Stochastic optimization algorithm with probability vector

Transcription

1 Proceedngs of the 4th WSEAS Internatonal Conference on COMPUTATIONAL INTELLIGENCE Stochastc optmzaton algorthm th probablty vector JAN POHL, VÁCLAV JIRSÍK, PETR HONZÍK Department of Control and Instrumentaton Brno Unversty of Technology Kolejní 2906/4, Brno CZECH REPUBLIC Abstract: It s ntroduced n the paper the nely developed optmzaton method the Stochastc Optmzaton Algorthm th Probablty Vector (PSV). It s related to Stochastc Learnng Algorthm th Probablty Vector for artfcal neural netorks. Both algorthms are nspred by stochastc terated functon system SIFS for generatng the statstcally self smlar fractals. The PSV s gradent method here the drecton of ndvdual future movement from the populaton s based stochastcally. PSV as tested on mathematcal functon mnmzaton and on the travellng sales man problem. Key Words: Stochastc, Optmzaton Algorthm, PSV, SIFS, Random Walk, Travelng Salesman Problem Introducton There are many optmzaton algorthms, e.g. gradent, stochastc, nspred by collectve behavor of bologcal ndvduals etc [8,9,0]. The PSV algorthm that s ntroduced n ths paper s a modfed verson of the stochastc learnng algorthm [,2]. The algorthm s nspred by the stochastc terated functon system SIFS for generatng the statstcally self smlar fractals [3,4]. It s based on the group of separate ndvduals, hch do not share nformaton th each other. It s based on the smlar prncple as the stochastc hll clmbng or random alk (RW). 2 Algorthm PSV PSV algorthm uses ndvduals from the group n the smlar ay as n the stochastc hll clmbng. There s defned a vector of transformatons (). ( t, t 2, ) T, = () L t n Each of them can modfy any parameter of an ndvdual n a specfc ay. In every step the transformaton s chosen randomly but th 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 alk (RW). ( p, p, 2 ) P, n = = (2) p L = p n (3) (,) 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 orst results ts probablty decreased. Let s have an ndvdualw k W. Ths ndvdual conssts of a set of parameters (7). (,, 2 ) W, = (7) k L n At frst the ftness functon of the selected ndvdual s evaluated. Secondly the roulette heel selecton s used to choose and apply the transformaton from (). ndvdual t ( ndvdual) = ne (8) Fnally the ftness functon of the ne ndvdual s evaluated. In the case the ne ndvdual has hgher ftness functon than the prevous the related probablty p s ncreased. ISSN: ISBN:

2 Proceedngs of the 4th WSEAS Internatonal Conference on COMPUTATIONAL INTELLIGENCE p = p + sgn( p )α _ ne (9) Symbol α s coeffcent used for ncreasng the probablty. If the ne ndvdual has loer ftness functon than the prevous the related probablty p s decreased. ne = p+ sgn ( p )( β) p _ (0) Symbol β s the coeffcent used for decreasng the probablty. Example of the basc transformatons s (). pλ + ( ) = ne = t _ () Transformaton () transforms parameter of the ndvdual W k n specfc drecton and the coeffcent λ s a learnng (for artfcal neural netorks) or movng rate. 3 Functon mnmzaton and travelng salesman problem We successfully tested the PSV method as the learnng algorthm for artfcal neural netorks [, 2]. In ths paper as 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 follong test functons:. st De Jong 2. Rodenbrock s saddle 3. 3rd De Jong 4. 4rd De Jong 5. Rastrgn s functon 6. Schefel s functon 7. Greangk s functon 8. Sne envelope ave functon 9. Stretched V sne ave functon 0. Test functon Ackley. Ackley s functon 2. Mchalecz s functon 3. Masters s cosne ave functon represents multmodal functon and Test functon Ackley s specal case of functon th 2 global mnmums. The other functons are n Tab. together th the best results found after steps. Every functon as tested th 50 ndvduals n steps th parameters α = 0.3, β= 0.4, λ= 0.. f.num. var. x var. x 2 ft.fun Tab. Functon results On the fgures n paragraph 3.. and 3..2 there are markers (ellpse and cross) hch represent estmated poston of global mnmum. 3.. Sne envelope ave functon Fg. Sne envelope ave functon Optmzaton of the travellng salesman problem as tested on the crcle and randomly deployed ctes. 3. Functon mnmzaton To specfc functons ere chosen for presentaton of the algorthm. The Sne envelope ave functon ISSN: ISBN:

3 Proceedngs of the 4th WSEAS Internatonal Conference on COMPUTATIONAL INTELLIGENCE 3..2 Test functon Ackley Fg.2 Default pont Fg.5 Test functon Ackley Fg.3 After 000 steps Fg.6 Default poston Fg.4 After steps Fg.7 After 000 steps ISSN: ISBN:

4 Proceedngs of the 4th WSEAS Internatonal Conference on COMPUTATIONAL INTELLIGENCE Fg.8 After steps Fg.0 After 000 steps 3.2 Travelng salesman problem The PSV algorthm as slghtly modfed for the purposes of the travelng salesman problem. Because the combnaton of the ctes s dscrete, the step λ equals and the probablty vector only determnes the drecton of the movement. Then the transformaton () s modfed to (2). sgn( p )λ + ( ) = ne = t _ (2) Parameters of the algorthm are α = 0.3, β = 0.3, λ =, number of ndvduals = 5. Fg. After steps 3.2. Random ctes poston Crcle ctes poston Fg.9 Random ctes poston Fg.2 Crcle ctes poston ISSN: ISBN:

5 Proceedngs of the 4th WSEAS Internatonal Conference on COMPUTATIONAL INTELLIGENCE Acknoledgements Ths ork as supported by the Mnstry of Educaton of the Czech Republc under Projects GA02/09/897 and MSM Fg.3 After 000 steps Fg.4 After steps 4 Concluson The nely developed algorthm PSV as ntroduced n ths paper. The am as to sho ts possble usage n to 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) ere shon n chapters 3. and 3.2. In chapter 3., 2D functons ere 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 hole optmzaton process. Future ork ll be focused on the extenson of the current algorthm th dstrbuton of the nformaton among ndvduals n the populaton and on a modfcaton of the algorthm th adaptve movng and adaptve momentum coeffcents λ, α, β. References: [] Pohl, J., Polách, P.: Stochastc Learnng Algorthm, 5 th Internatonal on soft computng Mendel conference 2009,ISSN , pp [2] Pohl, J.: Stochastcký algortmus učení pro umělé neuronové sítě s pravděpodobnostním směrovým vektorem, Časops pro elektrotechnku Elektrorevue, , ISSN [3] Zelnka, I.: Umělá ntellgence v problémech globální optmalzace, BEN Techncká lteratura, (2002), ISBN [4] Zelnka, I., Včelař, F., Čandlík, M.: Fraktální geometre prncpy a aplkace, BEN Techncká lterature, (2006), ISBN [5] JIAN YE, JUNFEI QIAO, MING-AI LI,XIAOGANG RUAN: A tabu based neural netork learnng algorthm, ScenceDrect NEUROCOMPUTING 70 (2007) , [6] SCHALKOFF, J. R.: Artfcal neural netorks, McGra-Hll nternatonal edtons Computer Scence Seres (997), ISBN [7] ŠÍMA, J., NERUDA, R.: Teoretcké otázky neuronových sítí. Praha: MATFYZPRESS, (996), 390 s. ISBN [8] Godfrey C. Onubolu., B.V. Babu.: Ne Optmzaton Technques n Engneerng, Sprnger- Hedelberg 2004, ISBN X [9] Alexandru-Lvu Olteanu.: Ant colony optmzaton meta-heurstc n project schedulng, (AIKED 09), 8th WSEAS nternatonal conference on Artfcal ntelgence, knoledge engneerng and data bases, Cambrdge UK, pages 29-34, (2009), ISSN [0] Jerzy Balck.: Tabu-based Evolutonary Algorthm th Negatv Selecton for PAreto-optmzaton n Dstrbuted Systems, (AIKED 08), 7th WSEAS nternatonal conference on Artfcal ntelgence, knoledge engneerng and data bases, Cambrdge UK, pages , (2008), ISSN ISSN: ISBN: