New method for solving nonlinear sum of ratios problem based on simplicial bisection

V Ù â ð f 33 3 Vol33, No3 2013 3 Systems Engineering Theory & Practice Mar, 2013 : 1000-6788(2013)03-0742-06 : O2112!"#$%&')(*)+),-))/0)1)23)45 : A 687:9 1, ;:<>= 2 (1?@ACBEDCFHCFEIJKLCFFM, NCO 453007; 2 PQCRESLTDCFHUCFM, PQ 710071) WYX[Z[\[][^[_a`[b[cedaf[geh[i[jakel[man[oepaqer, s c[get[u[v[w[x[y[z[{afe [}[~ [ a a b, ƒ[ [ [ [ [ˆ w[x[y, eš[ [Œa ež[ [ [, c[ [ge [ [ [ aqer[ [~[ [š[ feœ[ [ [š[ af[ž[ÿ, [ [ [ aqeraf n + 1 [ [, [ [ [ [ [ a eœ v[ [ af[ªe«[ [ a [ a f[ e [m, [±[² a e³[ [µ [ [ [ ¹ [}[~ {[º[ [ w[x[y[z[{ n[oep ; ; ; New method for solving nonlinear sum of ratios problem based on simplicial bisection WAN Chun-feng 1, LIU San-yang 2 (1 College of Mathematics and Information Science, Henan Normal University, Xinxiang 453007, China; 2 School of Science, Xidian University, Xi an 710071, China) Abstract For solving a special class of nonlinear sum of ratios problem arised in economy and finance, a global optimization algorithm is presented based on simplicial bisection In this algorithm, by constructing an initial simplex and using convex envelope theory, a new method to detere lower bound of the optimal value for the original problem is proposed With the deteration of the lower bound, n+1 feasible points of the original problem will be found, which can be used to improve upper bound Convergence of the algorithm is shown and some numerical examples are given to illustrate the feasibility and effectiveness of the presented algorithm Keywords global optimization; branch and bound; simplicial bisection; sum of ratios 1»½¼ ¾[ [À[Á[[[Å[[Ç[È[ÉaÊeÅ[Ë[Ì[Í (P) st f(x) = p i=1 c T i x + α i n i (x) p 2, c i R n, α i R, i = 1, 2,, p, A R mn, b R m Ð[Ñ[Ò[Ó[Ô D = {x, } Ç[Ö[[Ø[ Õ, n i (x) Ù[Ú[Û[Ü, Ý[Þ[ß x D, c T i x + α i 0, n i (x) > 0 ÊàÅáËáÌáÍáâáãáä áåáæèçêéáëáìáíáîáïáô áðáñáòá ô 1990 õáöá, øáù ÌáÍáúáûáüáý îáþáÿ Ï ÊeÅ[Ë[Ì[Í Ç [Ì[Í Ç [ Ê â[ì[í [Õ, î, ß[ö (P), c T i x + α i 0, Ý n i (x) Ù 0 Û[Ü, ü![ " # [1 4] $ n i (x) Ù % ì 0 Û[Ü, ü Ø * # ¾[ [À[Á [Ç - [5 8] & ' ( ) ' +[Ù 0 Ú[Û[Ü, ', / 0 1 Ì[Í 4 2 3[ø[ù ö 5[ß À[Á[Ì[Í[ ß 6, $ 3 7 8 9[Þ[ø[ù Ì[Í ¾[, & : <? "@# ü * #[ D E[É ; = > Þ ', A B[ [ 5 ' C F * I J K[ü * # ë [Ü H Ò[Ó É L[![É MONQPSR : 2010-10-08 TOUOVQW : XOYQZS[O\O]O^O_ (11171094); ÒaObOcOdO]OeOfO\OgOhOiOjOk (qd12103) lomonoo : poqor, s, t, uob, govowqx : yozo{o O}OwO~OO O

È È â ' û â H ) L 3 poqor, : ^OƒO O O O OÔ OŠO OŒQ SŽO O OkO O O O OwO~ 743 2 š œ ë : Ì[Í ž = Ÿ, (P) Ò[ö [Ù ì Ì[Í : p y i i=1 n i (x) st y i + c T i x = α i, i = 1, 2,, p â F Ì[Í[â ª[Ì[Í,, (P) 5 ç Ã[Ø[ [ Ï «[ [Ì[Í ¾[ Ù (P), : Ø * # â * #aï ; = > ' ( ), = > Þ ' 5 ç <[¾ ² ³ 21 µ ¹ º» ¾[ ¼[[ [9] Ïe " # ½ ¾ À D Á  = > S 0 à [ : œ * { n } γ = max x r x D, γ r = {x r x D}, r = 1, 2,, n ) Å S 0 Ù S 0 Ù = > À[ Ç Þ ' [ D = > [V s, V s ] aï Ê, ie Þ [ß, Ý S 0 = r=1 { Î É Ê x R n x r γ r, r = 1, 2,, n, } n x r γ, r=1 Ù {V 1, V 2,, V n+1 }, V 1 = (γ 1, γ 2,, γ n ), V j+1 = (γ 1, γ 2,, γ j 1, τ j, γ j+1,, γ n ), τ j = γ r j γ r, j = 1, 2,, n : Ë S = [V 1, V 2,, V n+1 ] Ì Í Î Ï Ð ' V s V s = Ù R n Ï Ñ : Õ Ü È {S 1, S 2 } Ö[Ù S = > Ø Ù S 0, = > max { V j V j }, j,j=1,2,,n+1 Þ ', S 1, S É Ê 2 ' [Ù Ã[Ø, c Ù S Ñ Ò {V 1, V 2,, V s 1, c, V s+1,, V n+1 }, {V 1, V 2,, V s 1, c, V s+1,, V n+1 } Â Þ [9] Ú, = > Þ ' [ [Õ Û Ü, Ý {S r } Ì Í 5 ß ' ( [ [ß à[ : á â ã ä r, S r+1 S r ), È x R n å æ S r = {x} r 22 ç è :  ë * â, Î é ê f(x) S [Ã[Ø D 5 LB(S), S = [V 1, V 2,, V n+1 ] Ì Í Á  = > S 0 ì ÿ í, = > : [ [Õ ¾[ * #[ É ï ð Î Ã î, [ ñ ò )[ ô É Ê 1 Ë U Ì Í[ö V 1, V 2,, V n+1 Ù ä ½ à[ õ ö, e = (1, 1,, 1) R n+1, Ý[Þ j à [È[É [Ì[Í[ {1, 2,, n + 1}, θ j Ù H : p y i i=1 n i (V j ) st AUλ b (LP) Uλ 0 j y i + c T i Uλ = α i, i = 1, 2,, p eλ = 1 â λ 0 f(x) S [Ã[Ø Ã [[È[É ø D 5 LB(S) Ò[ë LB(S) = n+1 θ j λ j j=1 st AUλ b (LP) Uλ 0 eλ = 1 λ 0 Õ ± Ó Ô (ie

È Ë Ù 0 ü " Ø î ) Ò 5 Ø 744 ùûúûüûýû û}ÿþ 33 Â Þ Ö (LP) Ò[Ó[Ô[Ù, È Ë LB(S) = + ) Å[Û[Ü g : R n R Ù { p y i g(x) = ξ,y n i=1 i (x) y i + c T i ξ = α i, i = 1, 2,, p, ξ S } D, $ S D, g(x) Õ : ½ ¾ x Ú[Û[Ü B [Ò â g(x) S 5 δ(x) : n+1 δ(x) = g(v j )λ j, Ç j=1 λ = (λ 1, λ 2,, λ n+1 ) Uλ = x, eλ = 1, λ 0 [ ) Å, Þ[ß x S, δ(x) g(x), [ø Ø Ù { p c T i x + α i n i=1 i (x) x S } { p y i D = n i=1 i (x) y i + c T i x = α i, i = 1, 2,, p, x S } D { { p y i x S D ξ,y n i (x) y i + c T i ξ = α i, i = 1, 2,, p, ξ S }} D = x S g(x) D x S δ(x)(2) D, (2) Å, Ò [à ä i=1 à x S D λ {λ AUλ b, Uλ 0, eλ = 1, λ 0}, y i + c T i ξ = α i y i + c T i Uλ = α i, i = 1, 2,, p * â f(x) S [Ã[Ø D 5 LB(S): n+1 LB(S) = x S δ(x) = g(v j )λ j AUλ b, Uλ 0, eλ = 1, λ 0 D, j=1 Þ j {1, 2,, n + 1}, { p } g(v j y i ) = n i=1 i (V j ) AUλ b, y i + c T i Uλ = α i, i = 1, 2,, p, Uλ 0, eλ = 1, λ 0 = θ j à * # ø ) [Ã[Ø ã ä {LB k } Õ = ô È 2 Ë S, S Õ ± n = >, Ý S S LB(S) LB( S) S = [V 1, V 2,, V n+1 ], S = [ V 1, V 2,, V n+1 ] È (i) S D =, LB( S) = +, 2 3 (ii) S D Ë g(x), ḡ(x) ' [Ù (1) Å ø [â ) S, S 5 Ú[Û[Ü, δ(x), δ(x) ' [Ù [Ù S S, ß[ö, Þ [ß x S, ḡ(x) g(x) δ(x) δ(x) 5[Ò Ú à LB( S) = { δ(x) x S D} {δ(x) x S D} {δ(x) x S D} = LB(S) (1) g(x), ḡ(x) 23 è * # [ Þ n = > S, LB(S), È * # [Ì[Í Î (P) : «* # Ò[Ó! [Ó, æ û " Ò[Ó, Ò[Ó Ø[ 5 8 ; à # : Ð[Ñ UB Ù $ $ %, È[É (λ j, y j ) Ù â (LP) j (j = 1, 2,, n + 1) = > S 5, È x j = Uλ j (j = 1, 2,, n + 1) Ì[Í (P) Ò[Ó ø, Ë λ Ù È[É [Ì[Í â (LP) = > S 5, È x = Uλ Õ Ì[Í (P) Ò[Ó â *[Ã[Ø & eø[ò Ú LB(S), [Ò[ö æ û : Ê [Ò[Ó F (S) = {x 1, x 2,, x n+1, x }, [Ã[Å 8 ; 5 B [Ò 3 ')()*)+),)-/ $ 1 2 3 4, 5 6 7 8 9 : ; # < = >? @ A B C D E F UB = {f(x 1 ), f(x 2 ),, f(x n+1 ), f(x ), UB} 0 H I ɛ 0 J K L M N O (P) P Q R D < n S T U V S 0 R n ; W 9 0 f(x) S 0 D 4 < 5 X LB(S 0 ); Y Z [ \ P Q ] F (S 0 ) S 0 D; ^ F = F (S 0 ), LB 0 = LB(S 0 ), UB 0 = {f(x) x F }; H

Ñ Ñ 0 _ 3 ` acbcd, e : fcgchcicjckclcmcncocpcqsrutcvcwcxcyczc{c ~}cc 745 I x 0 F ƒ f(x 0 ) = UB 0 UB 0 LB 0 ɛ, : x 0 N O (P) < ɛ- ˆ, Š UB 0 ɛ- ˆ Œ, ^ P 0 = {S 0 }, k = 1, Ž 1 1 T U V S k 1 š T U V S k,1, S k,2 2 i = 1, 2, W 9 0 f(x) S k,i D < 5 X LB(S k,i ), Ž Y Z [ \ ] F (S k,i ) S k,i D 3 ^ F = F {F (S k,i ) i = 1, 2}; UB k = {f(x) x F }; H I x k F ƒ f(x k ) = UB k 4 ^ P k = P k 1 \ {S k 1 } {S k,i i = 1, 2, LB(S k,i ) < UB k } 5 ^ LB k = {LB(S) S P k }, Ž œ S k P k ž LB k = LB(S k ) < T U V UB k LB k ɛ, : x k N O (P) < ɛ- ˆ, Š UB k ɛ- ˆ Œ, ^ k = k + 1, 1 5 6 Z Ÿ 7 8 9 : < C 3 (a) 9 : [ \, 9 : P ƒ N O (P) < ɛ- ª ˆ (b) 9 : «\, Z {S q } 9 : < «\ ± ² ³, Š ž q=1 Sq = { x}, µ x D P Q ³ {x q } < N O (P) < ª ˆ ¹ º (a)» 9 : [ \, ¼ ½ ¾ (b)»à9à:à«à\à À À À, À ÀÁÀ q, œ x qj (j = 1, 2,, n+1)  x q ÀÃÀ À À (LP) j (j = 1, 2,, n+ 1)  0 (LP) T U V S q 4 ˆ ƒ Å < P Q Ç È q, S q Å x, É Ç È q, [ x qj x(j = 1, 2,, n + 1) Ê x q x œ V qj (j = 1, 2,, n + 1) Ë Ì S q < Í É 4 Î ½ P Ï, V qj V j = x(j = 1, 2,, n + 1), Ð [ lim g(v qj ) = g(v j ) = f( x)(j = 1, 2,, n + 1) Ò, lim LB q = lim q q q LB(Sq ) = n+1 j=1 g(v j )λ j = f( x) n+1 j=1 λ j = f( x), Ó Ï ¼ ½ (b) Ø [ lim q (UB q LB q ) = lim q UB q f( x) = lim q f(xq ) f( x) = 0 Ô Õ 1 Ö [10], 4 Ù)Ú)Û)Ü Ý Þ 9 : < P Q Ê [ ß, à á â ã ä å æ Ý ç ³ è é ê Matlab 71, å æ Ý IV (306 HZ) ë ì 4 Ó Q 9 : < í î N O T U V ï : 0 ð ä å æ Ý, ñ ò I ɛ = 10E 3 1 [5] x 1 + 3x 2 + 2 st 4x 1 + x 2 + 3 + 4x 1 + 3x 2 + 1 x 1 + x 2 + 4 x 1 x 2 1 x 1 0, x 2 0 2 [11] x 1 + 2x 2 + 2 3x 1 4x 2 + 5 + 4x 1 3x 2 + 4 2x 1 + x 2 + 3 st x 1 + x 2 15 x 1 x 2 0 0 x 1 1, 0 x 2 1 3 [4,5] 3x 1 5x 2 3x 3 50 3x 1 + 4x 2 + 5x 3 + 50 + 3x 1 4x 3 50 4x 1 + 3x 2 + 2x 3 + 50 + 4x 1 2x 2 4x 3 50 5x 1 + 4x 2 + 3x 3 + 50 st 6x 1 + 3x 2 + 3x 3 10 10x 1 + 3x 2 + 8x 3 10 x 1 0, x 2 0, x 3 0 4 [12] st 5 c T i x + d i i=1 e T i x + f i Pentium

ð _ 746 ô õ ö ø ù/ú û ü µ c 1 = (0, 01, 03, 03, 05, 05, 08, 04, 04, 02, 02, 01) T, d 1 = 146, e 1 = ( 03, 01, 01, 01, 01, 04, 02, 02, 04, 02, 04, 03) T, f 1 = 142, c 2 = ( 02, 05, 0, 04, 01, 06, 01, 02, 02, 01, 02, 03) T, d 2 = 71, e 2 = (0, 01, 01, 03, 03, 02, 03, 0, 04, 05, 03, 01) T, f 2 = 17, c 3 = (01, 03, 0, 01, 01, 0, 03, 02, 0, 03, 05, 03) T, d 3 = 17, e 3 = (08, 04, 07, 04, 04, 05, 02, 08, 05, 06, 02, 06) T, f 3 = 81, c 4 = (01, 05, 01, 01, 02, 05, 06, 07, 05, 07, 01, 01) T, d 4 = 4, e 4 = (0, 06, 03, 03, 0, 02, 03, 06, 02, 05, 08, 05) T, f 4 = 269, c 5 = ( 07, 05, 01, 02, 01, 03, 0, 01, 02, 06, 05, 02) T, d 5 = 68, e 5 = (04, 02, 02, 09, 05, 01, 03, 08, 02, 06, 02, 04) T, f 5 = 37, 18 22 08 41 38 23 08 25 16 02 45 18 46 20 14 32 42 33 19 07 08 44 44 20 37 28 32 20 37 33 35 07 15 31 45 11 06 06 25 41 06 33 28 01 41 32 12 43 18 16 45 13 46 33 42 12 19 24 34 29 05 41 17 39 01 39 15 16 23 23 32 39 03 17 13 47 09 39 05 12 38 06 02 15 A = 05 42 36 06 48 15 03 06 36 02 38 28 01 33 43 24 41 17 10 33 44 37 11 14 06 22 25 13 43 29 41 27 08 29 35 12 43 19 40 26 18 25 06 13 43 23 41 11 00 04 45 44 12 38 19 12 30 11 02 25 01 17 29 15 47 03 42 44 39 44 47 10 38 14 47 19 38 35 15 23 37 42 27 01 02 01 49 09 01 43 16 26 15 10 08 16 33ý 5 b = (157, 318, 364, 385, 403, 100, 898, 58, 27, 163, 146, 727, 577, 345, 691) T 6 þ ÿ 5 6 < Ç ì N O 3x 1 + 4x 2 + 10 x 2 1 + 2x 1 x 2 2 + 15 + 3x 1 + 5x 2 + 6 2x 1 + 4x 2 + 3 + 2x 1 + 3x 2 + 1 x 2 1 + 18 st x 1 + x 2 4 x 1 + x 2 4 x 1 0, x 2 0 st p c T i x + d i i=1 e T i x + f i µ c i, e i R n, d i, f i R 0 [ 05, 05] 4 Ç ì 0, A R mn, b R m [0, 05] 4 Ç ì < 1 5 < W 9 ¼ Ë 1 6, Á, à á â 10 Ç ì æ Ý, I W 9 ¼ < 6 < = > W 9 ¼ Ë 2 Ð ä å æ Ý ¼ < P É 8, à á < ï : [ ß P Q <

_ 3 ` acbcd, e : fcgchcicjckclcmcncocpcqsrutcvcwcxcyczc{c ~}cc 747 1 1 5 c!c{!" #$%& 1 [5] (10, 00) 1428571 10 ours (10, 00) 14286 1 2 [11] (00, 0283935547) 1623183358 71 ours (00, 02813) 16232 42 3 [4] (00, 333333, 00) 3002924 66 [5] (00, 333329, 00) 3000042 30 ours (00, 33333, 00) 30029 26 4 [12] (6224297,20059738,3774868,5947937,00,7456478, 00,23312241,0000204,41031278,00,3171060) 16077978 11 ours (62237,200603,37747,59478,00,74567, 00,233126,00,410318,00,31711) 160780 1 5 ours (00, 40) 239094 1 2 6 p (m, n) '()*+, (h- ) '(#$%& 3 (5,10) 12318 2 (10,20) 132568 41 6 (5,10) 24282 24 (10,20) 144695 43 9 (5,10) 151346 71 (10,20) 290426 98 / 0 1 2 [1] Konno H, Yamashita H Minimizing sums and products of linear fractional functions over a polytope[j] Naval Research Logistics, 1999, 46: 583 596 [2] Konno H, Abe N Minimization of the sum of three linear fractional functions[j] Journal of lobal Optimization, 1999, 15: 419 432 [3] Kuno T A branch-and-bound algorithm for maximizing the sum of several linear ratios[j] Journal of lobal Optimization, 2002, 22: 155 174 [4] Wang Y J, Shen P P, Liang Z A A branch-and-bound algorithm to globally solve the sum of several linear ratios[j] Applied Mathematics and Computation, 2002, 168: 89 101 [5] Ji Y, Zhang K C, Qu S J A deteristic global optimization algorithm[j] Applied Mathematics and Computation, 2002, 185: 382 387 [6] Wang C F, Shen P P A global optimization algorithm for linear fractional programg[j] Applied Mathematics and Computation, 2002, 204: 281 287 [7] Benson H P A simplicial branch and bound duality-bounds algorithm for the linear sum-of-ratios problem[j] European Journal of Operational Research, 2007, 182: 597 611 [8] acbcd, 34, 5768 pcqsrutcvcwcxc cycz!9:c [J] ;<=>?@@A : B7CD@E, 2010, 38: 4 7 Wang C F, Li J, Shen P P A global optimization algorithm for sum of linear ratios problem[j] Journal of Henan Normal University: Natural Science, 2010, 38: 4 7 [9] Horst R, Pardalos P M, Thoai N V Introduction to global optimization[m] Kluwer, Dordrecht, Netherlands, 1995 [10] Horst R, Tuy H lobal optimization: Deteristic approaches[m] Springer, Berlin, 1996 [11] Jiao H W A branch and bound algorithm for globally solving a class of nonconvex programg problems[j] Nonlinear Analysis: Theory, Methods and Applications, 2009, 70: 1113 1123 [12] Phuong NTH, Tuy H A unified monotonic approach to generalized linear fractional programg[j] Journal of lobal Optimization, 2003, 26: 229 259