Multiobjective Evolutionary Optimization

Transcription

1 Multibjective Evlutinary Optimizatin Pressr Qingu ZHANG Schl Cmputer Science and Electrnic Engineering University Essex CO4 3SQ, Clchester UK

2 Outline Multibjective Optimisatin Paret Dminance Based Algrithms Decmpsitin Based Algrithms Estimatin Distributin Algrithm r Multibjective Optimizatin Cnclusin

3 Outline Multibjective Optimisatin Paret Dminance Based Algrithms Decmpsitin Based Algrithms Estimatin Distributin Algrithm r Multibjective Optimizatin Cnclusin 3

4 Why Multibjective? Many real-wrld applicatins invlves mre than ne bjective Dmain Cniguratin Objectives Ruting A rute rm A t B Travel time saety Manuacturing Design parameters cst quality Machine Learning Mdel, eatures simplicity accuracy Finance Prtli Management expected return risk 4

5 min F( x) = ( ( x), ( x), subject t x D Prblem Frmulatin K, m ( x)) bj vectr, image x a slutin slutin (search) space, x-space bj unctin Cmbinatrial MOP D: inite set x: rute, permutatin, timetable, tree. Cntinuus MOP n x R, D = { x g ( x) 0, i and g j are cntinuus. j j =, K, k} = D x = D F F F (D) 5

6 Dminatin: which slutin is better? y dminates x i y is n wrse than x in any bjs, and y is better than x in at least ne bj. C A B x-space F(c) dminates slutins in slutins in dminate F(c) F(c) cannt cmpare with slutins in F(A) F(B) F(C) F-space 6

7 Paret Optimal Slutins= Best Trade- Candidates x is Paret ptimal i n ther slutin dminates it. Paret set (PS): all the Paret ptimal slutins in the decisin space. Paret rnt (PF): the image the PS in the bjective space. A (ratinal) decisin maker desn t like nn-paret ptimal slutins. Mathematically, all the Paret ptimal slutins are equally gd. Paret Set x-space F-space Paret rnt 7

8 A decisin maker ten wants: a number representative Paret ptimal slutins (a gd apprximatin t the PF r PS ) r a decisin maker Task Multibjective Evlutinary Algrithm Paret Set (PS) F F ( D ) Paret Frnt (PF) 8

9 Outline Multibjective Optimisatin and Evlutinary Algrithms Paret Dminance Based Algrithms Aggregatin Based Algrithms New Tls in Evlutinary Multibjective Optimizatin Cnclusin 9

10 Intr crssver mutatin Ppulatin spring slutins Selectin Ppulatin cpy Ppulatin Suggestin by Gldberg (989) Classic EA ramewrk. Selectin mainly guided by Paret dminance. as clse t the PF as pssible as diverse as pssible. Three representatives: NSGA-II (00), SPEA-II (00 ), PAES (999). 0

11 + and - crssver mutatin Ppulatin spring slutins Selectin Ppulatin cpy Ppulatin Simple and easy t use Very general, dn t need very strng assumptins abut the prblem. Hwever Perrmance is nt excellent. T much randmness/nt very ratinal. dn t make use traditinal pt ideas. That was als sad, because thse methds take many decisins randmly.. M. J. D. Pwell, 007.

12 Outline Multibjective Optimisatin Paret Dminance Based Algrithms Decmpsitin Based Algrithms Estimatin Distributin Algrithm r Multibjective Optimizatin Cnclusin

13 Intr Aggregatin/decmpsitin is a majr strategy r dealing with MOP in traditinal ptimizatin. Decmpsitin based EAs use aggregatin unctins t guide the search. Tw Representatives: MOGLS (998, Ishibuchi et al, 996, A. Jaszkiewicz et al) and MOEA/D (007, Q. Zhang et al) 3

14 ,(z,(λ MOEA/D Backgrund strategy in traditinal math prgram methds: Aggregatin ( λ, λ ) Weight Sum Apprach min where g ws ( x, λ) = λ λ + λ = and ( x) + λ λ, λ ( x) 0. * F I the PF is cnvex, Fr any Paret ptimal slutin x*, there is a weight vectr such that x* is the ptimal slutin t the abve prblem. Fr any weight vectr, the ptimal slutin t the abve prblem is a Paret ptimal slutin. I the PF is nt cnvex, it might nt wrk. Q. Zhang and H. Li, MOEA/D, IEEE Trans n EC, 007 4

15 ,(z,(λ,(z,(λ λ, λ ) ( Techbyche Apprach min g T g T ( x, λ) * ( x, λ) = max{ λ ( x) z, λ * * z : the minimal value, z : the minimal value * * ( z, z ) ( x) z * }. * F Fr any Paret ptimal slutin x*, there is a weight vectr such that x* is the ptimal slutin t the abve prblem. ( λ, λ ) Aggregatin is still a very active research tpic in traditinal ptimizatin K. M. Mittinen, Nnlinear Multibjective Optimizatin, Kluwer, 999. G. Eichelder, Adaptive Scalarizatin Methds in Multibjective Optimizatin, Springer,

16 ,(z,(λ min g( x, λ ) Finding a set N representative Paret ptimal slutins min g( x, λ ) N prblems. Nt a N-bj pt prblem! min g( x, λ N ) min( + 0 ) min( ) + min( ) λ λ = = M (, 0), (0.9, 0.) g( x, g( x, λ ) = λ ) = λ = (0,) g( x, λ ) = 0 + min( ) min( 0 + ) Traditinal ptimizatin slves these N prblems ne by ne. The distributin inal slutins culd be very unirmly distributed i g( ) and λ i are prperly chsen. 6

17 Idea These prblems are related with each ther. I λ i and λ j i j are clse enugh, we can call min g( x, λ ) and min g( x, λ ) neighburs. neighburing prblems shuld have similar slutins. g( x, λ An example r deining neighburhd g( x, λ ) = 0 ( x) + g( x, λ ) = 0.0 ( x) M 00 ) = 0.99 ( x) g( x, λ ) = ( x) + 0 ( x) ( x) ( x) ( x) λ = (0,) λ λ λ = 00 0 (0.0, 0.99) = = (0.99, 0.0) (, 0) i min g( x, λ ) i i λ is amng the irst 4 clsest weight vectrs t g( x, λ )' s g( x, λ ), g( x, λ ), g( x, λ ), g( x, λ ), 30 g( x, λ )' s is a neighbur λ neighbrhd : neighbrhd : g( x, λ ), g( x, λ ), g( x, λ ), g( x, λ ), 7 j. min g(x,λ j )

18 min N g ( x, λ ) min g ( x, λ ) min g ( x, λ ) N agents are used r slving these N prblems During the search, neighburing agents can help each ther. 8

19 MOEA/D Algrithm Framewrk x x x N min g ( x, λ ) min g ( x, λ ) min g ( x, λ N ) x Agent i recrds, the best slutin he has und s ar r his prblem. At each generatin, each agent i des the llwing: Generate a new slutin. x i i y y i i i Replace by i g( y, λ ) < g( x, λ ). Pass y t all r sme its neighburs. They replace their best slutins by y i it is better than their current best slutins (measured by their individual bjectives). 9

20 x x x N min g ( x, λ ) min g ( x, λ ) min g ( x, λ N ) Hw agent i generate a new slutin (many pssible ways). y Randmly select several neighburs and btain their current slutins. Apply genetic peratrs n these slutins and generate a new slutin y. i Apply single pt. lcal search n y t ptimise its bj g ( x, λ ) and btain y. (ptinal) 0

21 Mre Details MOEA/D Initialisatin: each agent can initialise randmly r by using prblem-speciic knwledge, e.g. Randmly generate a pint in the decisin space and then use a single bj LS t imprve it. Aggregatin Methd: Any methds shuld d. We have tried weight sum apprach, Tchebyche apprach, Penalty based bundary intersectin (PBI) apprach and NBI style Tchebyche apprach. N The setting λ, K λ : which shuld be unirmly distributed in x i { m λ = (( λ, K, λ m ) i = λ i = ; λ 0} Other methds can als be cnsidered, e.g., dynamically tuning weight vectrs. Reerence pint (needed in PBI and Tchebyche appraches) is estimated rm the previus search.

22 Remarks n MOEA/D Diversity: Diversity amng subprblems will lead t the N diversity amng { x, x, K, x }. Mating Strategy: Slutins have a chance t mate nly when they are neighburs (can be relaxed). Cmplexity: much lwer than NSGA-II (detailed analysis can be und in the paper). Suppse that bth NSGA-II and MOEA/D uses the same ppulatin size, then at each generatin, the rati cmputatinal cmplexity between NSGA-II and MOEA/D is: where T is the number T O( ) Pp _ Size neighburs r each subprblem.

23 Dynamical resurce allcatin in MOEA/D dierent subprblems (agent) requires dierent amunt cmputatinal resurces. Each subprblem (agent) has a utility value, which measures the likelihd urther imprvement. The amunt imprvement btained Utility = The amunt cmputatinal resuces used At each generatin, a small number agents are selected based n utility value and d updating. This versin MOEA/D has wn CEC 009 uncnstrained MOEA cmpetitin. 3

24 Applicatins: Scheduling Machine learning Antennas design LT cde design Analg cell sizing Brain Cmputer Intelligence Rbt Path Planning Wireless netwrk design Missile Cntrl.. Variants: MOEA/D+ACO MOEA/D+PSO Parallel MOEA/D MOEA/D r expensive MOP MOEA/D r many-bjective pt MOEA/D+Simulated Annealing MOEA/D+NSGA-II MOEA/D with tw dierent aggregatin unctins/neighbrhd sizes Mre than ne slutins r each subprblem MOEA/D web: 4

25 + and - It is a bridge between traditinal math prgramming and ppulatin algrithms. Decisin maker s preerence inrmatin can be easily embedded in it. Hwever, I the PF shape is dis-cnneted, sme cmputatinal ert will be wasted. 5

26 Outline Multibjective Optimisatin Paret Dminance Based Algrithms Decmpsitin Based Algrithms Estimatin Distributin Algrithm r Multibjective Optimizatin Cnclusin 6

27 Machine Learning in MOEAs Optimizatin is a ML prblem. Objective reductin n line/ line. (Manild Learning) Preerence mdelling (SVM) Cmputatinal Resurce Allcatin (bandit learning). Estimatin distributin prmising slutins RM-MEDA (Regularity Mdel Based Multibjective Estimatin Distributin Algrithm). 7

28 RM-MEDA: Mtivatins x Paret set (PS) Regularity cntinuus MOP: Under certain cnditins, the PS (PF) is a (m-)-dimensinal piecewise cntinuus manild in decisin (bjective) space. Where m is the # the bjs. F x This prperty has been ignred by MOEA researchers. F(D) Hw can we deal with a cntinuus MOP i its PS is (m-)-d piecewise cntinuus manild? Paret rnt (PF) Q. Zhang et al, RM-MEDA, IEEE Trans n EC, 008 8

29 Suppse we use the llwing cmmnly-used ramewrk: Why cmmnly-used genetic peratrs d nt wrk well r cmplicated PSs? Ppulatin crssver mutatin spring slutins Selectin cpy Ppulatin PS in decisin Space. When tw parents are in the PS, their spring may nt be clse t the PS. The PS is nt an equilibrium. Ppulatin S we resrt t EDA (Mdelling and Sampling). 9

30 Basic Idea Ppulatin? spring slutins Selectin cpy Ppulatin In the case bjs The PS is a -D curve. I the algrithm wrks well. Ppulatin PS Ppulatin The principal (central) curve the ppulatin culd be an apprximatin t the PS. 30

31 Ppulatin? cpy spring slutins Ppulatin Selectin Ppulatin Each pint in the current ppulatin is regarded as a sample x = ξ + ε where ξ is unirmly distributed n a - D curve C. ε is a n - D Gaussian nise. Hw t mdel C and ε? 3

32 ε Mdelling: Hw t mdel C and? We assume that: The central curve C cnsists several line segments. This assumptin makes C cmputable. Hw t mdel C Divide the ppulatin int several clusters by lcal PCA. Cmpute the central line each cluster. Hw t mdel ε the deviatin the pints in each cluster t its central line. : pint in the current ppulatin. : central curve: C simpliicatin The number clusters needs t be preset. 3

33 Sampling Hw t sample new slutins: The number new slutins sampled arund : L( Ci ) # new _ slutins L( C ) + L( C ) + L( C Sampling arund Unirmly randmly pick a pint x in x~n(0, ), x=x+x. C i C i ε i I n n C i 3 ) 33

34 Test Instance PS: RM-MEDA>thers, Why I all the parents are in the PF, In RM-MEDA, the spring will be very clse t the PF. In all the ther algrithms, the spring will be ar away rm the PF. 34

35 Outline Multibjective Optimisatin and Evlutinary Algrithms Paret Dminance Based Algrithms Aggregatin Based Algrithms New Tls in Evlutinary Multibjective Optimizatin Cnclusin 35

36 Cnclusins Multibjective pt is everywhere. EMO is a very yung and prmising research area. Ideas rm nature, math prgramming, machine learning, statistics can be used in MOEA. MOEA+Machine Learning. 36

37 Reerences: Q. Zhang and H. Li, MOEA/D: A Multi-bjective Evlutinary Algrithm Based n Decmpsitin, IEEE Trans. n Evlutinary Cmputatin, vl., n. 6, pp ( awarded The IEEE TEVC Outstanding Paper Award). Q. Zhang, A. Zhu and Y. Jin, RM-MEDA: A Regularity Mdel Based Multibjective Estimatin Distributin Algrithm, IEEE Trans. n Evlutinary Cmputatin, vl., n., pp 4-63, 008 H. Li and Q. Zhang, Multibjective Optimizatin Prblems with Cmplicated Paret Sets, MOEA/D and NSGA-II, IEEE Trans n Evlutinary Cmputatin, n., 009. A. Zhu, Q. Zhang and Y. Jin, Apprximating the Set Paret Optimal Slutins in Bth the Decisin and Objective Spaces by an Estimatin Distributin Algrithm, IEEE Trans n Evlutinary Cmputatin, n. 5., 009. Q. Zhang, W. Liu, E. Tsang and B. Virginas, Expensive Multibjective Optimizatin by MOEA/D with Gaussian Prcess Mdel, IEEE Trans n Evlutinary Cmputatin, 00 S-Z Zha, P N Suganthan, and Q Zhang, MOEA/D with an Ensemble Neighburhd Sizes, IEEE Trans n Evlutinary Cmputatin, 0. Accepted. D. Saxena, J. A. Dur, A. Tiwari, K. Deb, and Q. Zhang, Objective Reductin in Many-bjective Optimizatin: Linear and Nnlinear Algrithms, IEEE Trans n Evlutinary Cmputatin, 0. Accepted. The paper/cde can be dwnladed rm: Thanks! 37

38 Thanks! 38