Unimodality Tests for Global Optimization of Single Variable Functions Using Statistical Methods

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1 Malaysa Umodalty Joural Tests of Mathematcal for Global Optmzato Sceces (): of 05 Sgle - 5 Varable (007) Fuctos Usg Statstcal Methods Umodalty Tests for Global Optmzato of Sgle Varable Fuctos Usg Statstcal Methods Ismal b Mohd, Sska Cadra Ngsh & Yosza Dasrl 3 Departmet of Mathematcs, Faculty of Scece ad Techology, Uverst Malaysa Tereggau (UMT), Megabag Telpot, 030 Kuala Tereggau, Malaysa E-mal: smalmd@kustem.edu.my, sskagsh@yahoo.com, 3 yosza@kustem.edu.my ABSTRACT Ths paper dscusses the radomess ad ormalty tests to the set of pots whch are obtaed by splttg a terval uder cosderato to several subtervals for verfyg that the optmzato problem costtutes a Weer process. The, by usg the optmzato probablstc algorthm, a best subterval ca be chose for umodalty test ad wll be cotued wth a smple local search method, startg at ay pot the such chose best terval. Furthermore, umercal results are gve to llustrate the tests. Keywords: Global optmzato, probablty, radomess test, ormalty test, ad umodalty test. INTRODUCTION The task of global optmzato of the optmzato problem z = f ( x) (.) subect to a x b (.) where f : R R, s to fd a soluto ts feasble soluto set for whch the obectve fucto acheved ts smallest value, the global mmum. Therefore, the global optmzato ams determg ot ust a local mmum but the smallest local mmum wth respect to ts feasble soluto set whch s defed by { } F = x a x b, x R. (.3) I (Ismal, 005) ad (Sska, 006) we have show how to use the radomess ad ormalty tests to aalyze the data obtaed by splttg the terval [ ab, ] X Rto several subtervals = [ x, x ]( = 0,, ) wth x0 = a ad x = b for verfyg + that the optmzato problem costtutes a Weer process. I ths paper, we wll descrbe how to use the umodalty test to the chose subterval whch s obtaed by usg the optmzato probablstc algorthm (Ismal, 005), (Sska, 006) before the (smple) local search procedure s employed. Malaysa Joural of Mathematcal Sceces 05

2 Ismal b Mohd, Sska Cadra Ngsh & Yosza Dasrl. STOCHASTIC PROCESS We shall cosder the Weer process (Archett, 978) ad Gve f = f x = µ 0 0 ( σ ) f x f y N 0, x y for x, y X. f = f x =,,, the dstrbuto of f ( x ), codtoed by (,,,, ) z = x f x f = 0, wth f x for x X [ x x] (.) s ormal wth some expected value E f ( x) z = µ x ad varace ( var f x z ) = σ ( x) gve by ad respectvely. x+ x x x 0,, ( 0,, ) f + f+ f = µ x = µ x = x+ x x+ x f ( x) = [ x, x],ad x, = x x x+ x ( 0,, ( 0,, ) ) σ f = µ x = σ ( x) = x+ x σ ( x x) ( = [ x, x],ad x, = ) Theorem. (Archett, 980) If f ( x), x [ a, b], s a Weer process such that f ( a) = f a, ad by cosderg the dstrbuto the f ( z) = P{m f () t z f ( b) = h}, a t b ( ) z m fa, f f ( z) = ( fa z)( fb z) exp < m, z fa f σ ( b a) ( ( b) ) b. 06 Malaysa Joural of Mathematcal Sceces

3 Umodalty Tests for Global Optmzato of Sgle Varable Fuctos Usg Statstcal Methods 3. STATISTICAL TESTS Gve f : S R R ad the pots x ( =,, ) splt the search terval X S equal parts. Let = ( = ) (3.) z f f,, The goodess of ft of the Weer process for f ca be checked by testg that whether z ( =,, ) s a radom sample draw from a ormal dstrbuto where the detals are as follows. Radomess Test The radomess of z ( =,, ) ca be checked by the smple correlato coefcet where R = z = SS z zk (3.) z = = + z = z, S = ( z z), K =, ad S = If z ( =,, ) are depedet o, the the radom varable T = R ( p) ( ) R (3.3) follows a t-dstrbuto wth ν = p degrees of freedom. The tabulated t-dstrbuto ca be used to costruct the statstcal tests for dfferet sgfcace levels α ad sample szes. The observatos z ( =,, ) are depedet f T < T αν,. Normalty Test I order to exame the ormalty of z ( =,, ), we frstly compute: { max F0 z() ; z; Sz,, F0( z )} () ; z; Sz σ = (3.4) where z() ( =,, ) are values z (,, ) F = rearraged creasg order, ad 0 x; x; s s the stadard ormal dstrbuto fucto gve by ( x x) / s u / F0 ( x; x; s) = e du, π (3.5) Malaysa Joural of Mathematcal Sceces 07

4 Ismal b Mohd, Sska Cadra Ngsh & Yosza Dasrl ad secodly, we compute D = max =,, ( σ ) (3.6) The varables are draw from a ormal dstrbuto f D <,. D α 4. THE PARAMETERS The followg parameters are eeded our algorthm. x x µ = f ( x0), = 0, x = x0 + ( =,, ) (4.) ad σˆ, the maxmum lkelhood estmate of σ, s computed as σˆ ( f ) f = x x = (4.) 5. STOPPING CRITERION The stoppg crtero for choose the best subterval s provded by Theorem.. Let L be the sum of the tervals of local adequacy of the model, after fucto evaluatos, performed x ( = 0,..., ) wth x x + ( = 0,..., ). At the ext step, f(x) s assumed umodal ay of the tervals of L ad a sample path of the Weer process for x X L. Defe the followg probablty We have P m, {. [, ] } = P f x f x x ε f f + x+ (5.) ( f ε f+ )( f ε f) exp, P = ( + ) σ x x 0, ([ x x+ ] L) ([ x x+ ] L) (5.) where f = m { f, = 0,, } the probablty ad ε s the prefxed accuracy level. Fally, we compute { PF = P m f x > f ε z} = ( P). (5.3) x X = 0 08 Malaysa Joural of Mathematcal Sceces

5 Umodalty Tests for Global Optmzato of Sgle Varable Fuctos Usg Statstcal Methods The algorthm wll termate whe PF > PT where PT s a gve probablty level, ad as example PT = If PF PT, the chage the value of. 6. THE CONTROL OF PARAMETER The { γ } sequece of parameters for cotrollg the covergecy of the algorthm, ca be determed by Theorem. gve by the followg probablty. { Pγ = P m f x > f }. γ z (6.) x X If The parameter γ s kept costat f Pγ P T where P T s a prefxed probablty level. Pγ > P, the compute T γ + = / γ. (6.) A subterval 7. THE CHOSEN SUBINTERVAL p s chose such that the probablty P m f x < f { γ z = max P m f x < f } γ z x p x = max P m f x < f γ f, f { } { } + x ( p ) ( f γ f )( f γ f ) + = max exp σ ( x+ x) = sze (7.) where f = m ( f0, f,, f ), ad γ s some postve value. Now, dvde sze( X ) by sze( p ) to obta subterval = x, x+ ( = 0,, m) where x x0 m =. sze( p ) Suppose that we have (7.) f = f x = m f, f,, f. (7.3) t t 0 m Therefore, the (best) chose subterval s a terval whch cotas x t ad ca be expressed by [ ] I = x, x. t t+ (7.4) Malaysa Joural of Mathematcal Sceces 09

6 Ismal b Mohd, Sska Cadra Ngsh & Yosza Dasrl 8. UNIMODALITY TEST Defto 8. If x ( a, b) s a mmzer of f over ( ab, ), f( x) f( x ) ( x< x ), ad f ( x ) f ( x) ( x < x) for x [ a, b], the f s sad to be umodal [ ab, ]. Suppose that f umodal I = [ xt, xt+ ] where the subterval I = [ xt, xt+ ] s chose as descrbed Secto 7. Deote x ( 0) = ad x ( 0) = x + ad compute ( 0) ( 0) ( 0) ( 0) 3 = 3 ad f4 f x4 ( 0) ( 0) ( 0) ( 0) 3 4 f f x xt = for ay par x ( 0), x ( 0) x < x < x < x. Based o the values of [, ] s s+ 3 4 ( 0) f ad 3 I = x x ca be reduced accordg to the followg algorthm. Algorthm (Umodalty Test) ( 0) ( 0) ( 0) ( 0) ( 0) ( 0) ( 0) ( 0) 0 0 Data: x, x, x < x3 < x4 < x, f3 = f x, ad f4 = f ( x4 ). ( 3 ) s ( 0, 0 ) x x whch satsfy ( 0) f (Ismal, 989), the wdth of 4. case true of ( 0) ( 0) () () ( 0) ( 0) ( 0) ( 0).. f3 < f4 : x, x = x, x 4 // The mmzer, x x, x 4 ( 0) ( 0) () () ( 0) ( 0) ( 0) ( 0).. f3 > f4 : x, x = x3, x // The mmzer, x x3, x ( 0) ( 0) ( 0) ( 0) ( 0) ( 0).3. f3 = f4 : x, x = x3, x 4 // The mmzer, x x3, x 4.4. default: {}! A error. retur. However, we do ot kow exactly about the umodalty property of f the ( 0) ( 0), = s, s+ subterval x x x x eve though ths chose terval produced by our algorthm s assumed has fullflled the umodalty property. Therefore, order to obta more evdece, durg mplemetato, f for the sequeces of pots ( 0) () ( ) x, x, x, or ( 0 ) () ( x, x, x ), we ca get () ( + ) f f > or () ( + ) > f respectvely, the t ca we assumed that f s umodal that subterval where () () f = f x = 0,,, ; =,. Statstcally, ths crtera ca be acheved sce 0 0 the subterval x, x = xt, x t+ has bee tested. Ths statstcal tests have bee doe before the umodalty test. f 9. TEST EXAMPLES The statstcal ad umodalty tests have bee tested o fve fuctos as follows: 0 Malaysa Joural of Mathematcal Sceces

7 Umodalty Tests for Global Optmzato of Sgle Varable Fuctos Usg Statstcal Methods Example : Mmze f ( x) = s ( x) e x subect to 0 x 0 where ts graph s gve by Example : Mmze f ( x) = /cos( x) + /3s( x) subect to 0 x 5 where ts graph s gve by Example 3: Mmze ( 0.75) π f x = x + s 8π x graph s gve by subect to 0 x where ts Malaysa Joural of Mathematcal Sceces

8 Ismal b Mohd, Sska Cadra Ngsh & Yosza Dasrl Example 4: Mmze f ( x) = s( x) 4xcos( x) subect to 0 x 5 where ts graph s gve by Example 5: Mmze f x = xcos x x s ( x) subect to 0 x 5 where ts graph s gve by 0. NUMERICAL RESULTS The results of the statstcal ad umodalty tests for the cosdered examples are gve the Tables.,.,.,.,, 5.. I Tables.,., 3.,, ad 5., deotes the umber of samples, T ad D are computed usg equatos (3.3) ad (3.6) respectvely. The values of T0.05, T0.0, D 0.05,, ad D 0.0, are obtaed from the t-dstrbuto table ad the table of Crtcal Values for the Llefors test for Normalty (Shesk, 000). Tables., TABLE. Results of radomess ad ormalty tests for Example T T 0.05 T 0.0 D D 0.05, D 0.0, Malaysa Joural of Mathematcal Sceces

9 Umodalty Tests for Global Optmzato of Sgle Varable Fuctos Usg Statstcal Methods., 3.,..., 5. cota the results of the umodalty test up to four teratos for each example. TABLE. Results of umodalty test for Example Iterato 0 3 () () () () (3) (3) TABLE. Results of radomess ad ormalty tests for Example T T 0.05 T 0.0 D D 0.05, D 0.0, TABLE. Results of umodalty test for Example Iterato 0 3 () () () () (3) (3) TABLE 3. Results of radomess ad ormalty tests for Example 3 T T 0.05 T 0.0 D D 0.05, D 0.0, Malaysa Joural of Mathematcal Sceces 3

10 Ismal b Mohd, Sska Cadra Ngsh & Yosza Dasrl TABLE 3. Results of umodalty test for Example 3 Iterato 0 3 () () () () (3) (3) TABLE 4. Results of radomess ad ormalty tests for Example 4 T T 0.05 T 0.0 D D 0.05, D 0.0, TABLE 4. Results of umodalty test for Example 4 Iterato 0 3 () () () () (3) (3) TABLE 5. Results of radomess ad ormalty tests for Example 5 T T 0.05 T 0.0 D D 0.05, D 0.0, Malaysa Joural of Mathematcal Sceces

11 Umodalty Tests for Global Optmzato of Sgle Varable Fuctos Usg Statstcal Methods TABLE 5. Results of umodalty test for Example 5 Iterato 0 3 () () () () (3) (3) CONCLUSION From the Tables.,., 3.,..., 5., for the gve fuctos, we ca say that varables are radom varables sce T < T αν,, ad the values of D < D α, suggests that the varables are draw from a ormal dstrbuto. The Tables.,., 3.,, 5. ad the gve fuctos cofrm that the best subterval determed by the suggested probablstc algorthm, s umodal.. ACKNOWLEDGEMENT The authors wsh to express ther apprecato to Uverst Malaysa Tereggau ad the Govermet of Malaysa for ther facal support ths proect. REFERENCES ARCHETTI, F. ad B. BETRO A Probablstc Algorthm For Global Optmzato. ARCHETTI, F Aalyss of Stochastc Strateges For The Numercal Soluto of The Global Optmzato Problem. Workg paper Numercal Techques For Stochastc System North- Hollad. ISMAIL, B.M., D. YOSZA, ad C.N. SISKA Probablstc Algorthm for Optmzato Usg Newto s Method, Refereed Coferece Proceedgs of Australa Operatos Research Socety ad th. Australa Optmzato Day, ISMAIL, B.M., ad H.H. MALIK Itroducto to Optmzato Methods, Uverst Pertaa Malaysa, Serdag. SISKA, C.N., D. YOSZA, ad B.M. ISMAIL Probablstc Algorthm Optmzato, MATEMATIKA, (). SHESKIN, J. D Hadbook of Parametrc ad Noparametrc Statstcal Procedures. Lodo. Malaysa Joural of Mathematcal Sceces 5