Bloch Quantum-behaved Pigeon-Inspired Optimization for Continuous Optimization Problems*

Transcription

1 Proceedngs of 4 IEEE Chnese Gudance, avgaton and Control Conference August 8-, 4 Yanta, Chna Bloch Quantum-behaved Pgeon-Insred Otmzaton for Contnuous Otmzaton Problems* Honghao L, and Habn Duan, Senor Member, IEEE Abstract In ths aer, a novel hybrd Pgeon Insred Otmzaton () and quantum theory s roosed for solvng contnuous otmzaton roblem. Whch s called Bloch Quantum-behaved Pgeon-Insred Otmzaton (BQ for abbrevaton). Quantum theory s adoted to ncrease the local search caacty as well as the randomness of the oston. As a consequence, the mroved BQ can avod the remature convergence roblem and fnd the otmal value correctly when solvng multmodal roblems. An emrcal study was carred out to evaluate the erformance of the roosed algorthm, whch s comared wth Partcle Swarm Otmzaton (), basc, and Quantum-behaved Partcle Swarm Otmzaton (Q). The comaratve results demonstrate that our roosed BQ aroach s more feasble and effectve n solvng comlex contnuous otmzaton roblems comared wth other swarm algorthm. I. ITRODUCTIO Many otmzaton roblems can be solved by oulaton-based swarm ntellgence algorthms. The key ssue of these meta-heurstc swarm ntellgence algorthms are exloraton and exlotaton as t makes few or no assumtons about the roblem beng otmzed and can search very large saces of canddate solutons [, ]. Quantum-behaved Partcle Swarm Otmzaton (Q) s a modfed Partcle Swarm Otmzaton () whch ntroduce the quantum theory [3]. Ths algorthm can evade the shortcomngs whch the standard has. In ste of that, Q stll have some roblems n feasblty and effectveness to solve some comlcated multmodal functon. Pgeon-Insred Otmzaton (), a oulaton-based swarm ntellgence algorthm, s a novel swarm ntellgence otmzaton technque roosed by H. B. Duan n 4 [4]. It s an effectve algorthm, but t wll become undeendable esecally when the oulaton sze of the geon swarm s small, whch I wll show n the Secton V. To resolve these shortcomngs, a new swarm ntellgence algorthm wth a Bloch Shere encodng mechansm wll be roosed n ths aer, whch name s Bloch Quantum-behaved Pgeon-Insred Otmzaton (BQ). *Ths work was artally suorted by atonal Key Basc Research Program of Chna (973 Project) under grant #4CB464, atonal Magnetc Confnement Fuson Research Program of Chna under grant # GB6, and Aeronautcal Foundaton of Chna under grant #384. Honghao L s wth Scence and Technology on Arcraft Control Laboratory, School of Automaton Scence and Electrcal Engneerng, Behang Unversty, Bejng, Chna (e-mal: lhonghao993@hotmal.com). Habn Duan s wth Scence and Technology on Arcraft Control Laboratory, School of Automaton Scence and Electrcal Engneerng, Behang Unversty, Bejng, Chna(e-mal: hbduan@buaa.edu.cn; hone: ). The remander of the aer s organzed as follows. The next secton ntroduces the man rocess of. Secton III resents the basc mathematcal model of BQ, and Secton IV secfes the detaled mlementaton rocedure of BQ. Subsequently, a seres of comarson exerments on several benchmark otmzaton roblems are conducted, and the comaratve results together wth the analyss wll be gven n Secton V. Our concludng remarks and drectons for future research are contaned n fnal secton. II. PIGEO-ISPIRED OPTIMIZATIO The homng geon has an nborn homng ablty to fnd ts way home over extremely long dstances by usng three homng tools: magnetc feld, sun and landmarks. Scentsts have found that on the to of geon's beak, massve ron artcles was found whch onts to the north just as artfcal comass. Whch hels geon to determne ts way home. Gulford and hs colleagues argue that homng geons n dfferent arts of the journey may use dfferent navgaton tools. As geons start ther journey, they may be more deendent on comass-lke tools. Whlst n the latter art of ther journey, they swtch to usng landmarks tools when they need to reassess ther route []. Insred by the above homng behavors of geons, a novel bo-nsred swarm ntellgence otmzer whch s named has been nvented n 4 [4]. In order to dealze some of the homng characterstcs of geons, two oerators are desgned by usng some rules, ma and comass oerator model s resented based on magnetc feld and sun, whle landmark oerator model s desgned based on landmarks. ) Ma and Comass Oerator In the Ma and Comass Oerator, comuter-generated geons are used. The oston and the velocty of geon can be defned as and V, whch wll udate n each teraton n a d-dmenson search sace. And the new oston and velocty V of geon at the t-th teraton can be obtaned wth () and () [4]: ( t + ) = ( t ) + V ( t + ) () Rt V ( t + ) = V e + rand ( Pg ) () where R s the ma and comass factor. rand s a random number (generally s a number between and ). P g s the global oston of the frst t teraton, whch can be calculated by comarng all the ostons among the whole swarm. ) Landmark Oerator In the Landmark Oerator, we assume the geons are stll dstant from the destnaton, and obvously they are /4/$3. 4 IEEE 634

2 unfamlar to the landmarks. All geons are ranked accordng ther ftness values. Then half of the number of geons ( /) s decreased accordng to (3). ( t) (3) ( t + ) = We then fnd the center geon from the left geons at the t-th teraton, whose oston ( c ) s the desrable destnaton. Ths can be descrbed by (4). And the new oston of other geons can be calculated by (). ( t ) = c ( t ) = ( t ) ( t ) ftness ( t ) = ( ftness ( ( t + ) = + rand ( ) where ftness( ) s consdered the ftness value of the each ftness( () t ) = fmn geon n the swarm. We can choose ( () t ) + ε for the mnmum otmzaton roblems. III. BLOCH QUATUM-BEHAVED PIGEO-ISPIRED OPTIMIZATIO BQ s an algorthm combned standard algorthm and a Bloch Shere encodng mechansm. In ths algorthm, every sngle artcle have quantum behavor. A. Quantum Evolutonary Theory The robablty amltudes of quantum bt encode the oston of artcles whle the quantum rotaton gates erform the changng of the oston whch acheve artcles searchng [3, 6, 7], and the quantum bt chromosome can be decreased accordng to (6) as a strng of m quantum bts: α α α3 αm q =... (6) β β β3 βm where α + β =, =,...,m. m reresents the quantty of quantum bts as well as the strng length of the quantum bt ndvdual. α reresents the chance that the quantum bt wll be found n the state. On the contrary, β shows the chance that the quantum bt wll be found n the state. The lnear sueroston of all ossble solutons can be reresented by quantum bt chromosome and the dversty of t s better than the classc one, then we brng a random number between [, ] to get a classcal chromosome and α comare t wth, f t s bgger, ths bt n the classcal chromosome s, otherwse [7-]. c (4) () - Fgure. Polar lot of the rotaton gate for quantum bt chromosome The tycal mutaton oeraton s comletely random wthout any drectons, and the seed of convergence s slowed down. However, the quantum bt reresentaton can be regarded as a mutaton oerator [3, 8, ]. Drected by the current ndvdual, quantum mutaton s comleted through the quantum rotaton gate, and [ α ] T β can be udated by ' α cosθ snθ α = (7) ' β snθ snθ β The olar lot of the rotaton gate for quantum bt chromosome can be descrbed as Fg. above. The θ I can be determned by quantum chromosome and classcal chromosome. Whch s showed n a table of [9]. B. Bloch Quantum-behaved Pgeon-Insred Otmzaton In order to mrove search ablty and otmzaton effcency and to avod remature convergence for, a novel algorthm called BQ s roosed for contnuous sace otmzaton. The classcal quantum bts n the Hlbert sace s unt crcle have only one varable. So the quantum roertes of t are greatly weakened. To solve ths roblem, s evolved by combnng wth a Bloch Shere encodng mechansm. As shown n Fg., a ont P s able to fx by the angle of θ and ϕ n a stuaton of 3D Bloch shere.every quantum bt has a corresondng ont n the Bloch Shere, hence the artcles ostons can be drectly erformed n the Bloch Shere coordnates. Assumng that Pgeon s the -th geon among the swarm. Then the Bloch Shere encodng rocess s descrbed as follows [7]: Pgeon cos ϕ sn θ = sn ϕ sn ϕ cos θ - Δ θ ' ' β ( α, ) α, β ) (... cos ϕ d sn θ d... sn ϕ d sn ϕ d... cos θ d (8) where ϕ j = rand, θ j = rand, rand s a random number between [,] =,,, ; j =, d; s the oulaton sze of the swarm and d s the sace dmenson. 63

3 Fgure. Quantum bt Bloch Shere Just lke, ths algorthm conssts of two oerators: ma and comass oerator and landmark oerator. In ma and comass oeraton, the geon s oston and velocty can be udated accordng to (9) to (): m ( t + ) = P (9) = where m (t) s the average value of the otmal oston of each geon at the t-th teraton. For each ndvdual geon (geon for examle), the otmal oston of the t-th teraton can be denoted wth P (t) []. We set a 3D mutaton oerator to show the ablty of quantum non-gate n the Hlbert sace s unt crcle to the Bloch Shere [7, ]. The mutaton oerator satsfes the followng matrx (): cos ϕ j ( t ) sn θ j ( t ) cos( π / ϕ j sn( π / θ j B sn ϕ j ( t ) sn ϕ j ( t ) = sn( π / ϕ j sn( π / θ j cos θ j ( t ) cos( π / θ j () And the 3D mutaton oerator we dscussed above can be descrbed as (). B = cot θ j ( t ) cot θ ( t ) j tan θ ( t ) j () Then a new geon oulaton was roduced. PP ( t + ) = f ( t + ) P( t) + ( f ( t + )) P () g t ω ( t ) = ωmax ( ωmax ωmn) (3) t max If f ( t + ). Then ( t + ) = P P ( t + ) + ω( t + ) m Else g g ( t + ) ln f ( t + ) (4) ( t + ) = P Pg ( t + ) ω ( t + ) m ( t + ) ln u( t + ) () where P (t) s the ersonal soluton found so far by an ndvdual geon whle P g (t) reresent the current global oston of the frst t teraton, whch can be calculated by comarng all the ostons among the whole swarm. For a sngle geon, the random oston between the otmal oston of tself and the global oston s P P g (t). f(t+) and u(t+) are factors of random varables drawn wth unform robablty from [,], redrawn for each geon n every teraton [3]. ω s the constrcton factor whch can decreases lnearly from ω max to ω mn, slowng the geons as the algorthm s carryng on, so that fner exloraton s acheved. In the ntal stage, a bgger ω stmulate the search, whle n the later erod, a smaller ω contrbute to the ablty of exlotaton, so that we can have a hgh convergence rate [4]. Then we oerate the Landmark Oerator. The Landmark Oerator s just the same as the Landmark Oeraor of, half of the number of geons s decreased accordng to (3). We then fnd the center geon of the left geons accordng to (4), the evoluton equaton can be descrbed as (). Fnally, we can get the soluton arameters and the cost value. IV. IMPLEMETATIO PROCEDURE OF BQ The detaled mlementaton rocedure of BQ for otmzaton can be descrbed as follows: Ste : Intalze arameters of BQ algorthm, such as soluton sace dmenson D, the oulaton sze, the ma and comass factor R, and the number of teraton t and t, where the t s consdered the teratons of the ma and comass oerator, and t wll be the total teratons of the whole algorthm. Ste : Set each sngle geon wth a randomzed velocty and oston. Then we can calculate the ftness value ftness ( ) of the each geon n the swarm. Ste 3: For each ndvdual geon, f the artcle s current oston s better than ts own-memory otmal oston P, then relace the latter wth current oston. Comare the ftness value of all geons, f the current global otmal oston s sueror to the otmal global oston P g ever searched, then relace the latter wth the current global otmal oston. Ste 4: Oerate the ma and comass oerator. Frstly, we udate the velocty and oston of every geon accordng to (9) to (). Then we comare all the geons ftness value and udate the new global oston P g. Ste : If t > t, sto the ma and comass oerator and turn to ste 6. If not, go to Ste 4. Ste 6: Oerate the landmark oerator. Rank all geons accordng ther ftness values, and wave half of geons whose ftness value are lower by usng (3). We then fnd the center geon of the left geons accordng to (4), whch s consdered the desrable destnaton. All geons wll fly to the 636

4 destnaton by adjustng ther flyng drecton accordng to (). ext, we can get the soluton arameters and the cost value. Ste 7: If t > t, sto the landmark oerator, and outut the results. Otherwse, go to Ste 6. V. RESULT FROM BECHMARK SIMULATIOS In order to nvestgate the feasblty and effectveness of the roosed BQ, a seres of exerments are conducted on several benchmark functon roblems: Shubert Functon, Rosenbrock Functon, Rastrgn Functon and Schaffer Functon. Shubert Functon (f ) s a comlex functon of two dmensons, whch have 76 extreme onts, acheve the mnmum at , and can be exressed as follows. mn f ( x, y) = { = cos[( + ) x + ]} { = cos[( + ) y + ]} (6) Rosenbrock Functon (f ) s a non-convex functon used as a erformance test roblem, the global mnmum, whose value s, s nsde a long, narrow, arabolc shaed flat valley. To converge to the global mnmum, however, s dffcult. The functon s defned by (7). mn f ( x ) = D = [ ( x x + ) + ( x ) ] (7) Rastrgn Functon (f 3 ) s a non-lnear functon wth multle eak values. It s qute a dffcult roblem to fnd the least value of ths functon, just because the huge searchng sace and a mass of local mnma. Ths functon can be defned as: mn D = f ( x ) = [ x cos( π x ) + ] (8) Schaffer Functon (f 4 ) s a two-dmensonal comlex functon, acheve the mnmum at. mn f ( x, x ) =. + (sn x ( +. ( x + x ) + x. )) (9) The test results and the comaratve evolutonary curves of the four functons are showed as TABLE, Fg. 3 to Fg. 6. For the and BQ, we set that soluton sace dmenson D s, the oulaton sze =, the ma and comass factor R=., and the number of teraton t =, t =. For the, we set that C =C =.496, ω=.9, D=, =. ftness value Q BQ Iteraton ftness value ftness value Fgure 3. Comaratve evolutonary curves of f Q BQ Iteraton Fgure 4. Comaratve evolutonary curves of f Q BQ Iteraton Fgure. Comaratve evolutonary curves of f 3 637

5 ftness value Q BQ Iteraton Fgure 6. Comaratve evolutonary curves of f 4 VI. COCLUSIO Wth the develoment of technology and the broadenng of the scoe of engneerng roblems, the scale and comlexty of the roblem s ncreasng. As a result, otmzaton results of tradtonal algorthms have many lmtatons, and the effectveness of mrovements for a defnte algorthm s very lmted. In ths work, we resented a novel hybrd and quantum theory for solvng contnuous otmzaton roblems, whch called Bloch Quantum-behaved Pgeon-Insred Otmzaton, BQ for short reference. Comaratve exermental results verfed the feasblty and effectveness of our roosed aroach. Our future work wll aly our roosed BQ model to solve the comlcated ssues of unmanned aeral vehchles [], and whch s a challengng ssue for bo-nsred comutaton , 7. [4] H. B. Duan, P.. Qao. Pgeon-nsred otmzaton: A new swarm ntellgence otmzer for ar robot ath lannng. Internatonal Journal of Intellgent Comutng and Cybernetcs, vol.7, no.,. 4-37, 4. [] T. Gulford, S. Roberts, D. Bro. Postonal entroy durng geon homng II: navgatonal nterretaton of Bayesan latent state models. Journal of Theoretcal Bology, vol. 7, no.,. -38, 4. [6] J. Onwunalu, L. Durlofsky, Alcaton of a artcle swarm otmzaton algorthm for determnng otmum well locaton and tye, Comut. Geosc, vol. 4, no., ,. [7] Y. Du, H. B. Duan, R. J. Lao, and. H. L. Imroved quantum artcle swarm otmzaton by bloch shere. Lecture otes n Artfcal Intellgence, vol.64, PART,.3-43,. [8] H. B. Duan, Z. H. ng. Imroved quantum evolutonary comutaton based on artcle swarm otmzaton and two-crossovers. Chnese Physcs Letters, vol.6, no.,. 34-3, 9. [9] R.C. Eberhart, Y.H. Sh. Comarng nerta weghts and constrcton factors n artcle swarm otmzaton. In: Proceedngs of IEEE Congress on Evolutonary Comutaton, Pscataway, IEEE Press, Los Alamtos,. [] P. C. L, S.Y. L. Quantum comutaton and quantum otmzaton algorthm,. 3 7.Harbn Insttute of Technology Press, 9. [] A.., Al-Rabad: ew dmensons n non-classcal neural comutng, art II: quantum, nano, and otcal. Internatonal Journal of Intellgent Comutng and Cybernetcs, vol., no.3, [] Y. C. Yu, H. B. Duan. A quantum-behaved artcle swarm otmzaton aroach to PID control arameters tunng. In: Proceedngs of the 3nd Chnese Control Conference, 'an, , 6-8 July 3. [3] R. Zhang, H. Gao. Imroved quantum evolutonary algorthm for combnatoral otmzaton roblem. In: Proceedngs of the 6th Internatonal Conference on Machne Learnng and Cybernetcs, Hong Kong,. 3 3, 7. [4] Y. H. Sh and R. Eberhart. A modfed artcle swarm otmzer. In: Proceedngs of IEEE Congress on Evolutonary Comutaton, , 998. [] H. B. Duan, and P. L. Bo-nsred Comutaton n Unmanned Aeral Vehcles. Srnger-Verlag, Berln, Hedelberg, ch., 4. REFERECES [] H. B. Duan,. Y. Zhang, and C. F. u. Bo-nsred Comutng. Bejng: Scence Press, ch. 4,. [] E. Bonabeau, M. Dorgo, and G. Theraulaz. Insraton for otmzaton from socal nsect behavor. ature, vol. 46, no. 6,. 39-4,. [3] P. C. L, S. Y. L. Quantum artcle swarms algorthm for contnuous sace otmzaton. Journal of Quantum Electroncs, vol. 4, no.4, TABLE I. THE TEST RESULTS Q Q f f f3 f4 = = = = = = = = Mean value Worst value Mean value 7.49E-4.E- 3.3E Worst value.3e E-.9E-4.E Mean value.8e-3.4 Worst value 6.4E-4.E-3.8E-4 7.7E-4.99 Mean value 7.93E- Worst value.48e E E