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

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1 V Ù â ð f 33 3 Vol33, No Systems Engineering Theory & Practice Mar, 2013 : (2013) : O2112!"#$%&')(*)+),-))/0)1)23)45 : A 687:9 1, ;:<>= 2 (1?@ACBEDCFHCFEIJKLCFFM, NCO ; 2 PQCRESLTDCFHUCFM, PQ ) 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 , China; 2 School of Science, Xidian University, Xi an , 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, $ [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 : TOUOVQW : XOYQZS[O\O]O^O_ ( ); ÒaObOcOdO]OeOfO\OgOhOiOjOk (qd12103) lomonoo : poqor, s, t, uob, govowqx : yozo{o O}OwO~OO O

2 È È â ' û â H ) L 3 poqor, : ^OƒO O O O OÔ OŠO OŒQ SŽO O OkO O O O OwO~ 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

3 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 ')()*)+),)-/ $ , : ; # < = 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

4 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, Z Ÿ : < 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 st 4x 1 + x x 1 + 3x x 1 + x x 1 x 2 1 x 1 0, x [11] x 1 + 2x x 1 4x x 1 3x x 1 + x st x 1 + x 2 15 x 1 x x 1 1, 0 x [4,5] 3x 1 5x 2 3x x 1 + 4x 2 + 5x x 1 4x x 1 + 3x 2 + 2x x 1 2x 2 4x x 1 + 4x 2 + 3x st 6x 1 + 3x 2 + 3x x 1 + 3x 2 + 8x 3 10 x 1 0, x 2 0, x [12] st 5 c T i x + d i i=1 e T i x + f i Pentium

5 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, A = ý 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 x x 1 x x 1 + 5x x 1 + 4x x 1 + 3x x 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 <

6 _ 3 ` acbcd, e : fcgchcicjckclcmcncocpcqsrutcvcwcxcyczc{c ~}cc c!c{!" #$%& 1 [5] (10, 00) ours (10, 00) [11] (00, ) ours (00, 02813) [4] (00, , 00) [5] (00, , 00) ours (00, 33333, 00) [12] ( , , , ,00, , 00, , , ,00, ) ours (62237,200603,37747,59478,00,74567, 00,233126,00,410318,00,31711) ours (00, 40) p (m, n) '()*+, (h- ) '(#$%& 3 (5,10) (10,20) (5,10) (10,20) (5,10) (10,20) / [1] Konno H, Yamashita H Minimizing sums and products of linear fractional functions over a polytope[j] Naval Research Logistics, 1999, 46: [2] Konno H, Abe N Minimization of the sum of three linear fractional functions[j] Journal of lobal Optimization, 1999, 15: [3] Kuno T A branch-and-bound algorithm for maximizing the sum of several linear ratios[j] Journal of lobal Optimization, 2002, 22: [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: [5] Ji Y, Zhang K C, Qu S J A deteristic global optimization algorithm[j] Applied Mathematics and Computation, 2002, 185: [6] Wang C F, Shen P P A global optimization algorithm for linear fractional programg[j] Applied Mathematics and Computation, 2002, 204: [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: [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: [12] Phuong NTH, Tuy H A unified monotonic approach to generalized linear fractional programg[j] Journal of lobal Optimization, 2003, 26:

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