Multiple Trajectory Search for Large Scale Global Optimization

Transcription

1 Multple Trajectory Search for Large Scale Global Optmzaton Ln-YuTsengandChunChen Abstract In ths paper, the multple trajectory search (MTS) s presented for large scale global optmzaton. The MTS uses multple agents to search the soluton space concurrently. Each agent does an terated local search usng one of three canddate local search methods. By choosng a local search method that best fts the landscape of a soluton s neghborhood, an agent may fnd ts way to a local optmum or the global optmum. We appled the MTS to the seven benchmark problems desgned for the CEC 008 Specal Sesson and Competton on Large Scale Global Optmzaton. I. INTROUCTION lobal numercal optmzaton s an mportant research Gssue because many real-lfe problems can be formulated as global numercal optmzaton. Many of numercal optmzaton problems cannot be solved analytcally, and consequently, numercal algorthms were proposed to solve these problems. Many of these algorthms are evolutonary algorthms. In CEC 005, a specal sesson of real-parameter optmzaton had been organzed. Several algorthms [][4][5][6][] were presented n ths specal sesson and all these algorthms were tested on a sute of 5 benchmark functons wth dmensons 0 and 30. Besdes these algorthms, some of evolutonary algorthms proposed recently are brefly surveyed n the followng. Kazarls et al. [] combned a standard GA wth a mcrogenetc algorthm (MGA) [GA wth small populaton and short evoluton] to solve the numercal optmzaton problem. In ther method, the MGA operator performs genetc local search. Leung and Wang [3] proposed an orthogonal genetc algorthm wth quantzaton for ths problem. They appled the quantzaton technque and orthogonal desgn to mplement a new crossover operator, such that the crossover operator can generate a small but representatve sample of solutons as the potental offsprng. Tsa et al. [8] hybrdzed the genetc algorthm and the Taguch method. The Taguch method s ncorporated nto the crossover operator and the mutaton operator. Tu and Lu [0] presented the stochastc genetc algorthm (StGA) for numercal optmzaton. They employed a stochastc codng method to code chromosomes. Each chromosome s coded as a Ln-Yu Tseng s wth the Insttute of Networkng and Multmeda and the epartment of Computer Scence and Engneerng, Natonal Chung Hsng Unversty, 50 Kuo Kuang Road, Tachung, Tawan 40, ROC (correspondng author; phone: ; e-mal: lytseng@cs.nchu.edu.tw). Chun Chen s a Ph student wth the epartment of Computer Scence and Engneerng, Natonal Chung Hsng Unversty, 50 Kuo Kuang Road, Tachung, Tawan 40, ROC (e-mal:phd940@cs.nchu.edu.tw). representatve of a stochastc regon descrbed by a multvarate Gaussan dstrbuton. Zhong et al. [] ntegrated multagent systems and genetc algorthms to form a new algorthm called the multagent genetc algorthm (MAGA) for numercal optmzaton. All above mentoned algorthms except the last one were tested on functons wth dmensons less than or equal to 00. The last algorthm (MAGA) was tested on functons wth dmensons from 0 to 0,000. In ths paper, the multple trajectory search (MTS) was presented to solve the large scale global optmzaton problem. The MTS had been used to solve the mult-objectve optmzaton problems and obtaned satsfactory results [9]. We used the same framework but mproved the local search methods for numercal optmzaton. When applyng the MTS to the seven benchmark problems desgned for the CES 008 Specal Sesson and Competton on Large Scale Global Optmzaton [7], the expermental results reveal that the MTS s effectve and effcent n solvng these problems. The remander of ths paper s organzed as follows. Secton II gves some defntons. Secton III descrbes the MTS algorthm. Secton VI presents the expermental results and Secton IV draws the concluson. II. PRELIMINARIES A. Problem efnton The global numercal optmzaton problem s defned as follows. Mnmze F(X) Subject to l X u where X=(x, x,,x N ) s a varable vector n R N space and F(X) s the objectve functon. l=(l, l,,l N )andu=(u, u,, u N ) defne the feasble soluton space, that s, a feasble soluton X=(x, x,,x N )mustsatsfesl x u for =,,, N. The feasble soluton space s denote by [l, u]. B. Orthogonal Array and Smulated Orthogonal Array We brefly ntroduce the concept of orthogonal arrays whch are used n expermental desgn methods. Suppose n an experment, there are k factors and each factor has q levels. In order to fnd the best settng of each factor s level, qk experments must be done. Very often, t s not possble or cost effectve to test all qk combnatons. It s desrable to sample a small but representatve sample of combnatons for testng. The orthogonal arrays were developed for ths purpose. In an experment that has k factors and each factor has q levels, an orthogonal array OA(n,k,q,t) s an array wth n rows and k columns whch s a representatve sample of n testng /08/$5.00 c 008 IEEE Authorzed lcensed use lmted to: UNIVERSITY OF NOTTINGHAM. ownloaded on ecember, 009 at 09:9 from IEEE Xplore. Restrctons apply.

2 experments that satsfes the followng three condtons. () For the factor n any column, every level occurs the same number of tmes. () For the t factors n any t columns, every combnaton of q levels occurs the same number of tmes. (3) The selected combnatons are unformly dstrbuted over the whole space of all the possble combnatons. In the notaton OA(n,k,q,t), n s the number of experments, k s the number of factors, q s the number of levels of each factor and t s called the strength. The orthogonal arrays exst for only some specfc n s and k s. So t s not approprate to use the OA n some applcatons. We proposed the smulated OA (SOA) n ths study. The SOA satsfes only the frst of the above mentoned three condtons, but t s easy to construct an SOA of almost any sze. Suppose there are k factors and each factor has q levels, an m k smulated orthogonal array SOA m k wth m beng a multple of q can be generated as follows. ForeachcolumnofSOA m k, a random permutaton of 0,,, q- s generated and denoted as sequence C. Then the elements n C are pcked one by one sequentally and flled n a randomly chosen empty entry of the column. If all elements n C were pcked, the process pcks elements agan from the begnnng of C. So n every column of SOA m k,eachofq elements wll appear the same number of tmes (condton ). III. MULTIPLE TRAJECTORY SEARCH In ths secton, we present the multple trajectory search (MTS) for the large scale global optmzaton problem. In the begnnng, the MTS generates M ntal solutons by utlzng the smulated orthogonal array SOA M N,where the number of factors corresponds to the dmenson N and the number of levels of each factor s taken to be M. So each of 0,,, M- wll appear once n every column. Usng SOA tends to make these M ntal solutons unformly dstrbuted over the feasble soluton space. The ntal search range for local search methods s set to half of the dfference between the upper bound and the lower bound. Afterwards, local search methods wll change the search range. The MTS conssts of teratons of local searches untl the maxmum number of functon evaluatons s reached. In the frst teraton, the MTS conducts local searches on all of M ntal solutons. But n the followng teratons, only some better solutons are chosen as foreground solutons and the MTS conducts local searches on these solutons. Three local search methods are provded for the MTS. The MTS wll frst test the performance of three local search methods and then choose the one that performs best, that s, the one that best fts the landscape of the neghborhood of the soluton, to do the search. After conductng the search on foreground solutons, the MTS apples Local Search to the current best soluton tryng to mprove the current best soluton. Before the end of an teraton, some better solutons are chosen as the foreground solutons for the next teraton. The multple trajectory search algorthm s descrbed n the followng Multple Trajectory Search /*Generate M ntal solutons */ Buld smulated orthogonal array SOA M N For =tom For j =ton X[j]=l +(u -l )*SOA[, j]/(m-) Evaluate functon values of X s For = to M Enable[] TRUE Improve[] TRUE SearchRangeX =(UPPER_BOUN-LOWER_BOUN)/ Whle ( #ofevaluaton predefned_max_evaluaton) For = to M If Enable[]=TRUE Then GradeX 0 LS_TestGrade 0 LS_TestGrade 0 LS3_TestGrade 0 For j = to #oflocalsearchtest LS_TestGrade LS_TestGrade+ LocalSearch(X, SearchRangeX) LS_TestGrade LS_TestGrade+ LocalSearch(X, SearchRangeX) LS3_TestGrade LS3_TestGrade+ LocalSearch3(X, SearchRangeX) Choose the one wth the best TestGrade and let t be LocalSearchK /* K may be,, or 3 */ For j = to #oflocalsearch GradeX GradeX+ LocalSearchK(X, SearchRangeX) For = to#oflocalsearchbest LocalSearch(BestSoluton, SearchRangeBestSoluton) For = to M Enable[] FALSE Choose #offoreground X s whosegradex are best among the M solutons and set ther correspondng Enable[]toTRUE End Whle In the MTS, three local search methods are used for searchng dfferent landscape of the neghborhood of a soluton. Local Search searches along one dmenson from the frst dmenson to the last dmenson. Local Search s smlar to Local Search except that t searches along about one-fourth of dmensons. In both local search methods, the search range (SR) wll be cut to one-half untl t s less than 0-5 f the 008 IEEE Congress on Evolutonary Computaton (CEC 008) 3053 Authorzed lcensed use lmted to: UNIVERSITY OF NOTTINGHAM. ownloaded on ecember, 009 at 09:9 from IEEE Xplore. Restrctons apply.

3 prevous local search does not make mprovement. In Local Search, on the dmenson concernng the search, the soluton s coordnate of ths dmenson s frst subtracted by SR to see f the objectve functon value s mproved. If t s, the search proceeds to consder the next dmenson. If t s not, the soluton s restored and then the soluton s coordnate of ths dmenson s added by 0.5*SR, agan to see f the objectve functon value s mproved. If t s, the search proceeds to consder the next dmenson. If t s not, the soluton s restored and the search proceeds to consder the next dmenson. Local Search and Local Search are lsted n the followng Functon LocalSearch(Xk, SR) If Improve[k]=FALSE Then SR = SR / If SR <e-5 Then SR (UPPER_BOUN-LOWER_BOUN) *0.4 Improve[k] FALSE For =ton Xk[] Xk[]- SR If Xk s better than current best soluton Then grade grade + BONUS If functon value of Xk s the same Then restore Xk to ts orgnal value If functon value of Xk degenerates Then restore Xk to ts orgnal value Xk[] Xk[]+ 0.5*SR If Xk s better than current best soluton Then grade grade + BONUS If functon value of Xk has not been mproved Then restore Xk to ts orgnal value grade grade + BONUS Improve[k] TRUE grade grade + BONUS Improve[k] TRUE return grade Functon LocalSearch(Xk, SR) If Improve[k]=FALSE Then SR=SR/ If SR < e-5 Then SR (UPPER_BOUN-LOWER_BOUN)*0.4 Improve[k] FALSE For l =ton For =ton r[] Random{0,,,3} [] Random{-,} For =ton If r[]=0 Then Xk[] Xk[]-SR*[] If Xk s better than current best soluton Then grade grade + BONUS If functon value of Xk s the same Then restore Xk to ts orgnal value f functon value of Xk degenerates Then restore Xk to ts orgnal value For =0toN If r[]=0 Then Xk[] Xk[]+0.5*SR*[] If Xk s better than current best soluton Then grade grade + BONUS If functon value of Xk has not been mproved Then restore Xk to ts orgnal value grade grade + BONUS Improve[k] TRUE grade grade + BONUS Improve[k] TRUE return grade Local Search 3 s dfferent from Local Search and Local Search. Local Search 3 consders three small movements along each dmenson and heurstcally determnes the movement of the soluton along each dmenson. In Local Search 3, although the search s along each dmenson from the frst dmenson to the last dmenson, the evaluaton of the objectve functon value s done after searchng all the dmensons, and the soluton wll be moved to the new poston only f the objectve functon has been mproved at ths evaluaton. Local Search 3 s descrbed n the followng. Functon LocalSearch3(X,SR) IEEE Congress on Evolutonary Computaton (CEC 008) Authorzed lcensed use lmted to: UNIVERSITY OF NOTTINGHAM. ownloaded on ecember, 009 at 09:9 from IEEE Xplore. Restrctons apply.

4 For =ton X X s th coordnate s ncreased by 0. Y X s th coordnate s decreased by 0. X X s th coordnate s ncreased by 0. If X s better than current best soluton Then grade grade + BONUS If Y s better than current best soluton Then grade grade + BONUS If X s better than current best soluton Then grade grade + BONUS = F(X)-F(X ) If >0 /*X s better than X*/ Then grade grade + BONUS = F(X)-F(Y ) If >0 /*Y s better than X*/ Then grade grade + BONUS 3 = F(X)-F(X ) If 3 >0 /*X s better than X*/ Then grade grade + BONUS a Random[0.4, 0.5] b Random[0., 0.3] c Random[0, ] X[]=X[]+a( - )+b( 3 - )+c If functon value of X has not been mproved Then restore X to ts orgnal value grade grade + BONUS return grade In our experments, M s set to 5 and #offoreground s set to 3. Hence three best solutons are n the foreground and the other two solutons are n the background. The MTS apples the local search method that best fts the landscape of a soluton s neghborhood to ths soluton. From teraton to teraton, a soluton n the foreground may be put n the background and vce verse. IV. EXPERIMENTAL RESULTS The proposed MTS was appled to the seven benchmark functons for CEC008 Specal Sesson and Competton on Large Scale Global Optmzaton. The PC confguraton and the parameter settng are stated n the followng. A. PC Confguraton For Problems -6 System: Wndows XP RAM: GB CPU: Intel Pentum.66GHz For Problem 7 System: Lnux RAM: GB CPU: Intel Xeon E530.6GHz Language: C++ B. Parameter Settng M=5 #offoreground=3 #oflocalsearchtest=3 #oflocalsearch=00 #oflocalsearchbest=50 BONUS=0 BONUS= a=random[0.4, 0.5], b=random[0., 0.3] c=random[0, ], C. Expermental Results The functon error value (f(x)-f(x*)) after FES/00, FES/0, and Max_FES are lsted n Table, and 3 respectvely for seven problems wth dmensons 00, 500 and 000. Seven benchmark problems are lsted n Table 4. The detals can be found n [7]. The convergence graphs for problems -6 and problem 7 wth dmenson =000 are depcted n Fgure and Fgure respectvely. V. CONCLUSIONS The multple trajectory search (MTS) was presented and appled to solve the seven benchmark problems provded for the purpose of competton n CEC 008. The seven problems are wth dmensons 00, 500 and 000, so the scalablty of algorthms can also be tested. When MTS searches the neghborhood of a soluton, t tests the performance of the three predefned local search methods frst and then chooses the best one to conduct the local search. Therefore, the MTS performs ts search automatcally adaptng to the landscape of the soluton s neghborhood. For problems -6 whose optmal solutons are already known, the MTS can fnd the optmal solutons of four problems wth FES/0 when dmenson s 00. Also, t can fnd the optmal solutons of two and three problems wth FES/0 when dmenson s 500 and 000 respectvely. ACKNOWLEGMENTS The authors gratefully acknowledge the support of Natonal Scence Councl of ROC under the contract NSC E MY3. REFERENCES [] P. J. Ballester, J. Stephonson, J. N. Carter and K.Gallagher, Real-parameter optmzaton performance study on the CEC-005 benchmark wth SPC-PNX, Proceedngs of 005 IEEE Congress on Evol. Comput.,.pp , 005. [] S. A. Kazarls, S. E. Papadaks, J. B. Theochars, and V. Petrds, Mcrogenetc algorthms as generalzed hll-clmbng operators for GA optmzaton, IEEE Trans. Evol. Comput., vol. 5, pp. 04 7, Jun. 00. [3] Y. W. Leungand Y. Wang, An orthogonal genetc algorthm wth quantzaton for global numercal optmzaton, IEEE Trans. Evol. Comput., vol. 5, pp. 4 53, Feb. 00. [4] P. Posk, Real-parameter optmzaton usng the mutaton step co-evoluton, Proceedngs of 005 IEEE Congress on Evol. Comput.,.pp , IEEE Congress on Evolutonary Computaton (CEC 008) 3055 Authorzed lcensed use lmted to: UNIVERSITY OF NOTTINGHAM. ownloaded on ecember, 009 at 09:9 from IEEE Xplore. Restrctons apply.

5 [5] J. Ronkkonen, S. Kukkonen and K.V. Prce, Real-parameter optmzaton wth fferental Evoluton, Proceedngs of 005 IEEE Congress on Evol. Comput.,.pp , 005. [6] A. Snha, S. Twar and K. eb, A populaton-base, state procedure for real-parameter optmzaton, Proceedng of 005 IEEE Congress on Evol. Comput., pp. 54-5, 005. [7] K. Tang, X. Yao, P. N. Suganthan, C.MacNsh, Y. P. Chen, C. M. Chen and Z. Yang, Benchmark functon for the CEC 008 Specal Sesson and Competton on Large Scale Global Optmzaton, Techncal Report, [8] J. T. Tsa, T. K. Lu, and J. H. Chou, Hybrd Taguch-genetc algorthm for global numercal optmzaton, IEEE Trans. Evol. Comput., vol. 8, pp , Aug [9] L. Y. Tseng and C. Chen, Multple trajectory search for multobjectve optmzaton, Proceedngs of 007 IEEE Congress on Evol. Comput.,.pp , 007. [0] Z. Tu and Y. Lu, A robust stochastc genetc algorthm for global numercal optmzaton, IEEE Trans. Evol. Comput., vol. 8, pp , Oct [] B. Yuan, M. Gallagher, Expermental result for the Specal Sesson on real-parameter optmzaton at CEC 005: a smple, contnuous EA, Proceedngs of 005 IEEE Congress on Evol. Comput.,.pp , 005. [] W. Zhong, J. Lu, M. Xue, and L. Jao, A multagent genetc algorthm for global numercal optmzaton, IEEE Trans. on system, man,and cybernetcs Part B, pp. 8-4, Apr Table Error values acheved for problems -6 and functon values for problem 7, wth =00 FES Prob th (best).7647e E+0.447E E E E E+03 7 th.054e E E E+0.553E+0.654E E+03 3 th (medan).5884e E+0.774E E+0.769E+0.336E E e+3 9 th.870e E E E+0.886E+0.379E E+03 5 th (worst).4348e E+0.03E E+0.808E+0.458E E+03 Mean.436E E E+0 4.E+0.69E+0.776E E+03 Std 5.665E E E E E+0.668E E+00 th (best) E+00.90E-0.369E E E E E+03 7 th E+00.68E E E E E E+03 3 th (medan) E E E E E E E e+4 9 th E E-0.333E E E E E+03 5 th (worst) E E-0.480E E E E E+03 Mean E E E E E E E+03 Std E E E-05.03E E E E+00 th (best) E E E E E E E+03 7 th E E E E E E E+03 3 th (medan) E E E E E E E e+5 9 th E E-.5000E E E E E+03 5 th (worst) E E E E E E E+03 Mean E E E E E E E+03 Std E E-.6085E E E E E IEEE Congress on Evolutonary Computaton (CEC 008) Authorzed lcensed use lmted to: UNIVERSITY OF NOTTINGHAM. ownloaded on ecember, 009 at 09:9 from IEEE Xplore. Restrctons apply.

6 Table Error values acheved for problems -6 and functon values for problem 7, wth =500 FES.50e+4.50e+5.50e+6 Prob th (best) E E+0.85E-0.379E E+0.353E E+03 7 th 9.75E E+0.535E E E E E+03 3 th (medan) E E+0.85E E E E E+03 9 th.0495e E+0.853E+0.49E E E E+03 5 th (worst).3e e+0.994e e e e e+03 Mean 9.956E E+0.330E E E E E+03 Std.0638E E E E E+0.63E E+0 th (best) E E E E E E- -7.3E+03 7 th E+00.97E E E E E E+03 3 th (medan) E E E E E E E+03 9 th E E E E E E E+03 5 th (worst) E E E E+0.39E E E+03 Mean E E E E E E E+03 Std E+00.50E E E E E-.490E+0 th (best) E E-06.46E E E E E+03 7 th E E E E E E E+03 3 th (medan) E E E E E E E+03 9 th E E E E E E E+03 5 th (worst) E+00.9E E E E E E+03 Mean E E E E E E E+03 Std E+00.48E E E E E E+0 Table 3 Error values acheved for problems -6 and functon values for problem 7, wth =000 FES 5.00e e e+6 Prob th (best).899e E E-0.885E E+0.358E E+04 7 th.977e E E E E E E+04 3 th (medan).00e E E E E E E+04 9 th.698e E E E E E E+04 5 th (worst).5055e e E E E+03.45E E+04 Mean.0957E E E E E E E+04 Std.85E E E E E+0.908E E+0 th (best) E E E E E E E+04 7 th E E+0.333E-0.996E E+00.68E E+04 3 th (medan) E E+0.333E E E-4.95E E+04 9 th.0687e E+0.333E E E E E+04 5 th (worst) 3.4E E+0.365E-0.494E+03.39E E E+04 Mean.359E E+0.980E E E E E+04 Std 6.435E E E E E E E+0 th (best) E E E E E E E+04 7 th E E E E E+00.08E E+04 3 th (medan) E E E E E+00.39E E+04 9 th E E E E E E E+04 5 th (worst) E E E E E E E+04 Mean E E E E E E E+04 Std E E E E E E E IEEE Congress on Evolutonary Computaton (CEC 008) 3057 Authorzed lcensed use lmted to: UNIVERSITY OF NOTTINGHAM. ownloaded on ecember, 009 at 09:9 from IEEE Xplore. Restrctons apply.

7 Table 4 Benchmark problems Optmal soluton Benchmark problems Search range value F(x*) F = z + f _ bas, z = x o = [-00,00] -450 o = [ o, o,..., o ] : the shfted global optmum. o = [ o, o,..., o F = max{ z, } + f _ bas, z = x o ]: the shfted global optmum. F3 = (00(z - z+ ) + (z -) ) + f _ bas3, z = x o + = o = [ o, o,..., o ]: the shfted global optmum. o = [ o, o,..., o F4 = (z = -0cos(πz ) + 0) + f _ bas, z = x o ] : the shfted global optmum. 4 [-00,00] -450 [-00,00] 390 [-5,5] -330 z z F5 = cos( ) + + f _ bas5, z = x o = 4000 = [-600,600] -80 o = [ o, o,..., o ] : the shfted global optmum. z = x o, o = [ o, o,..., o F6 = 0 exp( 0. z ) exp( cos(πz )) f _ bas6, F = 7 = twst ( y) = 4( y fractal( x) y fractal ( x 4 = wth equal probablt y from the 3 k = = ]: the shfted global optmum. k ran ( o ) + twst ( x doubledp ( x, ranl ( o), 6 4 ( 644 ( x c) ( x c) 39( x c) doubledp ( x, c, s) = 0, otherwse ranl ( o) : double, pseudorand omly chosen, wth seed o, nterval [0,]. ran(o) : nteget, pseudorand omly choose, wth seed o, wthequal probablt y from the set {0,,} 3 + y ) ( mod ) + )) k ) ( ranl ( o)) + ) s, 0.5 < x < 0.5 [-3,3] -40 [-,] unknown IEEE Congress on Evolutonary Computaton (CEC 008) Authorzed lcensed use lmted to: UNIVERSITY OF NOTTINGHAM. ownloaded on ecember, 009 at 09:9 from IEEE Xplore. Restrctons apply.

8 Log(f(x)-f(x*)) Convergence Graphs(000) f f f3 f4 f5 f FEs 0 5 Fgure Convergence graphs for problems -6 wth dmenson = F7 F7-000 f(x) FEs 0 5 Fgure Convergence graph for problem 7 wth dmenson = IEEE Congress on Evolutonary Computaton (CEC 008) 3059 Authorzed lcensed use lmted to: UNIVERSITY OF NOTTINGHAM. ownloaded on ecember, 009 at 09:9 from IEEE Xplore. Restrctons apply.