Genetic algorithms using multi-objectives(8)

SOPwithgivenvaluesoftheparameters,undercer-tainconditions,belongstotheParetofront;changingtheparametersoftheSOPleadsthesolutiontomoveonthefront.Inthesecondone,theMOPissolvedwithadirectapproachusingthedominancecriteria,sothatasetofParetooptimalsolutionsisdevelopedsimultaneously.Themainadvantageofthe rststrat-egyisthatSOPsaregenerally,verywell-studiedprob-lemsandmanyef cientmethodsareavailabletosolvethem.

6.1.EquivalentSOP1:TheweightingapproachFollowingtheweightingapproach[16],theMOP[42]ismadetocorrespondtothefollowingparametrizedSOP:P(w):minwT

f nx∈X

(x)=

wjfj(x),(4)

j=1

where

w∈W=

w|w∈Rn,wj(x) 0,

n j=1,...,nand

w

j=1j=1

,

(5)

thecorrespondencebetweentheMOPandtheSOPissubjecttosomerules.Ifx0isanoptimalsolutionofP(w0),thenitisalsoParetooptimalifoneofthetwofollowing0conditionsisveri ed:

xistheuniqueoptimalsolutiontoP(w0w0);isstrictlypositive.

ThisimpliesthatatleastsomeParetooptimalso-lutionscanbegeneratedbysolvingP(w)forsomeproperlychosenw,withoutanyhypothesisonthecon-vexityofXandf(X).Instead,someconvexityhy-pothesesareanecessitycondition.Therefore,ifbothXandf(X)areconvex,thenforanygivenParetoop-timalsolution,x ,itispossibleto ndaweightvectorw,notnecessarilyunique,suchthatx isasolutiontoP(w).Therefore,whentheseconvexityassumptionsareveri ed,allParetooptimalsolutionscan,inthe-ory,befoundbyvaryingwandsolvingP(w),whileiftheyarenotveri ed,someParetooptimalsolutionsmayneverbediscoveredbythisprocedure.

搜索“diyifanwen.net”或“第一范文网”即可找到本站免费阅读全部范文。收藏本站方便下次阅读，第一范文网，提供最新教学研究Genetic algorithms using multi-objectives(8)全文阅读和word下载服务。