Do Additional Objectives Make a Problem Harder? · Computer Engineering and Networks Laboratory Do...
Transcript of Do Additional Objectives Make a Problem Harder? · Computer Engineering and Networks Laboratory Do...
Computer EngineeringComputer Engineeringand and NetworksNetworks LaboratoryLaboratory
Do Additional Objectives Make a Problem Do Additional Objectives Make a Problem Harder?Harder?
Dimo Brockhoff, Tobias Friedrich,Dimo Brockhoff, Tobias Friedrich,Nils Hebbinghaus, Christian Klein,Nils Hebbinghaus, Christian Klein,
Frank Neumann, Eckart ZitzlerFrank Neumann, Eckart Zitzler
GECCO 2007GECCO 2007, 11 JulyJuly 20072007
Do Additional Objectives Make a Problem Harder? 2© Systems Optimization Group ETH Zurich
Starting Point
Most optimization problems are multiobjective in nature…→ Include as many objective as possible would be desirable
~1995: formulated as single-objective problem
~2005: formulated as two- or three-objective problem
Today: many-objective problems
Are the algorithms bad or are the problems more difficult or both?
We haven't understood yet how additional objectives affects the problem complexity!
Wagner et al. (2007)
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Adding Objectives: What Is Known Today…
problems may become harder
Fonseca and Fleming (1995), Deb (2001), Coello et al. (2002), and others:
conflicts between objectivesPareto front size # incomparable solutions
Winkler (1985):theoretical work for random objectives
problems may become easier
Knowles et al. (2001):multiobjectivization
Jensen (2004):helper-objectives
Scharnow et al. (2002), Neumann and Wegener (2006):
theoretical investigations2D faster than 1Ddecomposition
Statements are contradictory: some studies say that…
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Given:single-objective problemalgorithm for single- and multiobjective optimization: SEMO - (1+1)EA
Now:add an objective, but keep the search space the same
Question:What is the runtime complexity to generate all Pareto optima in comparison to the original problem?
Note:single-objective optimum is part of the Pareto-optimal set
Question Addressed In This Study
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General Considerations On Adding Objectives
Weak Pareto-dominance relation illustrated as relation graph
comparable
incomparable
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A Simple Evolutionary Multiobjective Optimizer
initially 1 solution
all solutions in P incomparablestandard bit mutation
with 1 objective: SEMO ≅ (1+1)EA
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Overview of the Problem Scenario & Basic Idea
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Adding ONEMAX
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Adding ONEMAX (Proof)
|Population| Initial point :
to reach Initial point :
SEMO produces trade-off solutionsexpected steps
necessary to increase maximal number of ones by one
Pareto-optimal
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Adding ZEROMAX
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Wait a minute…
Results:Added simple problem to a more difficult problem→ combination less complex ( + → )Other perspective: added difficult problem to an easy one→ more complex ( + → )
Question:What happens if we combine two equally complex problems?
In other words:Combination more complex or even less complex?
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Combining Objectives: Easier
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Harder already shown: Laumanns et al. (2002)
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Conclusions
General result:gained general understanding of how problem structure changes with
additional objectives (introducing plateaus, deceptive/helping)Introducing incomparable solutions is not always a problemPerformance of EA depends on the objectives chosen
Specific Results:Running time analyses for simple EAs
One and the same plateau function can become harder and easier with an additional objectiveExample, where the combination of two problems as a bi-objective problem is easier than solving the single problems individually
All proofs also for an asymmetric mutation operator
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Making comparable solutions incomparable
modified LOTZ with additional function (iii)2-dimensional modified LOTZ
modified LOTZ with additional function (iv)
More Objectives…
Brockhoff and Zitzler: "Offline and Online Objective Reduction in Evolutionary Multiobjective
Optimization Based on Objective Conflicts",TIK report no. 269, May 2007
www.tik.ee.ethz.ch/sop/