Heuristic Algorithms for Multiobjective Combinatorial Optimization - - PDF document

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HEURISTIC OPTIMIZATION

Heuristic Algorithms for Multiobjective Combinatorial Optimization

Adapted from a tutorial by Lu´ ıs Paquete 1 / 36

Introduction

Multiobjective Combinatorial Optimization Problems (MCOPs)

I Many real-life problems are multiobjective

• Logistics and transportation
• Timetabling and scheduling
• ... and many others

I But most MCOPs are NP-hard and intractable

How to design and analyze SLS algorithms for MCOPs?

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Train roundtrip through capitals of German federal states

The fastest roundtrip:

I take only ICE trains

The cheapest roundtrip:

I take only local trains

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The fastest The cheapest cost time

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The fastest The cheapest cost time

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The fastest The cheapest cost time

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cost time

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cost time

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Multiobjective Combinatorial Optimization Problem The set X of feasible solutions is finite and its elements have some combinatorial property (graph, tree, path, partition, etc.). The goal is to min

x2X f(x) = (f1(x), . . . , fQ(x)) I The objective function f maps x 2 X to RQ

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I Optimality depends of the decision maker’s preferences

(or lack of them).

I Pareto-optimality is based on component-wise order :

u  v ( ) u 6= v and ui  vi, i = 1, . . . , Q

I A solution x 2 X is efficient iff @ x0 2 X s.t. f(x0)  f(x) I Efficient set is the set of all efficient solutions I Nondominated set is the image of the efficient set in f

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cost time

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MCOPs and Solution Methods

I Most MCOPs are NP-hard

Decision version of MCOP (MCOP-D) [Serafini 1986]: Given z = (z1, . . . , zQ), does there exist a solution x 2 X s.t. f(x)  z or f(x) = z?

• 1. If the single-objective problem is NP-complete, then the

corresponding MCOP-D is also NP-complete.

• 2. If the single-objective problem is solvable in polynomial time,

the corresponding MCOP-D may still be NP-complete.

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Solution Methods to MCOPs

I Enumeration Methods

• Multiobjective Branch & Bound
• Multiobjective Dynamic Programming

I Scalarized Methods

• Solving several related single-objective problems
• Weighted Sum, ✏-constraint, etc.

I SLS Algorithms

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Weighted Sum

I min x2X Q

X

i=1

ifi(x)

I gives a search direction I An optimal solution with

> 0 is efficient.

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Weighted Sum

I min x2X Q

X

i=1

ifi(x)

I gives a search direction I An optimal solution with

> 0 is efficient.

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SLS Algorithms

SLS Algorithm design challenges for MCOPs

I How to attain more than one solution? I How to attain high quality solutions? I How to evaluate performance?

Rule of thumb

I Closeness to the nondominated set I Well-distributed outcomes I The more, the better

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Scalarized Acceptance Criterion (SAC) Model

I Weighted Sum

f (x) =

Q

X

i=1

ifi(x)

I Weighted Chebycheff

f (x) = max

i=1,...,Q

• i | fi(x) yi |
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SAC Search Model

————————————– input: weight vectors Λ for each 2 Λ do x is a candidate solution x0 = SolveSAC(x, ) Add x0 to Archive Filter Archive return Archive ————————————–

cost time

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SAC Search Model

———————————— input: weight vectors Λ for each 2 Λ do x is a candidate solution x0 = SolveSAC(x, ) Add x0 to Archive Filter Archive return Archive ———————————— AAA

I Search Strategy I Number of Scalarizations I Intensification Mechanism I Neighborhood

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SAC Search Model – EMO ————————————— input: candidate solution set Xn repeat Xr = Reproduce/Mutate(Xn) R = Rank(Xr, Xn) Xs = Select(Xr, Xn, R) Xn = Replace(Xs) return Xn —————————————

cost time

3 3 3 2 2 2 2 1 1 1 1 20 / 36

SAC Search Model – EMO ————————————— input: candidate solution set Xn repeat Xr = Reproduce/Mutate(Xn) R = Rank(Xr, Xn) Xs = Select(Xr, Xn, R) Xn = Replace(Xs) return Xn —————————————

cost time

5 5 6 3 2 2 3 1 1 1 1 21 / 36

SAC Search Model – EMO —————————————– input: candidate solution set Xn repeat Xr = Reproduce/Mutate(Xn) R = Rank(Xr, Xn) Xs = Select(Xr, Xn, R) Xn = Replace(Xs) return Xn —————————————– AA

I Component-wise order I Closeness I Performance indicators

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Multiobjective Local Search

——————————————————————– input:candidate solution x while x is not a local optimum do choose a neighbor x0 from x such that f(x0)  f(x) x = x0 return x ——————————————————————–

I What if f(x0) and f(x) are mutually nondominated? I How to obtain more than a single solution?

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CWAC Search Model ———————————— input: candidate solution x Add x to Archive repeat Choose x from Archive XN = Neighbors(x) Add XN to Archive Filter Archive until all x in Archive are visited return Archive ————————————

cost time

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CWAC Search Model ———————————— input: candidate solution x, ✏ Add x to Archive repeat Choose x from Archive XN = Neighbors(x) Add XN to Archive Filter Archive according to ✏ until all x in Archive are visited return Archive ———————————— Archive bounding [Angel et al. 2004]

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Hybrid Search Model ———————————— input: weight vectors Λ for each 2 Λ do x is a candidate solution x0 = SolveSAC(x, ) X 0 = CW(x0) Add X 0 to Archive Filter Archive return Archive ————————————

cost time

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Performance Assessment

Rules of Thumb: An algorithm performs better if

I It is closer to the nondominated set I It has better distributed outcomes I It has more solutions

Indicators of Performance

I Measure some property of the outcomes I Most of the indicators have limitations

[Knowles & Corne 2002, Zitzler et al. 2003]

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cost time

Many runs of Algorithms Blue and Red

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Another example

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I Better relations

[Hansen & Jaszkiewicz 1998, Zitzler et al. 2003]

cost time

Blue and Red are incomparable

cost time

Blue is better than Red

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I Unary Indicator: Hypervolume

[Zitzler and Thiele, 1998] H(B) = 45 H(W ) = 25 B is better than W = ) H(B) > H(W )

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I Unary Indicator: Hypervolume

[Zitzler and Thiele, 1998] H(B) = 37 H(W ) = 36 H(B) > H(W ) = ) B is not worse than W

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I Attainment Functions [V.G. da Fonseca et al. 2001]

AF: Prob. that an outcome set is better or equal to z. EAF: How many runs an outcome set is better or equal to z?

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I Attainment functions – Visualization of differences

EAFBlue EAFRed positive differences negative differences

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I Attainment functions – Statistical testing

K-S test statistic: max | EAFBlue EAFRed | positive differences negative differences

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References

I Textbooks: R.E. Steuer 1986, K. Miettinen 1999, M. Ehrgott 2005,

V.T’kindt et al. 2002, K. Deb 2002.

I Reviews: M. Ehrgott and X. Gandibleux 2000, 2002, 2004, 2009,

C.C. Coello 2000, D. Jones et al. 2002, J. Knowles and D. Corne 2004, L. Paquete and T. St¨ utzle 2007.

I Complexity and Approximation: P. Hansen 1979, P. Serafini 1986,

• M. Ehrgott 2000, C.H. Papadimitriou and M. Yannakakis 2000,
• E. Angel et al. 2007.

I Performance Assessment: E. Zitzler et al. 2003, 2008, V.G. da

Fonseca et al. 2001, 2010, M. Lop´ ez-Ib´ a˜ nez et al. 2010.

I Web material: PISA (http://www.tik.ethz.ch/~sop/pisa),

MOMH (http://home.gna.org/momh), ParadisEO (http://paradiseo.gforge.inria.fr)

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