On the Newton method for solving fuzzy optimization problems

Else ie Edi o ial Sys em( m) o Fuzzy Se s and Sys ems Manusc ip D a Manusc ip Numbe : FSS-D-14-00513 Ti le: On he New on me hod o sol ing uzzy op imiza ion p oblems A icle Type: Full Leng h A icle (FLA) Keywo ds: Fuzzy op imiza ion; gene alized Hukuha a di e en iabili y; New on me hod. Co esponding Au ho : D . Y. Chalco Cano, Co esponding Au ho 's Ins i u ion: Uni e sidad de Ta apacá Fi s Au ho : Y. Chalco Cano O de o Au ho s: Y. Chalco Cano; Ge aldo Sil a; An onio Ru ián-Lizana 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 In his a icle we conside op imiza ion p oblems whe e he objec i esa e uzzy unc ions ( uzzy- alued unc ions). Fo his class o uzzy op imiza ion p oblems we discuss he New on me hod o ind a non-domina ed solu ion. Fo his pu pose, we use he gene alized Hukuha a diffe en iabili y no ion, which is he mos gene al concep o exis ing diffe en iabili y o uzzy unc ions. This wo k imp o es and co ec he New on Me hod p e iously p oposed in he li - e a u e. 1 Abs ac 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 On he New on me hod o sol ing uzzy op imiza ion p oblems 4 Y. Chalco-Cano a,1,G.N.Sil a b,2,A.Ru i ´ an-Lizana c,3 aIns i u o de Al a In es igaci´on, Uni e sidad de Ta apac´a, Casilla 7D, A ica Chile bIns i u o de Biociˆencias, Le as e Ciˆencias Exa as, UNESP-Uni .Es adualPaulis a, Cˆampus de S˜ao Jos´e do Rio P e o, Depa amen o de Ma em´a ica Aplicada, S˜ao Jos´e do Rio P e o-SP, B asil. cDepa amen o de Es ad´ıs ica e I.O., Uni e sidad de Se illa, Spain. Abs ac In his a icle we conside op imiza ion p oblems whe e he objec i es a e uzzy unc- ions ( uzzy- alued unc ions). Fo his class o uzzy op imiza ion p oblems we discuss he New on me hod o ind a non-domina ed solu ion. Fo his pu pose, we use he gen- e alized Hukuha a diffe en iabili y no ion, which is he mos gene al concep o exis ing diffe en iabili y o uzzy unc ions. This wo k imp o es and co ec he New on Me hod p e iously p oposed in he li e a u e. Key wo ds: Fuzzy op imiza ion, gene alized Hukuha a diffe en iabili y, New on me hod. 1In oduc ion Fuzzy op imiza ion p oblems ha e been s udied by many esea che s in se e al di ec ions wi h a lo o applica ions. The collec ion o pape s on uzzy op imiza ion edi ed by Delgado e al. [11], Lodwick and Kacp zyk [19], Inuiguchi and Ram´ık 1Co esponding au ho , e-mail: ychalco@u a.cl 2e-mail: [email p o ec ed] 3e-mail: u i[email p o ec ed] 4The esea ch in his pape has been suppo ed by Fondecy -Chile, p ojec 1120665 and 1120674, by Minis e io de Ciencia y Tecnolog´ıa (Spain) h ough p ojec MTM 2010- 15383, by he B azilian Na ional Council o Scien i ic and Technological De elopmen - CNPq (B azil) unde G an numbe s 309335/2012-4 and 479109/2013-3 and by Sao Paulo S a e Founda ion (FAPESP) unde G an numbe 2013/07375-0 – h ough he Cen e o Ma hema ical Sciences Applied o Indus y – CeMEAI/CEPID. P ep in submi ed o Else ie 27 June 2014 *Manusc ip Click he e o iew linked Re e ences 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 [15], Rommel ange and Slowi´nski [26], Slowi´nski and Teghem [27], and he books by Lai and Hwang [16,17] p o ide e iews abou his opic oma e yb oadpoin o iew. I is usually difficul o de e mine he coefficien s o an objec i e unc ion as a eal numbe since, mos o en, hese possess inhe en unce ain y and/o inaccu- acy. Gi en ha his is he usual s a e, we conside uzzy- alued objec i e unc ion as one app oach o ackle unce ain y and inaccu acies in heobjec i e unc ion coefficien s o ma hema ical p og amming models (see, e.g., Lodwick [18]). Op i- miza ion p oblems wi h uzzy- alued objec i e unc ions we e s udied by many esea che s. Fo ins ance, see [5,8,23,20,30–37,39]. In pa icula , in [5,9,33,37] Ka ush-Kuhn-Tucke ype op imali y condi ions o his class o uzzy op imiza- ion p oblems we e ob ained. Mo e ecen ly, Pi zada and Pa hak in [21] p oposed a New on me hod o ind a non-domina ed solu ion o a uzzy op imiza ion p oblem using he Hukuha a diffe en iabili y o uzzy- alued unc ions. The concep o Hukuha a diffe en ibili y (H-diffe en iabili y, o sho ) o uzzy unc ions is e y es ic i e. Fo ins ance, F(x)=C·x,whe eCis any uzzy in e - al and xis a eal numbe , is no H-diffe en iable being ha Fis a gene aliza ion o alinea unc ion.Ingene al,a uzzy unc ionde inedbyF(x)=C·g(x), whe e gis a diffe en iable eal unc ion and Cis a uzzy in e al, is no always H-diffe en iable. Howe e , i is always gH-diffe en iable. I is well-known ha he concep o gH- diffe en iable uzzy unc ion (gene alized Hukuha a diffe en iable uzzy unc ion) is a mo e gene al concep han le el-wise diffe en iabili y [37,38], Hukuha a di - e en iabili y [14], and G-diffe en iabili y [1–3,6,7]. Thus, he mo e use ul concep o diffe en iabili y o uzzy unc ions is gH-diffe en iabili y. The condi ions imposed o implemen he New on me hod in oduced by Pi zada and Pa hak in [21] in Theo em 4.1 a e e y es ic i e because heyask ha in he neighbo hood o a nondomina ed solu ion, all poin s mus be compa able, bu he o de ela ion used is only pa ial. Mo eo e , i is also equi ed ha he nondom- ina ed solu ion is an ideal poin o he endpoin unc ions o heobjec i e uzzy unc ions. The examples p esen ed in he same pape do no obey he condi ions equi ed by Theo em 4.1. In addi ion, he objec i e unc ions o all examples hey conside a e no H-diffe en iable. E en so, hey apply he New on me hod o he ex- amples. No su p isingly, when hey apply o he Example 4.1, heyob ainapoin ha is no a non domina ed solu ion, al hough he au ho s claim i is. In his pape we o mula e he New on me hod o ind a non-domina ed solu- ion o uzzy op imiza ion p oblems wi hou he equi emen ha all easibleso- lu ions in he neighbo hood o a nondomina ed solu ion be compa able and use gH-diffe en ibili y ins ead o H-diffe en iabili y. Finally, we co ec he examples conside ed in [21]. 2 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 2No a ionand hespaceo uzzyin e als A uzzyse onRnis a mapping u:Rn→[0,1]. Fo each uzzy se u,wedeno e i s α-le el se as [u]α={x∈Rn|u(x)≥α} o any α∈(0,1]. The suppo o uis deno ed by supp(u), whe e supp(u)={x∈Rn|u(x)>0}.Theclosu eo supp(u) de ines he 0-le el o u,.i.e.[u]0=cl(supp(u)), whe e cl(M)means heclosu eo he subse M⊂Rn. De ini ion 1 A uzzyse uonRis said o be a uzzy in e al i : (1) u is no mal, i.e. he e exis s x0∈Rsuch ha u(x0)=1; (2) u is an uppe semi-con inuous unc ion; (3) u(λx+(1 −λ)y)≥min{u(x),u(y)},x,y∈R,λ∈[0,1]; (4) [u]0is compac . Le FCdeno e he amily o all uzzy in e als. So, o any u∈F Cwe ha e ha [u]α∈K C o all α∈[0,1], whe e KCdeno es he space o all compac in e als in R,and hus heα-le els o a uzzy in e al a e gi en by [u]α=!uα,uα",uα,uα∈R o all α∈[0,1]. I [u]1is a single on hen we say ha uis a uzzy numbe . T ian- gula uzzy numbe s a e a special ype o uzzy numbe s which a e well de e mined by h ee eal numbe s a≤b≤cand we w i e u=(a,b,c)and [u]α=[a+(b−a)α,c−(c−b)α], o all α∈[0,1]. Fo uzzy in e als u, ∈F C ep esen ed by !uα,uα"and ! α, α", espec i ely,and o any eal numbe λ,wede ine headdi ionu+ and scala mul iplica ion λuas ollows: (u+ )(x)=sup y+z=x min{u(y), (z)} (λu)(x)=⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ u'x λ(,i λ!0, 0,i λ=0. I is well known ha , o e e y α∈[0,1], [u+ ]α=!(u+ )α,(u+ )α"=!uα+ α,uα+ α"(1) and [λu]α=!(λu)α,(λu)α"=!min{λuα,λ uα},max{λuα,λ uα}".(2) Ac ucialconcep inob ainingause ulwo kingde ini iono de i a i e o uzzy unc ions is de i ing a sui able diffe ence be ween wo uzzy in e als. Towa d his end we ha e he ollowing de ini ion. 3 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 De ini ion 2 ([29]) Gi en wo uzzy in e als u, , he gene alized Hukuha a di - e ence (gH-diffe ence o sho ) is he uzzy in e al w, i i exis s, such ha u⊖gH =w⇔⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ (i)u= +w, o (ii) =u+(−1)w. I is easy o show ha (i)and(ii)a ebo h alidi andonlyi wis a c isp numbe . No e ha he case (i) is coinciden o Hukuha a diffe ence (see [14]) and so he concep o gH-diffe ence is mo e gene al han H-diffe ence. I u⊖gH exis s hen, in e ms o α-le els, we ha e [u⊖gH ]α=[u]α⊖gH [u]α=!min{uα− α},max{uα− α}", o all α∈[0,1], whe e [u]α⊖gH [u]αdeno es he gH-diffe ence be ween wo in e - als (see [28,29]). Gi en u, ∈F C,we de ine he dis ance be ween uand by D(u, )=sup α∈[0,1] H([u]α,[ ]α) =sup α∈[0,1] max )***uα− α***,|uα− α|+. So, (FC,D)isacomple eme icspace. 3Diffe en iable uzzy unc ions Hence o h, Kdeno es an open subse o Rn.A unc ionF:K→F Cis said o be a uzzy unc ion.Fo eachα∈[0,1], we associa e wi h F he amily o in e al- alued unc ions Fα:K→K Cgi en by Fα(x)=[F(x)]α.Fo anyα∈[0,1], we deno e Fα(x)=! α(x), α(x)". He e, he endpoin unc ions α, α:K→Ra e called uppe and lowe unc ions o F, espec i ely. Nex we p esen he concep o gH-diffe en iabili y o uzzy unc ions in he one dimensional case. De ini ion 3 ([3]) Le K ⊂Rwi h F :K→F Ca uzzy unc ionandx 0∈Kandh be such ha x0+h∈K. Then he gene alized Hukuha a de i a i e (gH-de i a i e, 4 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 o sho ) o F a x0is de ined as F′(x0)=lim h→0 F(x0+h)⊖gH F(x0) h.(3) I F′(x0)∈F Csa is ying (3) exis s, we say ha F is gene alized Hukuha a diffe - en iable (gH-diffe en iable, o sho ) a x0. The gH-de i a i e o an in e al- alued unc ion [28] is simila oDe ini ion3. Mo e p ecisely, an in e al- alued unc ion F:K→K Cis gH-diffe en iable a x0∈K,wi hgH-de i a i e F′(x0)∈K C,i (3)exis swi h espec o helimi in he me ic space (KC,H), whe e he diffe ence is gi en by he gH-diffe ence be ween in e als (see [28]). Theo em 1 Le F :K→F Cbe a uzzy unc ion. I F is gH-diffe en iable hen he in e al- alued unc ion Fα:K→K Cis gH-diffe en iable o each α∈[0,1]. Mo eo e ,F′(x)-α=F′ α(x).(4) P oo . The p oo is a consequence o he de ini ion o gH-diffe en iabili y. Example 1 Conside he uzzy mapping F :R→F Cde ined by F(x)=C·x, whe e C is a uzzy in e al and [C]α=[Cα,Cα]wi h Cα<Cα.No e ha Fisa gene aliza ion o a linea unc ion and o each α∈[0,1] we ha e Fα(x)=⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ !Cαx,Cαx"i x ≥0; !Cαx,Cαx"i x <0. Thus he endpoin unc ions αand αa e no diffe en iable a x =0.Howe e F is gH-diffe en iable on Rand F′(x)=C o allx∈R.Ingene al,i F(x)=C·g(x), whe e g :R→Ris a diffe en iable unc ion and C ∈F C, heni ollows ela i ely easy ha he gH-de i a i e exis s and i is F′(x)=C·g′(x),bu heendpoin unc ions αand αa e no necessa ily diffe en iable. In gene al we ha e he ollowing esul which connec s gH-diffe en iabili y o F and he diffe en iabili y o i s endpoin unc ions αand α. Theo em 2 Le F :K→F Cbe a uzzy unc ion. I F is gH-diffe en iable a x0∈K hen, o each α∈[0,1],oneo he ollowingcaseshold: (a) αand αa e diffe en iable a x0and [F′(x0)]α=!min )( α)′(x0),( α)′(x0)+,max )( α)′(x0),( α)′(x0)+"; 5 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 (b)( α)′ −(x0),( α)′ +(x0),( α)′ −(x0)and ( α)′ +(x0)exis and sa is y ( α)′ −(x0)=( α)′ +(x0) and ( α)′ +(x0)=( α)′ −(x0).Mo eo e [F′( 0)]α=!min )( α)′ −(x0),( α)′ −(x0)+,max )( α)′ −(x0),( α)′ −(x0)+" =!min )( α)′ +(x0),( α)′ +(x0)+,max )( α)′ +(x0),( α)′ +(x0)+" P oo . The p oo is a consequence o Theo em 9 in [6] and Theo em 1. Rema k 1 No e ha he gH-diffe en iabili y is coinciden wi h he H-diffe en ia- bili y (diffe en iabili y in he sense o Hukuha a in oduced by Pu i andRalescu [22] as a gene aliza ion o he Hukuha a de i a i e o se - alued unc ions [14]) only when αand αa e diffe en iable and ( α)′(x)≤( α)′(x) o all α∈[0,1]. Thus, he gH-diffe en iabili y is a mo e gene al concep o diffe en iabili y o uzzy unc ions han he H-diffe en iabili y. The gH-diffe en iabili y concep is also mo e gene al han G-diffe en iabili y, see [1]. We a e now going o de ine he pa ial de i a i e o a uzzy unc ion Fde ined on K⊂Rn,i.e.,F(x)=F(x1, ..., xn)∈F C o each x=(x1, ..., xn)∈K.Fo his, gi en a uzzy unc ion F:K→F C,wedeno e he uzzyin e alF(x)by F(x)=! (x), (x)"and, o each α∈[0,1], Fα(x)=! α(x), α(x)". De ini ion 4 Le F be a uzzy unc ion de ined on K ⊂Rnand le x0='x(0) 1, ..., x(0) n( be a ixed elemen o K. We conside he uzzy unc ion hi(xi)=F(x(0) 1, ..., x(0) i−1,xi,x(0) i+1, ..., x(0) n). I hiis gH-diffe en iable a x(0) i, henwesay ha Fhas hei hpa ialgH-de i a i e a x0(deno ed by (∂F/∂xi)(x0))and(∂F/∂xi)(x0)=(hi)′(x(0) i). De ini ion 5 Le F be a uzzy unc ion de ined on K and le x0='x(0) 1, ..., x(0) n(∈Kbe ixed.Wesay ha FisgH-diffe en iable a x0i all he pa ial gH-de i a i es (∂F/∂x1)(x0),..., (∂F/∂xn)(x0)exis on some neighbo hood o x0and a e con inuous a x0. No e ha i Fis gH-diffe en iable a x0, hen(∂F/∂xi)(x0)isa uzzyin e al.So, o each α∈[0,1], we deno e .∂F ∂xi (x0)/α =∂Fα ∂xi (x0)=⎡⎢⎢⎢⎢⎣ ∂Fα ∂xi (x0),∂Fα ∂xi (x0)⎤⎥⎥⎥⎥⎦. We ob ain '∂Fα/∂xi((x0)and'∂Fα/∂xi((x0) omTheo em2. Nex we p esen an in e es ing p oposi ion which will be used oob ainou main esul s. 6 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 P oposi ion 1 Le F :K→F Cbe a uzzy unc ion. I F is gH-diffe en iable a x0∈K hen, o eachα∈[0,1], he eal- alued unc ion α+ α:K→Ris diffe en iable a x0.Mo eo e , ∂Fα ∂xi (x0)+∂Fα ∂xi (x0)= ∂' α+ α( ∂xi (x0).(5) P oo . The p oo is a consequence o Theo em 2. F om p e ious de ini ion we can de ine he g adien o a uzzy unc ion as ollows. De ini ion 6 Gi en he uzzy unc ion F :K→F C, heg adien o Fa x 0,deno ed by ˜ ∇F(x0),isde inedby ˜ ∇F(x0)=66∂F ∂x17(x0), ..., 6∂F ∂xn7(x0)7,(6) whe e (∂F/∂xj)(x0)is he j h pa ial G-de i a i e o F a x0. No e ha ˜ ∇F(x)isan-dimensional uzzy ec o . Fo he g adien o a uzzy unc- ion we use he symbol ˜ ∇,whe eas o heg adien o a eal- alued unc ionweuse he symbol ∇. De ini ion 7 Le F :K⊂Rn→F Cbe a uzzy unc ion, whe e K ⊂Rnis an open se . Suppose now ha he e is x0∈Ksuch ha g adien o F,˜ ∇F, is i sel gH- diffe en iable a x0, ha is, o eachi, he unc ion∂F ∂xi:K→F Cis gH-diffe en iable a x0.Deno e hegH-pa ialde i a i eo ∂F ∂xiby D2 ijF(x0)o ∂2F ∂xixj (x0),i i !j, and D2 iiF(x0)o ∂2F ∂x2 i (x0),i i =j. I F is wice gH-diffe en iable a each x0in K, we say ha F is wice gH-diffe en iable on K, and i o each i,j=1,2, ..., n, he c oss-pa ial de i a i e ∂2F ∂xixjis con inuous unc ion om K o FC,wesay ha Fis wicecon inuouslygH-diffe en iable on K. We de ine a m- imes con inuously gH-diffe en iable uzzy unc ion in way simila o De ini ion 7, ha is, F:K→F Cis m- imes con inuously gH-diffe en iable on Ki and only i all o he pa ial gH-de i a i es o o de m∈Nexis and a e con inuous (in he sense o uzzy unc ion). I Fis gH-diffe en iable we ha e ha he endpoin unc ion αand αa e no necessa ily diffe en iable. Howe e , om P oposi ion 1, we ha e ha α+ αis 7 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 Re e ences [1] Bede B., Gal S. 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