MULTIMODAL OPTIMIZATION MIKE PREUSS. Multimodal Optimization 1 - - PowerPoint PPT Presentation

Multimodal Optimization Mike Preuss. 1 2014-09-14 Mike Preuss 2014-09-14

MIKE PREUSS.

MULTIMODAL OPTIMIZATION

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WHAT ARE WE DEALING WITH?

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• more questions t han answers in Mult imodal Opt imizat ion (MMO)
• field not well defined
• basic t erms not well defined
• similarit ies t o Mult i-Obj ect ive Opt imizat ion (MOO)
• huge bulk of lit erat ure
• Evolut ionary Comput at ion (EC) people focus on EC approaches
• consider t his as “ request for comment s”
• suggest ions for fut ure work appreciated
• bet t er: you st art t o do int erest ing MMO st uff

SOME GENERAL NOTES

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• why mult imodal opt imizat ion (MMO)?
• abst ract ion: niching and a model EA
• different scenarios and t heir measures
• t axonomy of met hods
• result s/ compet it ion/ soft ware
• t he fut ure

OUTLINE

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why multimodal optimization (MMO)?

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In a mult imodal opt imizat ion t ask, t he main purpose is t o f ind mult iple opt imal solut ions (global and local), so t hat t he user can have a bet t er knowledge about dif f erent opt imal solut ions in t he search space and as and when needed, t he current solut ion may be swit ched t o anot her suit able opt imum solut ion.

Deb, Saha: Multimodal Optimization Using a Bi-Objective Evolutionary Algorithm, ECJ, 2012

main t asks:

• alt ernat ive solut ions
• problem knowledge

ATTEMPTING A DEFINITION

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• many solutions invalid, looks like Rastrigin problem

Henrich, Bouvy, Kausch, Lucas, Preuss, Rudolph, Roosen. Economic optimization of non- sharp separation sequences by means of evolutionary algorithms. In Comput ers & Chemical Engineering, Volume 32, Issue 7, pp. 1411-1432. Elsevier, 2008.

REAL-WORLD EXAMPLES

SEPARATION PROCESS OPTIMIZATION

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Rudolph, Preuss, Quadflieg. Two-layered surrogate modeling for tuning metaheuristics. In ENBIS / EMS E Conference Design and Analysis of Comput er Experiment s, 2009

LINEAR-JET OPTIMIZATION

REAL-WORLD EXAMPLES

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Preuss, Burelli, Y

• annakakis. Diversified Virtual Camera Composition. In EvoApplications

2012, pp. 265-274. S pringer, 2012

REAL-WORLD EXAMPLES

CAMERA POSITIONING

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• in which sit uat ions are MMO met hods act ually bet t er t han

“ usual” EC opt imizat ion algorit hms?

• problems
• performance measures
• ext ernal condit ions, e.g. runt ime
• among different MMO met hods, which one shall we choose?
• what are t he limit s for furt her improvement ?

assumption: successful MMO needs dist ribut ion of solut ions int o different basins of at t raction, t his resembles t he niching idea

MAIN RESEARCH QUESTIONS

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abstraction: niching and a model EA

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“ Niching in EAs is a t wo-st ep procedure t hat a) concurrent ly or subsequent ly dist ribut es individuals ont o dist inct basins of at t ract ion and b) facilit at es approximat ion of t he corresponding (local) opt imizers.” (Preuss, BIOMA 2006)

NICHING

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NICHING/SPECIATION

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OPTIMIZATION PHASES

Redundancy for repeat ed local search and b basins (Beasley 1993):

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BASIN IDENTIFICATION/BASIN RECOGNITION

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BASIN IDENTIFICATION

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BASIN RECOGNITION

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• basin ident ificat ion relies on det ect ing if t wo solut ions are

locat ed in t he same basin (binary)

• basin recognit ion: is t he basin of a cert ain solut ion known?
• no perfect knowledge: probabilist ic approach
• t hese express sensit ivit y (we do not have informat ion about

unvisit ed areas)

PROBABILISTIC IDENTIFICATION/RECOGNITION

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SIMULATION

• quest ion: how many local searches necessary t o find t he global
• pt imum (t 2), or
• or t o visit all basins at least once (t 3)?

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given a set of 8 collect or’s cards, and we randomly get 3,

• how many it erat ions unt il we get one specific card?

(2.67)

• or obt ain all exist ing cards?

(6.58 it erat ions)

COUPON COLLECTOR‘S PROBLEM (CCP)

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• under t he assumpt ion of equal probabilit ies (for single

cards/ basins), t his can be comput ed

• formula of (S

tadj e. The collector’s problem with group drawings. Advances in Applied Probability, 22(4):866– 882, 1990):

• b = cards/ basins per drawing,
• c = number of cards/ basins
• n = desired element s of desired set , l = desired set size

P(BI) = 1, P(BR) = 0

EXACT RESULTS

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EXACT RESULTS

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• under t he equal basin size assumpt ion, obt aining t he global
• pt imum (t 2) needs on average b local searches!
• so basin ident ificat ion does not make sense?

but :

• what about basin recognit ion?
• equal basin sizes not realist ic
• we cannot know if we have reached t 2
• sit uat ion changes if we want mult iple solut ions

THIS IS SHOCKING!

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SUMMARIZING THE SIMPLE CASES

• we leave out perfect BR, no BI, seems unreasonable
• even under ideal circumst ances, not much gain for t 2
• but BI/ BR help for t 3:
• rat ionale for mult imodal opt imizat ion
• more complex cases (unequal basin sizes, PBI/ PBR not 0 or 1)

have to be simulat ed

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SIMULATION: EQUAL BASIN SIZES

P(BI) = 0, P(BR) = 0

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SIMULATION: EQUAL BASIN SIZES

P(BI) = 0.5, P(BR) = 0

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SIMULATION: EQUAL BASIN SIZES

P(BI) = 1, P(BR) = 0 (t his is t he t heoret ically t ract able case, t he difference comes from inst ant st opping when reaching t 2)

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SIMULATION: EQUAL BASIN SIZES

P(BI) = 0.5, P(BR) = 0.5

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• why should we care?
• because size differences grow exponent ially in dimensions
• 10D wit h 2:1 per dim makes a volume difference of 1024:1
• however, basin ident ificat ion/ basin recognit ion may be very

difficult wit h large size differences

• we simulat e abst ract 1:10 size difference

UNEQUAL BASIN SIZES?

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SIMULATION: UNEQUAL BASIN SIZES

P(BI) = 0, P(BR) = 0

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SIMULATION: UNEQUAL BASIN SIZES

P(BI) = 0.5, P(BR) = 0

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SIMULATION: UNEQUAL BASIN SIZES

P(BI) = 1, P(BR) = 0

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SIMULATION: UNEQUAL BASIN SIZES

P(BI) = 0.5, P(BR) = 0.5

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• t here are limit s t o possible improvement s
• for equal basin sizes, t2 cannot really be improved
• t 3 can be improved a lot
• for unequal basin sizes, t 2 and t 3 are improved by BI/ BR
• basin recognit ion (needs archive) is more import ant t han basin

ident ificat ion

MODEL EA FINDINGS

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different scenarios and their measures

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• ne-global: looking for t he global opt imum only

all-global: find all preimages of t he global opt imum

• t he problems of t he CEC 2013 niching compet it ion belong here

all-known: find all preimages of known opt ima, (local or global) good-subset : locat e a small subset of preimages of all opt ima t hat is well dist ribut ed over t he search space

MULTIMODAL OPTIMIZATION SCENARIOS

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• t he BBOB (black-box opt imizat ion benchmark) est ablished t he

expect ed runt ime (ERT)

• MMO not really well suit ed t o one-global scenario
• t his could also be applied t o ot her scenarios, need t o redefine

t arget s

ONE-GLOBAL

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2 main component s:

• subset select ion
• measuring

MEASURING PROCESS

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MEASURES

most ly used current ly in lit erat ure (also for CEC’ 2013):

• peak rat io (PR), but t his is problemat ic

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• S
• low-Polasky diversit y measure heavily dependent on crit ical

paramet er

• result set size t aken int o account by quant it y adj ust ment
• peak dist ance (PD) and averaged Hausdorff dist ance (AHD) can

be “ augment ed” by adding obj ect ive values as dimension

• AHD penalizes solut ions far away from any opt imum
• > t rend t o smaller result set s
• similar measures for basins (basin rat io, basin accuracy) can be

defined if basins are known

Preuss, Wessing. Measuring Multimodal Optimization S

• lution S

ets with a View to Multiobj ective Techniques. In EVOLVE IV, pp. 123– 137, S pringer, 2013

MUCH OF WHICH IS RELATED TO MULTI-OBJECTIVE MEASURING

RECENT FINDINGS ON MMO MEASURING

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• approximat ion set , parallel local search, maximal explorat ion
• not e t hat PR measures for t he left t wo are similar
• PR measure for t he right should be good if radius not t oo small

DIFFERENT SCENARIOS

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• several paramet ers have t o be set properly (e.g. radius)
• aggregat ion of binary measure (gradual improvement not

rewarded)

• does not respect result set dist ribut ion (reached opt ima may

all be in a small region)

• does not penalize huge result set s

we need alt ernat ives

PEAK RATIO CRITIQUE

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PEAK DISTANCE (PD)

• int roduced in slight ly different form in

S toean, Preuss, S toean, Dumitrescu. Multimodal optimization by means of a topological species conservation algorithm. IEEE TEC 14(6) (2010) 842-864

• for every opt imum, looks for nearest element in populat ion P
• similar t o invert ed generat ional dist ance as known in MOO
• large result set s are not penalized (needs subset select ion)
• no paramet er needed, gradual improvement measured

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• we set p=1 here (paramet er used t o penalize out liers)
• max of peak dist ance and reverse component (for every

solut ion, find nearest opt imum)

• originally int roduced for mult i-obj ect ive opt imizat ion (MOO) in

S chütze, Esquivel, Lara, Coello Coello: Using the averaged hausdorff distance as a performance measure in evolutionary multiobj ective optimization. IEEE Transactions on Evolutionary Computation 16(4) (2012) 504-522

AVERAGED HAUSDORFF DISTANCE

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taxonomy of methods

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GENERAL METHOD OVERVIEW

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• we assume t hat some sort of niching is necessary for MMO
• niching is meant as paradigm used t o “ organize search wit h

respect t o basins of at t ract ion”

• it helps t o avoid 2 problems:

“ Type I Error, Local search will be repeat ed in some region of at t ract ion. Type II Error, Local search will not st art in some region of at t ract ion even if a sample point has been locat ed in t hat region of at t ract ion.”

t his st at ement comes from an early global opt imizat ion work:

Ali, S

• torey. Topographical multilevel single linkage. Journal of Global Optimization,

5(4):349– 358, 1994.

WHAT NICHING CAN DO

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A. Explicit basin ident ificat ion: mapping from search space t o basins for det ermining t he basin any locat ion in t he search space belongs t o B. Basin avoidance (implicit basin ident ificat ion or basin recognit ion): avoid search in known regions C. Diversit y maint enance: spread out search while ignoring t opology. Also const rained informat ion exchange wit hout explicit relat ion bot basins, e.g., by subpopulat ions or mat ing rest rict ions

NICHING BASED CLASSIFICATION

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NICHING BASED TAXONOMY I

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NICHING BASED TAXONOMY II

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• many early “ niching met hods” are not class A niching met hods
• t he number of used t echniques is limit ed: single-link, densit y

based clust ering, t opological met hods, archives appear oft en

• t here are many A met hods using dist ances, obj ect ive values

and can handle a variable number of opt ima/ basins

• early global opt imizat ion met hods (e.g. Timmers’ mult i-level

single linkage) may make good MMO algorit hms

• t here is not hing like BBOB (many algorit hms comparisons) here

SOME FINDINGS

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• parallelizes in t ime (sequent ial)
• basically rest art ed local search
• modifies obj ect ive funct ion t o avoid known basins (derat ing)
• relat ed t o “ t unneling”
• comes wit h t he same problems: basins are not exact ly known
• opt ima may not be complet ely hidden
• new opt ima may be int roduced unint endedly

Beasley, Bull, Martin. A sequential niche technique for multimodal function optimization. Evolutionary Computation, 1(2):101– 125, 1993

SEQUENTIAL NICHING

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• Niching Evolut ion S

t rat egy (or Niching-CMA-ES ) as example

• uses DPI (dynamic peak ident ification), fit t est first ordering
• for every search point , we check if dist ance t o any exist ing

peak is < preset radius

• is execut ed for every peak (in parallel)
• fixed number of niches
• ext ensions: shape learning, st ep size / radius coupling

S

• hir. Niching in Derandomized Evolution S

trategies and its Applications in Quantum

• Control. PhD thesis, Universiteit Leiden, 2008

RADIUS-BASED APPROACHES

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• mult i-level single-linkage (MLS

L) uses a method very similar t o DPI, but more t han 10 years earlier

• a t heoret ically mot ivat ed radius separat es “ species”
• from an init ial sample, local searches are execut ed t o find t he
• pt ima t hat belong t o t he st art ing set samples
• “ det ect s” t he number of opt ima by it self
• only used as global opt imizat ion algorit hm, not for MMO

Rinnooy Kan, Boender, Timmer. A stochastic approach to global optimization. Technical Report WP1602-84, 1984.

EARLY GLOBAL OPTIMIZATION METHODS

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• t opographical global opt imizat ion (TGO) does away wit h radius
• uses t he k-t opograph (connect each point t o all of k nearest

neighbors t hat are worse) inst ead

• point s wit hout incoming connect ions are seen as near t o local
• pt ima, used as st art point s for local search
• k usually > 8, so t hat only few local opt ima can be ident ified
• some published improvement s, never used for MMO

Törn, Viitanen. Topographical global optimization. In Recent Advances in Global Optimization, pp. 384–

• 398. Princeton University Press, 1992

MORE GLOBAL OPTIMIZATION METHODS

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• uses obj ective values and distances to detect basins
• best known heuristic by Ursem: hill-valley method
• needs additional function evaluations
• limitation: all geometric methods bad in dimensions (>>10D)
• Ursem. Mult inat ional evolut ionary algorit hms. In Proceedings of t he Congress of Evolut ionary

Comput at ion (CEC-99), pp. 1633– 1640, 1999. IEEE Press

TOPOLOGICAL SEPARATION

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• connect every solut ion t o nearest one t hat is bet t er
• longest edges are connect ions bet ween opt ima

NEAREST-BETTER CLUSTERING

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• works wit h clust ered (left ) and randomized (right ) samples
• needs heurist ic t o remove “ t he right ” longest edges

NEAREST-BETTER CLUSTERING

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NBC ALGORITHM WITH RULE 2

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NBC EXAMPLE CLUSTERING

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ITERATED SEQUENTIAL ALGORITHM TYPE

NICHING EVOLUTIONARY ALGORITHM 2

• most flexible wit h it erat ions of clust ering + local opt imizat ion
• can be improved e.g. wit h archive, but not always successfull
• for real-valued opt imizat ion, CMA-ES

is used

• not very dependent on paramet ers

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• DE/ rand/ 1 already shows abilit y t o “ hold” many opt ima in t he

populat ion

• inst ead of an individual we employ it s nearest neighbor as base

Epitropakis, Plagianakos, Vrahatis. Finding multiple global optima exploiting differential evolution's niching capability. 2011 IEEE S ymposium on Differential Evolution (S DE)

WITH MATERIAL PROVIDED BY MICHAEL EPITROPAKIS

DE -> DE/NRAND/1

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• addit ion of a paramet er adapt at ion met hod for F and CR, t aken

from JADE

• addit ion of dynamic archive:
• put only bet t er solut ions in
• if near bet t er cont ained,

re-init ialize individual

• ident ificat ion radius R

adapt ed during run

• much bet t er performance

Epitropakis, Li, Burke. A Dynamic Archive Niching Differential Evolution Algorithm for Multimodal Optimization. CEC 2013

PARALLEL METHOD

DE/NRAND/1 -> DADE/NRAND/1

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results/ competition/ software

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• BBOB collection for global optimization: http:/ / coco.gforge.inria.fr/
• CEC 2013 Niching Competition Problems (20)
• a collection of known problems in different dimensions, 1D to 20D

http:/ / goanna.cs.rmit.edu.au/ ~xiaodong/ cec13-niching/ competition/

• Preuss/ Lasarczyk generator: mixture of polynomials

Preuss, Lasarczyk. On the importance of information speed in structured

• populations. In Proc. PPS

N VIII, pp. 91– 100, 2004, S pringer

• Gallagher/ Yuan generator: mixture of gaussian distributions

Gallagher and B. Yuan. A general-purpose t unable landscape generat or. IEEE Trans. Evolut ionary Comput at ion, 10(5):590– 603, 2006

TEST PROBLEMS/BENCHMARK SETS

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• MMO algorit hm can be bet t er t han CMA-ES

if t opology suit able

• however, classical GO met hods oft en bet t er in t hese cases
• for global opt imizat ion, MMO algorit hms not t he right t ool

SELECTED MULTIMODAL BBOB FUNCTIONS

ONE-GLOBAL CASE

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• t ask: find all global opt ima (1 t o 20D) wit h given accuracy level

FROM THE CEC 2013 NICHING COMPETITION

ALL-GLOBAL CASE

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MORE ACCURATE, PLEASE

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• t ight race bet ween NEA2

and dADE/ nrand/ 1

• won by t he sequent ial

met hod (t his t ime)

• result depends very much on

experiment al set up

• crit ique t owards PR as basic

performance measure

OVERALL ASSESSMENT

many thanks to the CEC 2013 niching competition team: Michael Epitropakis, Xiaodong Li and Andries Engelbrecht

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the future

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• define MMO, t asks and scenarios
• improve problem libraries
• set up benchmarks for different scenarios
• agree on proper performance measures for t hese
• real-world mot ivat ed benchmarks?
• work on MMO algorit hms, recombine component s?
• MMO algorit hms for non real-valued represent at ions?

THINGS TO DO

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S pringer book “ Mult imodal Opt imizat ion by Means of Evolut ionary Algorit hms” (monograph on base of my dissert at ion) coming out soon!

WHERE IS THE MATERIAL FROM?

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S 4.2 Proc. p. 141, Tuesday 11:00 Wessing, Preuss, Traut mann: S t opping Crit eria for Mult imodal Opt imizat ion

MMO STOPPING CRITERIA?

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A RECENT EXAMPLE FROM COMPUTATIONAL INTELLIGENCE IN GAMES

MMO FOR NON REAL-VALUED PROBLEMS

• design t ool for map sket ches: diverse but good set needed

here

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(original, vertical balance of impassables+left half concentration of impassables, horizontal balance of resources+top half concentration of resources, diagonal concentration of impassables, impassable segments+largest segment)

VISUAL IMPRESSION MAP DISTANCE

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good for small maps good for large maps

• AVG. 6 OBJECTIVES AGAINST AVG. MIN. VISUAL IMPRESSION DISTANCES

COMPARISON TO RESTART ES/MC/NOVELTY

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TAKE HOME

• FIELD MUS

T BE DEFINED MUCH BETTER (PROBLEMS , MEAS URES )

• LOOK INTO GLOBAL OPTIMIZATION WORK (TOERN, RINNOY KAN,

ALI) TO FIND MANY US EFUL CLUES

• MMO METHODS

NOT REALL Y US EFUL FOR GLOBAL OPTIMIZATION

• BUT US

EFUL FOR S ET OPTIMIZATION

• UNCOORDINATED RES

TARTED LOCAL S EARCH GOOD BAS ELINE

• NEA2 AND DADE/ NRAND/ 1 GOOD METHODS

FOR MMO

• UNEXPLOITED CONNECTIONS

TO MULTI-OBJECTIVE OPTIMIZATION

• APPL

Y MMO TO MORE NON REAL-VALUED REPRES ENTATIONS !