ANALYSIS OF TWO ALGORITHMS FOR MULTI-OBJECTIVE MIN-MAX OPTIMIZATIONSimone Alicino

Prof. Massimiliano Vasile

Department of Mechanical and Aerospace Engineering

University of Strathclyde, Glasgow, UK

BIOMA 2014

13th September 2014, Ljubljana, Slovenia

2

Model of the System

d, u f(d,u)

f

u

Bel/Pl

f

Design under uncertainty

u

pdf

aleatory

m1

m2

m3

u

bbaepistemic

3

Belief and Plausibility that f(u) < ?

Bel(f < ) = m(q1) + m(q2) = 0.4

Pl(f < ) = m(q1) + m(q2) + m(q3) = 0.8

q1 q2 q4

m(q1) = 0.3 m(q3) = 0.4 m(q4) = 0.2

q3

m(q2) = 0.1

Evidence theory

4

MACS2• Population-based search• Pareto ranking + Tchebycheff scalarization• Exploitation: sampling of neighborhood• Exploration: differential evolution• ArchiveIDEA• Population-based search• Hybrid DE + MBH• Improves local convergence• Avoids stagnation

CROSS-CHECKS• To rank and check and • Increase prob. of finding global maxima

Methodology

ν𝑚𝑎𝑥=mind∈𝐷 [maxu∈𝑈

𝑓 1 (d ,u ) ,maxu∈𝑈

𝑓 2 (d , u ) ,…,maxu∈𝑈

𝑓 𝑞 (d ,u ) ]MACS2ν𝑚𝑎𝑥=min

d∈𝐷[ 𝑓 𝑚𝑎𝑥

1 (d , u ) ,…, 𝑓 𝑚𝑎𝑥𝑞 (d ,u ) ]

d 𝑖

IDEA

5

Individualistic Actions

Subproblem selection

Initialization

Cross-Check

Cross-Check

Archive resize

Validation

Social Actions

All other individuals

Social individuals

Min-Max Selection

Cross-Check

Min-Max Selection

SOCIAL ACTIONS Child generated by interaction (DE) of agents with neighbours or global archive.

SUBPROBLEM SELECTIONupdate of the composition of

the social population and their associated scalar

subproblems.

GLOBAL ARCHIVEan external repository in

which non-dominated solutions are stored. The

archive is kept below a maximum size.

INDIVIDUALISTIC ACTIONS Child generated by random moves (pattern search) of each agent.

INITIALIZATIONInitial population is randomly

generated (LHS) in the search domain D.

MACS

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MACS: Cross-check

f2

f1

𝑓 (d ,u )

𝑓 𝑎(d 𝑎 ,u 𝑎)

𝑓 12(d ,u 𝑎)

𝑓 21(d 𝑎 , u)

If agent in the population dominates or is dominated by the archive

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MACS: Min-max selection

f

Uunewu

f

Dd dnew

otherwise

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Run global optimization over U

until

MACS: Validation

f2

f1

𝑓 𝑙= 𝑓 𝑙

𝑓 2

9

SO GLOBAL MAXIMIZATION Performed by IDEA on u space, for each di solution of MO global minimization

ARCHIVE MAXIMAStore in U-archive solution of

IDEA only if it is better than solution of MACS2

(maximization might fail to find global optimum)

FINAL CROSS-CHECKLocal search to refine accuracy of U-archive

MO GLOBAL MINIMIZATION Performed by MACS2 on d space, and uses u’s stored in U-archive (internal cross-check).

INITIALIZATIONInitial population is randomly

generated (LHS) andU-archive is initialized.

MACSminmax

SO maximizationsIDEA

Archive min solutions

Archive max solutions

MO minimizationMACS2

Initialization

Cross-Check

Cross-Check

Dominance

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MACSminmax: restoration

Archived maximum

Candidate minimum in d

Solution

Selected minimum in d

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MACSminmaxMACSν

Individualistic Actions

Subproblem selection

Initialization

Cross-Check

Cross-Check

Archive resize

Validation

Social Actions

Min-Max Selection

Cross-Check

Min-Max Selection

SO maximizationsIDEA

Archive min solutions

Archive max solutions

MO minimizationMACS2

Initialization

Cross-Check

Cross-Check

Dominance

Comparison

Global vs. local search, same purpose: make sure that each d is associate to a global

maximum u

Local search vs. cross-checkfor every agent of the minimization

Both implement similar mechanisms to

increase probability of archiving global

maxima

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Performance metrics

• Convergence

• Spreading

• Success rate

𝑀 𝑠𝑝𝑟=1𝑀𝑝

∑𝑖=1

𝑀𝑝

min𝑗∈𝑁 𝑝

100‖ 𝑓 𝑗−𝑔𝑖

𝑔𝑖‖

𝑀 𝑐𝑜𝑛𝑣=1𝑁𝑝

∑𝑖=1

𝑁 𝑝

min𝑗∈𝑀 𝑝

100‖𝑔 𝑗− 𝑓 𝑖𝑔 𝑗

‖

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Test cases

Settings

MACS2• 200n function evaluations• 10 agents• 5 (1/2) social agents• F = 1• CR = 0.1

IDEA• 200n function evaluations• 5 agents• F = 1• CR = 0.1

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Test case 1𝑓 1=∑

𝑖=1

𝑛=2

𝑑𝑖𝑢𝑖2

𝑓 2=∑𝑖=1

𝑛=2

(5−𝑑𝑖 ) (1+cos𝑢𝑖 )+(𝑑𝑖−1 ) (1+sin𝑢𝑖 )

Max f1 Max f2 Mconv Mspr pconv / tconv pspr / tspr

MACS 100% 100% 0.2 1.7 100 / 0.5 79 / 2

MACSminmax 100% 100% 0.2 1.3 100 / 0.5 100 / 2

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Test case 2𝑓 1=∑

𝑖=1

𝑛=8

(𝑑𝑖−𝑢𝑖 )2

𝑓 2=∑𝑖=1

𝑛=8

(2𝜋 −𝑢𝑖 )cos (𝑢𝑖−𝑑𝑖 )

Max f1 Max f2 Mconv Mspr pconv / tconv pspr / tspr

MACS 100% 65% 0.5 16.1 100 / 1 0 / 2

MACSminmax 100% 60% 0.6 2.0 100 / 1 64 / 2

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Test case 3𝑓 1=∑

𝑖=1

𝑛=8

(𝑑𝑖−𝑢𝑖 )2

𝑓 2=∑𝑖=1

𝑛=8

(𝑢𝑖−3𝑑𝑖 ) sin𝑢𝑖+(𝑑𝑖−2 )2

Max f1 Max f2 Mconv Mspr pconv / tconv pspr / tspr

MACS 100% 100% 0.6 7.5 46 / 0.5 3 / 2

MACSminmax 100% 100% 0.1 0.3 100 / 0.5 100 / 2

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Test case 4𝑓 1=∑

𝑖=1

𝑛=2

(2𝜋−𝑢𝑖 )cos (𝑢𝑖−𝑑𝑖 )

𝑓 2=∑𝑖=1

𝑛=2

(𝑑𝑖−𝑢𝑖 ) cos (3𝑑𝑖−5𝑢𝑖 )

Max f1 Max f2 Mconv Mspr pconv / tconv pspr / tspr

MACS 100% 91.3% 0.3 0.9 83 / 0.5 97 / 2

MACSminmax 100% 85.7% 0.4 1.0 77 / 0.5 91 / 2

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Test case 5𝑓 1=∑

𝑖=1

𝑛=4

(2𝜋−𝑢𝑖 )cos (𝑢𝑖−𝑑𝑖 )

𝑓 2=∑𝑖=1

𝑛=4

(𝑢𝑖−3𝑑𝑖 ) sin𝑢𝑖+(𝑑𝑖−2 )2

Max f1 Max f2 Mconv Mspr pconv / tconv pspr / tspr

MACS 98.6% 54.1% 1.2 5.8 48 / 1 60 / 6

MACSminmax 92.8% 87.6% 2.7 8.0 24 / 1 42 / 6

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Test case 6𝑓 1=∑

𝑖=1

𝑛=1

(𝑑𝑖+𝑢𝑖 )cos [3𝑑𝑖−𝑢𝑖 (5|𝑑|+5 ) ]

𝑓 2=∑𝑖=1

𝑛=1

(𝑑𝑖−𝑢𝑖 )cos (3𝑑𝑖−5𝑢𝑖 )

Max f1 Max f2 Mconv Mspr pconv / tconv pspr / tspr

MACS 100% 100% 0.3 1.2 95 / 0.5 97 / 2

MACSminmax 100% 100% 0.3 2.0 91 / 0.5 63 / 2

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Test case 7

Max f1 Max f2 Max f2 Mconv Mspr pconv / tconv pspr / tspr

MACS 100% 100% 95.3% 5.0 9.3 50 / 5 8 / 5

MACSminmax 100% 100% 98.3 4.6 2.1 66 / 5 100 / 5

𝑓 1=∑𝑖=1

𝑛=8

(𝑑𝑖−𝑢𝑖 )2

𝑓 2=∑𝑖=1

𝑛=8

(2𝜋 −𝑢𝑖 )cos (𝑢𝑖−𝑑𝑖 )

𝑓 2=∑𝑖=1

𝑛=8

(𝑢𝑖−3𝑑𝑖 ) sin𝑢𝑖+(𝑑𝑖−2 )2

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Conclusions

• Worst-case design– Evidence Theory to model epistemic uncertainty– Maximization of Belief function: worst-case scenario design

• Two multi-objective algorithms– Cross-checks to increase probability to find global maximum– MACS: bi-level algorithm, modification of MACS2– MACSminmax: restoration methodology, works with any MO/SO algorithm

• Test cases– 6 bi- and 1 three- objective cases, with different dimensions and complexity– Global fronts identified, with good to excellent accuracy– Comparable performance between MACS and MACSminmax

• Limitations– Limited number of cases, objectives, and dimensions– Test suite: neither fronts, nor global maxima analytically known (difficult to

assess performance)

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Belief and Plausibility that f(u) < ?

Evidence theory

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1. Worst-case solution (Bel = 1) (best d that gives the minimum of the maxima of f over u)• Above this point the design

is certainly feasible given the current information.

1

2

Bel 3

Pl

Computational approach

2. Best possible solution (Pl = 0) • Below this point the design

is certainly not possible

3. Belief and Plausibility of every intermediate solution between best and worst• Trade-off curve

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Results

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Fronts