Stochastic optimization algorithm with probability vector in mathematical function minimization and travelling salesman problem

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1 Stochastc optmzaton algorthm wth probablty vector n mathematcal functon mnmzaton and travellng salesman problem JAN POHL, VÁCLAV JIRSÍK, PETR HONZÍK Department of Control and Instrumentaton Brno Unversty of Technology Kolejní 2906/4, Brno CZECH REPUBLIC xpohlj00@stud.feec.vutbr.cz, jrsk@feec.vutbr.cz, honzkp@feec.vutbr.cz Abstract: In the paper there s ntroduced the newly developed optmzaton method the Stochastc Optmzaton Algorthm wth Probablty Vector (PSV). It s related to Stochastc Learnng Algorthm wth Probablty Vector for artfcal neural networks. Both algorthms are nspred by stochastc terated functon system SIFS for generatng the statstcally self smlar fractals. The PSV s gradent method where the drecton of ndvdual future movement from the populaton s based stochastcally. PSV was tested on mathematcal functon mnmzaton and on the travellng sales man problem. The nfluence of the quantty of ndvduals upon the best acheved ftness functon was also tested on the mathematcal functons mnmzaton. Key Words: Stochastc, Optmzaton Algorthm, PSV, SIFS, Random Walk, Travellng Salesman Problem 1 Introducton There are many optmzaton algorthms, e.g. gradent, stochastc, nspred by collectve behavour of bologcal ndvduals etc [1, 3, 4, 9 and 10]. The PSV algorthm that s ntroduced n ths paper s a modfed verson of the stochastc learnng algorthm [5, 6]. The algorthm s nspred by the stochastc terated functon system SIFS for generatng the statstcally self smlar fractals [11, 12]. It s based on the group of separate ndvduals, whch do not share nformaton wth each other. It s based on the smlar prncple as the stochastc hll clmbng [2, 7] or random walk (RW). 2 Algorthm PSV PSV algorthm uses ndvduals from the group n the smlar way as n the stochastc hll clmbng. There s defned a vector of transformatons (1). ( t, t 2, ) T, = (1) 1 t n Each of them can modfy any parameter of an ndvdual n a specfc way. In every step the transformaton s chosen randomly but wth regard to the probablty n the probablty vector (2) that has to satsfy the condtons (3) and (4). Ths step s smlar to the algorthm random walk (RW) [7]. ( p, p, 2 ) P, = (2) 1 p n n = 1 p = 1 (3) ( 1,1) p 0 p (4) Each value n the probablty vector can be nterpreted as the probablty (5) and the drecton of movement (6). p probablty (5) sgn( p ) drecton (6) If the chosen transformaton s accepted, then the probablty of ths transformaton ncreases. Transformaton s accepted f the ftness functon gves better result after ts applyng. On the other hand f the transformaton gves worse results ts probablty decreases. Let s have an ndvdualw k W. Ths ndvdual conssts of a set of parameters (7). ( w, w, 2 ) W, = (7) k 1 w n At frst the ftness functon of the selected ndvdual s evaluated. Secondly the roulette wheel selecton s used to choose and apply the transformaton from (1). ISSN: Issue 7, Volume 7, July 2010

2 ndvdual t ( ndvdual) = new (8) Fnally the ftness functon of the new ndvdual s evaluated. In the case the new ndvdual has hgher ftness functon than the prevous the related probablty p s ncreased. p = p + sgn( p )α _ new (9) Symbol α s coeffcent used for ncreasng the probablty. If the new ndvdual has lower ftness functon than the prevous the related probablty p s decreased. new = p+ sgn ( p )( β) p _ (10) Symbol β s the coeffcent used for decreasng the probablty. Example of the basc transformatons s (11). w pλ + ( ) = w w 1 w new = t _ (11) Transformaton (11) transforms parameter w of the ndvdual W k n specfc drecton and the coeffcent λ s a learnng (for artfcal neural networks) or movng rate. 3 Functon mnmzaton and travellng salesman problem We successfully tested the PSV method as the learnng algorthm for artfcal neural networks [5, 6]. In ths paper was ntroduced the modfed PSV algorthm. It s used to solve the travellng salesman problem and to fnd extremes of selected mathematcal testng functons. We used the followng test functons: 1. 1st De Jong 2. Rodenbrock s saddle 3. 3rd De Jong 4. 4rd De Jong 5. Rastrgn s functon 6. Schwefel s functon 7. Grewangk s functon 8. Sne envelope wave functon 9. Stretched V sne wave functon 10. Test functon Ackley 11. Ackley s functon 12. Mchalewcz s functon 13. Masters s cosne wave functon Optmzaton of the travellng salesman problem was tested on the crcle and randomly deployed ctes. 3.1 Functon mnmzaton The examples of the tested functons are n Tab. 1 together wth the best results found after steps. Every functon was tested wth 50 ndvduals and dfferent sets of the varables wth parameters α, β, λ. f.num. var. x 1 var. x 2 ft.fun Tab. 1 Functon results On the fgures n paragraph there are examples of the some functon from the Tab.1. Markers (ellpse and cross) represent estmated poston of global mnmum n the specfc nterval. The red dot represents the ndvdual wth the best ftness functon Rastrgn s functon Tested wth parameters α = 0.3, β= 0.4, λ= 0.1. Fg.1 Rastrgn s functon ISSN: Issue 7, Volume 7, July 2010

3 Fg.2 Rastrgn s functon startng poston Fg.5 Schwefel functon startng poston Fg.3 Rastrgn s functon poston after steps Schwefel s functon Tested wth parameters α = 0.3, β= 0.4, λ= 0.1. Fg.6 Schwefel functon poston after steps Grewangk s functon Tested wth parameters α = 0.3, β= 0.4, λ= 1. Fg.4 Schwefel functon Fg.7 Grewangk s functon ISSN: Issue 7, Volume 7, July 2010

4 Fg.8 Grewangk s functon startng poston Fg.11 Ackley functon startng poston Fg.9 Grewangk s functon poston after steps Ackley functon Tested wth parameters α = 0.1, β= 0.9, λ= 1. Fg.12 Ackley functon poston after steps Sne envelope wave functon Tested wth parameters α = 0.3, β= 0.4, λ= 1. Fg.10 Ackley functon Fg.13 Sne envelope wave functon ISSN: Issue 7, Volume 7, July 2010

5 Fg.14 Sne envelope wave functon startng poston Fg.17 Test functon Ackley startng poston Fg.15 Sne envelope wave functon poston after steps Test functon Ackley Tested wth parameters α = 0.3, β= 0.4, λ= 0.1. Fg.18 Test functon Ackley poston after steps Mchalewcz functon Tested wth parameters α = 0.3, β= 0.4, λ= Fg.16 Test functon Ackley Fg.19 Mchalewcz functon ISSN: Issue 7, Volume 7, July 2010

6 Fg.20 Mchalewcz functon startng poston Fg.23 Masters s cosne wave functon startng poston Fg.21 Mchalewcz functon poston after steps Master s cosne wave functon Tested wth parameters α = 0.3, β= 0.4, λ= 0.1. Fg.24 Master s cosne wave functon poston after steps 3.2 Travellng salesman problem The PSV algorthm was slghtly modfed for the purposes of the travellng salesman problem. Because the combnaton of the ctes s dscrete, the step λ equals 1 and the probablty vector only determnes the drecton of the movement. Then the transformaton (11) s modfed to (12). w sgn( p )λ + ( ) = w w 1 w new = t _ (12) Parameters of the algorthm are α = 0.3, β = 0.3, λ = 1, number of ndvduals = 5. Fg.22 Master s cosne wave functon ISSN: Issue 7, Volume 7, July 2010

7 3.2.1 Random ctes poston Crcle ctes poston Fg.25 Random ctes poston Fg.28 Crcle ctes poston Fg.26 After 1000 steps Fg.29 After steps Fg.27 After steps Fg.30 After steps 4 Influence of the quantty of ndvduals upon the best acheved ftness functon At the same tme wth functon mnmzaton the nfluence of the ndvduals quantty upon the best reached ftness functon was tested. Every test had to fulfl the same rule: ftness functon callng. It s steps (ftness functon callng) for populaton of ISSN: Issue 7, Volume 7, July 2010

8 one ndvdual and 2000 steps per ndvdual for populaton of the sze 50. The varables α, β, λ were the same as n examples The colour lnes on the examples are average values from the ten experments of evoluton of the ftness functon for every populaton (1, 10, 30, and 50). Red (dash-dot) lne represents evoluton of the ftness functon for populaton of the 1 ndvdual. Green (dashed) lne represents evoluton of the ftness functon for populaton of the 10 ndvduals. Blue (dotted) lne represents evoluton of the ftness functon for populaton of the 30 ndvduals and magenta (sold) lne represents evoluton of the ftness functon for populaton of the 50 ndvduals Rastrgn s functon Tested wth parameters α = 0.3, β= 0.4, λ= 0.1. Fg.33 Grewangk s functon Ackley functon Tested wth parameters α = 0.1, β= 0.9, λ= 1. Fg.31 Rastrgn s functon Schwefel s functon Tested wth parameters α = 0.3, β= 0.4, λ= 0.1. Fg.34 Ackley functon Sne envelope wave functon Tested wth parameters α = 0.3, β= 0.4, λ= 1. Fg.32 Schwefel functon Grewangk s functon Tested wth parameters α = 0.3, β= 0.4, λ= 1. Fg.35 Sne envelope wave functon ISSN: Issue 7, Volume 7, July 2010

9 4.1.6 Test functon Ackley Tested wth parameters α = 0.3, β= 0.4, λ= 0.1. Fg.38 Master s cosne wave functon Fg.36 Test functon Ackley Mchalewcz functon Tested wth parameters α = 0.3, β= 0.4, λ= Fg.37 Mchalewcz functon Master s cosne wave functon Tested wth parameters α = 0.3, β= 0.4, λ= Concluson The newly developed algorthm PSV was ntroduced n ths paper. The am was to show ts possble usage n two dfferent felds of optmzaton (fndng extreme and permutaton problem). Algorthm testng s stll at the begnnng phase. The man advantages of the PSV algorthm are ts mplementaton smplcty and adaptablty to dfferent types of optmzaton problems. PSV can treat both the ntegers and the real numbers. The examples of solvng functon mnmzaton problem (real number codng) and permutaton problem (postve nteger codng) were shown n chapters 3.1 and 3.2. In chapter 3.1, 2D functons were chosen because t s possble to dsplay them and t s possble to dsplay the movement of ndvduals durng optmzaton n 2D state space. The dot (green) represents the best ftness functon reached by sngle ndvdual durng whole optmzaton process and the red dot represents the best ftness functon reached n populaton. In chapter 4 the nfluence of the populaton quantty upon the best reached ftness functon was tested. As we expect n most of the examples the populaton wth several ndvduals s more successful. Future work wll be focused on the extenson of the current algorthm wth dstrbuton of the nformaton among ndvduals n the populaton and on a modfcaton of the algorthm wth adaptve movng and adaptve momentum coeffcents λ, α, β. Acknowledgements Ths work was supported by the Mnstry of Educaton of the Czech Republc under Projects GA102/09/1897 and MSM ISSN: Issue 7, Volume 7, July 2010

10 References: [1] Balck. J.: Tabu-based Evolutonary Algorthm wth Negatv Selecton for PAreto-optmzaton n Dstrbuted Systems, (AIKED 08), 7th WSEAS nternatonal conference on Artfcal ntelgence, knowledge engneerng and data bases, Cambrdge UK, pages , (2008), ISSN [2] Jan Ye, Junfe Qao, Mng-A L,Xaogang Ruan: A tabu based neural network learnng algorthm, ScenceDrect NEUROCOMPUTING 70 (2007) , [3] Olteanu A. L.,: Ant colony optmzaton metaheurstc n project schedulng, (AIKED 09), 8th WSEAS nternatonal conference on Artfcal ntelgence, knowledge engneerng and data bases, Cambrdge UK, pages 29-34, (2009), ISSN [4] Onwubolu. C. G., Babu B.V..: New Optmzaton Technques n Engneerng, Sprnger-Hedelberg 2004, ISBN X [5] Pohl, J., Polách, P.: Stochastc Learnng Algorthm, 15 th Internatonal on soft computng Mendel conference 2009,ISSN , pp [6] Pohl, J.: Stochastcký algortmus učení pro umělé neuronové sítě s pravděpodobnostním směrovým vektorem, Časops pro elektrotechnku Elektrorevue, , ISSN [7] Schalkoff, J. R.: Artfcal neural networks, McGraw-Hll nternatonal edtons Computer Scence Seres (1997), ISBN [8] Šíma, J., Neruda, R.: Teoretcké otázky neuronových sítí. Praha: MATFYZPRESS, (1996), 390 s. ISBN [9] Slušný S., Neruda R., Vnderová P.: Behavour Patterns Evoluton on Indvdual and Groupe Level, (CIMMACS 07), 6th WSEAS Internatonal Conference on computatonal ntellgence, man-machne systems and cybernetcs, pages 23-28, ISSN [10] Tng-Yu Chen, Y Lang Chen, Global Optmzaton Usng Hybrd Approach, (SMO'07), 7th WSEAS Internatonal Conference on smulaton, modellng and optmzaton, pages , ISSN [11] Zelnka, I.: Umělá ntellgence v problémech globální optmalzace, BEN Techncká lteratura, (2002), ISBN [12] Zelnka, I., Včelař, F., Čandlík, M.: Fraktální geometre prncpy a aplkace, BEN Techncká lterature, (2006), ISBN ISSN: Issue 7, Volume 7, July 2010