A hybrid global optimization algorithm based on parallel chaos optimization and outlook algorithm

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1 Avalable onlne Journal of Chemcal and Pharmaceutcal Research, 2014, 6(7): Research Artcle ISSN : CODEN(USA) : JCPRC5 A hybrd global optmzaton algorthm based on parallel chaos optmzaton and outlook algorthm Xaolan Wu 1 and Gufang Guo 2 1 School of Mechancal and Electrcal Engneerng, X'an Unversty of Archtecture and Technology, X an, Chna 2 School of Informaton Engneerng, Tbet Unversty for Natonaltes, Xanyang, Chna ASTRACT ased on the analyss of the propertes of tent map, parallel chaos optmzaton algorthm (PCOA) usng logstc map and outlook algorthm, a hybrd global optmzaton algorthm (PCOOA) s presented. The algorthm s structured n two stages. The frst stage uses parallel chaos optmzaton based on tent map for global search, whle outlook algorthm s employed n the second stage for local search. The smulaton results show that the algorthm, wth hgh rate of convergence, optmzaton effcency and strong robustness etc., s an effectve technque to solve nonlnear programmng (NLP) problems. Keywords: parallel chaos optmzaton; outlook algorthm; global optmzaton; tent map INTRODUCTION Chaos s a unversal phenomenon exstng n many systems n all areas of scence [1]. It has three mportant dynamc propertes: the ntrnsc stochastc property, ergodcty and regularty [2]. A chaotc movement can go through every state n a certan area accordng to ts own regularty, and every state s obtaned only one. Takng advantage of chaos, a new searchng algorthm called chaos optmzaton algorthm (COA) s presented [3]. It can more easly escape from local mnma compared wth the extng stochastc searchng algorthm. However, the COA has the shortage of the senstve dependence on ntal condton, tny dfference n ntal value, there may be carryng completely searchng process. Some states may be reached costng longer tme. Therefore, the search tme wll be very long, f the global optmal soluton appears ncely n these states. Parallel chaos optmzaton algorthm was proposed to solve ths problem by searchng synchronously from several ntal ponts [4].Whereas, further research show that ths method has the lower searchng effcency near the optmum pont owng to stochastc property of chaotc movement [5]. Outlook algorthm s proposed accordng to common knowledge that one decdes the hghest pont of mountans by outlook. It can solve global optmzaton problem by employng supervson mechansm of outlook, strateges of generatng outlook ponts and mechansms of constructng and solvng local problems [6]. Outlook algorthm has hgh rate of convergence, fast search velocty and strong robustness, etc. In ths paper, we propose a hybrd algorthm based on PCOA and outlook algorthm. The algorthm s structured n two stages. The frst stage uses parallel chaos optmzaton based on tent map for global search. To accelerate local 1884

2 Xaolan Wu and Gufang Guo J. Chem. Pharm. Res., 2014, 6(7): convergence and rapdly generate an exact soluton, outlook optmzaton algorthm s subsequently employed n the second stage. The smulaton results show that the algorthm s more effectve than PCOA. Ths paper s organzed as follows. Secton II formulates nonlnear programmng problem. Secton III summarzes frstly PCOA and outlook algorthm, then presents the hybrd global optmzaton algorthm based on PCOA and outlook algorthm. In Secton IV, some examples are tested by the proposed method and the results are compared wth PCOA.. Conclusons n Secton V close the paper. EXPERIMENTAL SECTION Problem formulaton A NLP problem wth nequalty constrants s a problem that can be put nto the form: mnmze f ( x) sbuect to g ( x) 0, = 1, 2,, m x [ a, b ], = 1,2,, n Where x s a vector of n component x1,, xn, f ( x ) s the obectve functon and g ( x ) s the nequalty constrants. Hybrd global optmzaton algorthm A. PCOA PCOA was proposed by Lang [4]. Its man dea s searchng the soluton space by dfferent several group chaos sequences. Frstly, use the carrer wave method to make optmzaton varables vary to chaos varables. Secondly, amplfy the ergodc area of chaotc moton to the varaton ranges of every controllable varable. Fnally, use the chaos search method to optmzaton problem. The method s summarzed as followng: Intalzaton. Set n = 1, x(,, n+ = 4 x(,, n) (1 x(,, n) ) Where = 1,, p, represents the dfferent ntal startng ponts of classes, = 1,, N, expresses the varable number ncluded n the optmzed problem, n s the nteraton tmes. 2) Carryng out the frst carrer wave. The chaos varables are mported nto the optmzed varables, moreover, the change range of the chaos varables are separately amplfed the correspondng value range of optmzed varables. ' x = c + d x (,, n+ (, ) (, ) (,, n+ Where (,, n x + s chaos varable, c(, ) and d (, ) are constants, x ' (,, n+ s varable used for optmzed problem. 3) Carryng out teraton search. In each generaton, set the optmal soluton of all classes as the current soluton. If no better soluton s found after N searches, the second carrer wave wll be executed accordng to the followng equaton: '' x = x + α x (, n+ (, n+ Where x s the current soluton, α s regulaton constant, x (, n+ s chaos varable. 4) Performng teraton search. If no better soluton s found after M searches, stopng search and output current optmal soluton.. Outlook algorthm Outlook algorthm was proposed by Ca [6]. It s composed by three parts: supervson mechansm of outlook, strateges of generatng outlook ponts and mechansms of constructng and solvng local problems. It can solve global optmzaton problem accordng to the followng route: 1885

3 Xaolan Wu and Gufang Guo J. Chem. Pharm. Res., 2014, 6(7): Conformng basc pont by supervson mechansm of outlook; 2) Generatng outlook pont of base pont by strateges of generatng outlook ponts; 3) Choosng outlook pont accordng to gven standard by supervson mechansm of outlook; 4) Constructng the local problem of outlook pont and solvng t by local optmzaton algorthm; 5) After gettng all the solutons of local problems chosen, conformng next base pont and begn a new teraton untl satsfyng end condton and put out soluton. C. Chaos varables Tent map has better ergodcty unformty than logstc map, so the COA based on tent map has better optmzaton effcency [6]. In addton, tent map has smple structure and nteraton process s ft for computng by computer [7]. In ths paper chaos varables are generated by tent map. The tent map s defned by: 2 γ ( k) 0 γ ( k) 1/ 2 γ ( k + = 2(1 γ ( k)), 1/ 2 < γ ( k) 1 ( After shft transformng, t can be expressed as the followng equaton: γ ( k + = (2 γ ( k)) mod 1 (2) Its output s lke a stochastc output, no value of γ ( k) s repeated and the determnstc equaton s senstve to ntal condtons. Those are the basc characterstcs of chaos. Moreover, the essental of carryng out tent map by computer s left shft wthout symbol about bnary dgt of decmal fracton. Ths operaton use adequately computer s characterstc and more suts for large magntude data sequence. D. Proposed algorthm Frst, usng PCOA based on tent map for global search. It s easy to reach the area near global optmzaton soluton owng to the ergodcty. However, local searchng speed become very slowly and t s dffcult to get the hgh precson optmzaton soluton due to the stochastc property of algorthm. Thus the outlook optmzaton algorthm s employed n the second stage for local search. Hgh searchng effcency s obtaned after untng PCOA wth outlook algorthm. The method s presented as followng: Step Intalze chaos varable γ (0), 0 γ (0) 1, ( = 1,2,, n = 1,2,, p), by means of stochastc way, whch have small dfferences. There wll generate p n chaos varables havng dfferent track. The postve ntegers N 1, N 2 are specfed. Let flag = 1, C = 0, k = 0 ; where flag s outlook symbol, C s base pont counter, k s teraton tmes. Step 2) Chaos varable γ (0) s mapped nto the varance ranges of optmzaton varables by the followng equaton: x (0) = a + γ (0)( b a ) ( = 1,2,, n = 1,2,, p) Let f = f ( X (0)), X = X (0), f = mn( f ), X = X, where X s the best soluton of the team, Step 3) Carry out chaos search by usng the carrer wave: (3) X s the global best soluton. Repeat γ ( k + = (2 γ ( k)) mod

4 Xaolan Wu and Gufang Guo J. Chem. Pharm. Res., 2014, 6(7): x ( k + = a + ( b a ) γ ( k + ( = 1,2,, n = 1, 2,, p) If Then f ( X ( k + ) < f, X = X ( k +, f = f ( X ( k + ) Else f f ( X ( k + ) f, Then gve up X ( k + If mn f Then f f <, f =, Else do nothng. Let k k + 1 untl X = X f does not mprove after N 1 searches. Step 4) Set X = X, where X s outlook base pont. Step 5) If flag = 1 and C < N 2 Then carry out Step 6) Else go to step 7). Step 6) Generatng outlook pont of base pont X o ( = 1, 2,, m) accordng to strateges of generatng outlook ponts provded by paper [6] (ths paper use strateges of generatng outlook ponts based on square body). o Step 7) Whle f ( X ) f ( X ) Carry out local search and get local optmum soluton search method such as smplex search, gradent search etc. l If mn f ( X ) < f ( X ), l l X from the pont o X.The search strateges can use local Then X = X, flag = 1, return to step 7). Else carry out step 8). Step 8) Stop the search process and put out X = X as the best soluton, f = f ( X ) as the best value. RESULTS AND DISCUSSION Test functons To evaluate the effcency and effectveness of the algorthm for nonlnear programmng problems, we choose the followng four complex test functons for smulaton. These functons entrap nto local optmum easly. There are numerous local optma around the global optmum. The optmal soluton and exact optmum of functons n theory s shown n Table F1: mn f ( X ) = 100( x x ) + (1 x ), x 2.048, = 1, F2 : mn f ( X ) = [1 + ( x + x + (19 14x + 6x x x )] [30 + (2x 3 x ) (18 32x + 12x + 48x x2 36x x + 27 )] 2 x 2, = 1,

5 Xaolan Wu and Gufang Guo J. Chem. Pharm. Res., 2014, 6(7): x F3: mn f ( X ) = (4 2.1 x + x / 3) x + x x + ( x ) -2 x 2, = 1, 2 x1 + x2 x1 x2 sn 0.5 F4 : mn f ( X ) = 0.5, ( ( + )) 4 x 4, = 1, 2 Table-1 The optmal soluton and exact optmum of functons n theory (- OPT. SOLUTION (1, ,0.7126) (0, (0, - (0.0898,- 0) ) EXACT OPT Numercal results Ths paper computes separately the above four test functons 100 runs usng the proposed method (PCOOA) and PCOA [2].The platform used s a AMD 2.19GHz PC. The convergence rate of two algorthm are shown n table-2, the average teraton tme s shown n table-3, the average computng tme of fndng the optmum s shown n table- 4. Table-2 Convergence rate of two algorthms Algorthm Convergence rate (%) PCOA PCOOA The data of table-2 show that both of the PCOA and the PCOOA can fnd the global optmal solutons as convergence rate 1. Table-3 Iteraton tmes of two algorthms Algorthm Iteraton tmes PCOA PCOOA Table-4 Computng tme of two algorthms algorthm Computng tme (sec) PCOA PCOOA The data of Table-3 and Table-4 show that the PCOOA uses less tme and fewer nteraton tmes than PCOA for fndng the global optmum solutons of NLP. It s apparent that the proposed algorthm s superor to PCOA n computng tme and searchng effcency. CONCLUSION Ths paper proposed a hybrd algorthm based on parallel chaos optmzaton and outlook algorthm for solvng NLP problems. PCOA performs a global search n the frst stage. To accelerate local convergence n the second stage, the outlook algorthm s subsequently adopted n the second stage. A comparatve study between the hybrd algorthm and PCOA s conducted. The results show that the hybrd algorthm wth hgh rate of convergence and search effcency, s an effectve method to solve NLP problems. REFERENCES 1888

6 Xaolan Wu and Gufang Guo J. Chem. Pharm. Res., 2014, 6(7): [1] Jefferes D J, Deane J H, Johnstone G G. An ntroducton to chaos [J]. Electroncs & Communcaton Engneerng Journal, 1989, 1(3): [2] Wu X X, Chen Z. Introducton of chaos theory [J]. Shangha Scence and Technology blographc Publshng House, Shangha, [3] JIANG L I W. Optmzng complex functons by chaos search [J]. Cybernetcs & Systems, 1998, 29(4): [4] Lang H Y, Gu X S. A novel chaos optmzaton algorthm based on parallel computng [J]. Journal of East Chna Unversty of Scence and Technology (Natural Scence Edton), 2004, 30(4): [5] We T. Chaotc Optmzaton Method ased on Power Functon Carrer and Its Applcatons [J]. Control and Decson, 2005, 9: 016. [6] Yanguang Ca, Jxn Qan, and Youxan Sun. The global optmzed outlook algorthm [J]. Journal of Guangdong Unversty of Technology, 2006, 23(2):1-10. [7] Shan L, Qang H, L J, et al. Chaotc optmzaton algorthm based on Tent map [J]. Control and Decson, 2005, 20(2):

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