conclusion
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Stable marriage problem
◦ Stable marriage problem (SMP) [2] is a problem of finding a stable
matching between two sets of matching agents, given each
matching agent has a preference list over all matching agents in
the other set.
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A Prize Winning Algorithm
◦ Lloyd Shapley, Nobel Prize
Winner 2012 in economics,
shared with Alvin E Roth
Photo Bengt Nyman
http://en.wikipedia.org/wiki/File:Lloyd_Shapley_2_2012.jpg
Source: www.cs.uu.nl/docs/vakken/an/an-stablemarriage.ppt
The Stable Marriage Problem
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Stable marriage problem
Not a stable matching
Because 𝑚2 prefers 𝑤3 to 𝑤1
and 𝑤3 prefers 𝑚2 to 𝑚3 .
This kind of situation is
unstable and should be avoid.
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Stable marriage problem
A stable matching solution
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Stable marriage problem
◦ Stable matching algorithm proposed in [2]
Note: the set that propose the matching request should be no more than the other set
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Stable marriage problem
◦ Stable marriage problem (SMP) is a one-to-one matching problem
◦ Matching problem can be generalized into many different
problems
◦ College admission problem/hospitals-residents problem (many-to-one
matching)
◦ Stable roommate problem (only one set of matching agents)
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Stable Matching-Based Selection
◦ Model the selection in MOEA/D as a SMP
◦ Subproblems and solutions are treated as two sets of matching agents
◦ Each subproblem has a preference list over all solutions
◦ Each solution has a preference list over all subproblems
◦ Preference value of 𝑝 on x
◦ ∆p 𝑝, x = 𝑔(𝑥|𝑤, 𝑧 ∗ )
◦ ∆p 𝑝, x is aggregated objective of x for subproblem 𝑝
◦ Subproblem prefers solution that can minimize its single objective
◦ Preference value of x on 𝑝
◦ ∆x x, 𝑝 = 𝐹 𝑥 −
𝑤𝑇𝐹 𝑥
𝑤𝑇𝑤
Convergence
𝑤
◦ ∆x x, 𝑝 is the distance between solution x and subproblem 𝑝 in objective space
◦ Solution prefers subproblem close to itself in objective space
Diversity
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Stable Matching-Based Selection
◦ Model the selection in MOEA/D as a SMP
◦ The selection is achieved by computing a stable matching solution to the
modeled SMP using proposed algorithm in [2]
◦ Since the preferences are defined considering convergence and diversity
◦ A stable matching solution to the SMP is regarded as a balance between
convergence and diversity
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Stable Matching-Based Selection
◦ MOEA/D-STM [3]
◦ K. Li, Q. Zhang, S. Kwong, M. Li, and R.Wang, Stable matching-based
selection in evolutionary multiobjective optimization. IEEE Trans. Evol.
Comput., vol. 18, no. 6, pp. 909-923, Dec. 2014.
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Contents
◦ Introduction
◦ Stable Matching-Based selection
◦ Two-Level Stable Matching-Based selection
◦ Adaptive Two-Level Stable Matching-Based selection
◦ Conclusion
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Stable Matching-Based Selection
◦ Problem
◦ A stable matching solution cannot always ensure that subproblem select a
solution close to itself
◦ Experiments [4] show that for MOP test instances, MOEA/D-STM cannot maintain
diverse solutions over the entire Pareto front
File result of MOEA/D-STM
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Stable Matching-Based Selection
◦ E.g.
Subproblems’ preference lists
Solutions’ preference lists
Reason:
Some part of Pareto front can be located
easily, while other part is not. All
subproblems prefer solutions close to
the PF.
Outcome:
Stable matching cannot ensure that the
selected solutions are matched to their
closest subproblems.
Diversity of the population is destroyed.
Stable matching based selection result
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Stable marriage problem with incomplete
lists
◦ Very common for an SMP
◦ Some men or women refuse to be matched to some candidates. Then they
just remove these candidates in their preference lists. A matching will never
to made to these rejected candidates.
◦ These kind of SMP is referred as stable marriage problem with incomplete
lists (SMP-IC)
◦ A matching agent does not need to have a complete preference list over all
matching agents in the other sets
◦ A matching agent can only be matched to one of the matching agents present in
its preference list
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Stable marriage problem with incomplete
lists
◦ So if we only keep a few most preferred subproblems on a
solution’s preference list, then
◦ A solution is only allowed to be matched to its one of most preferred
subproblems
◦ We can ensure that all selected solutions are matched to its closest
subproblems after stable matching
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Stable marriage problem with incomplete
lists
◦ E.g.
Subproblems’ preference lists
Solutions’ preference lists
The length of solution’s preference list is
reduced to 2.
Only the top two preferred subproblems
remained.
Diverse population is selected.
Stable matching with incomplete preference lists
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Stable marriage problem with incomplete
lists
Note: a stable matching for an SMP-IC might not match a stable solution for all subproblems
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Two-Level Stable Matching-Based Selection
◦ Frist Level (SMP-IC)
◦ The preference lists of solutions are set
to a limited length 𝑟
◦ Only the 𝑟 most preferred subproblems
remains in solution’s preference list
Subproblems
with complete
preference list
Solutions with
incomplete
preference list
First level
stable matching
◦ Second level (original SMP)
◦ Since the stable matching of a SMP-IC
might not match a stable solution for
all subproblems
◦ The remaining subproblems and
solutions are matched using complete
preference lists
Matched
subproblems
and solutions
Unmatched
subproblems
and solutions
with complete
preference list
Second level
stable matching
34
Two-Level Stable Matching-Based Selection
◦ MOEA/D-STM2L [6]
◦ M. Wu, S. Kwong, Q. Zhang, K. Li, R.
Wang and B. Liu, "Two-Level Stable
Matching-Based Selection in MOEA/D,"
Systems, Man, and Cybernetics, 2015
IEEE International Conference on, pp.
1720-1725, 2015.
◦ r is set to be 8 for UF test instances and
4 for MOP test instances
◦ Provides promising performance for
MOP test instances
◦ Inherits the main ability of MOEA/DSTM for UF test instances
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Contents
◦ Introduction
◦ Stable Matching-Based selection
◦ Two-Level Stable Matching-Based selection
◦ Adaptive Two-Level Stable Matching-Based selection
◦ Conclusion
36
Adaptive Two-Level Stable Matching-Based
Selection
◦ Parameter sensitivity studies of r
◦ For some test instances, performance is very sensitive to the setting of r
◦ If r is too large, the advantage of incomplete lists degenerates
◦ If r is too small, diversity is paid on too much attention while the convergence is
affected
◦ r is a problem dependent parameter
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Adaptive Two-Level Stable Matching-Based
Selection
◦ E.g.
Different settings of r affect
the selection.
Larger r, better convergence;
Smaller r, better diversity.
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Adaptive Two-Level Stable Matching-Based
Selection
◦ MOEA/D-ASTM (current work)
◦ An adaptive approach for setting 𝑟 is proposed
◦ Local domination information is extracted from the distribution of current
population to help determine 𝑟 for each single solution adaptively
◦ Experimental studies show that proposed approach is effective for different
test problems
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Contents
◦ Introduction
◦ Stable Matching-Based selection
◦ Two-Level Stable Matching-Based selection
◦ Adaptive Two-Level Stable Matching-Based selection
◦ Conclusion
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Conclusion
◦ Stable marriage model provides a different perspective to handle the
selection in EMO.
◦ The requirements of convergence and diversity are represented in the
preferences between subproblems and solution.
◦ Experimental studies proved its effectiveness
◦ Two-level stable matching-based selection ensures the diversity by
limiting the preference lists of solutions without much influence on
convergence
◦ Future work:
◦ For some extreme situations (e.g. UF10), the convergence ability of the stable
matching based-selection still needs to be further enhanced
◦ For many-objective optimization problems, the effectiveness of stable matching
based-selection needs to be further studied
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References
◦ [1] Q. Zhang, and H. Li, MOEA/D: A Multiobjective evolutionary algorithm
based on decomposition. IEEE Trans. Evol. Comput., vol. 11, no. 6, pp. 712-731,
Dec. 2007.
◦ [2] D. Gale and L. S. Shapley, “College admissions and the stability of
marriage,” Amer. Math. Mon., vol. 69, no. 1, pp. 9–15, 1962.
◦ [3] K. Li, Q. Zhang, S. Kwong, M. Li, and R.Wang, Stable matching-based
selection in evolutionary multiobjective optimization. IEEE Trans. Evol.
Comput., vol. 18, no. 6, pp. 909-923, Dec. 2014.
◦ [4] H. Liu, F. Gu, and Q. Zhang, Decomposition of a multiobjective optimization
problem into a number of simple multiobjective subproblems. IEEE Trans. Evol.
Comput., vol. 18, no. 3, pp. 450-455, Jun. 2014.
◦ [5] K.Iwama , D.Manlove, S.Miyazaki, and Y. Morita, Stable marriage with
incomplete lists and ties. ICALP, vol. 99, pp. 443-452, July, 1999.
◦ [6] M. Wu, S. Kwong, Q. Zhang, K. Li, R. Wang and B. Liu, "Two-Level Stable
Matching-Based Selection in MOEA/D," Systems, Man, and Cybernetics, 2015
IEEE International Conference on, pp. 1720-1725, 2015.
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Thank you
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