Neural Networks for Nonlinear Fractional Programming

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1 Iteratoal Joural o Scetc & Egeerg Research, Volue, Issue, Deceber- ISSN Neural Networks or Nolear Fractoal Prograg S.K Bso, G. Dev, Arabda Rath Abstract - Ths paper presets a eural etwork or solvg o-lear a ultobectve ractoal prograg proble subect to olear equalty costrats. Neural odel s desged or optzato wth costrats codto. Methodology s based o the lagrage ultpler wth saddle pot optzato. Key words: Multobectve, Fractoal prograg, saddle pot, Lagrage ultpler, varatoal equalty, proecto.. Itroducto Optzato probles arse a wde varety o scetc ad egeerg applcatos cludg sgal processg, syste detcato, lter desg, ucto approato, regresso aalyss ad so o. Here, we peror a rgorous aalyss o a eural etwork or solvg o-lear ractoal prograg probles ad preset a saddle pot optalty theory or the a ractoal prograg proble. Varous uercal procedures have bee preseted over decades or solvg lear ad olear optzato probles. Tak ad Hop eld [5] 986 rst proposed a eural etwork or lear prograg. Keedy ad Chua [,] eteded ad proved the tak ad Hopeld etwork by usg pealty ethod or solvg olear prograg proble. Varety o attepts to avod usg pealty paraeters have bee ade. Radrguez Vazquez et al.[] proposed a swtched capactor eural etwork or solvg a class o costraed o-lear cove optzato probles. Neural etwork odels or optzato proble have bee vestgated tesvely sce the poeer work o Hop eld see [8,6,,4], or the ege value proble, Xa [4] gave a prosg eural etwork odel whch was proved to have global covergece wth respect to the probles easble set. Xa ad Wag [8] gave a geeral eural etwork odel desgg ethodology whch put together way gradet based etwork odels or solvg the cove prograg probles wth globally coverget stablty. Neural etwork or quadratc ad olear optzato wth terval costrats were developed by Bouzerdor, pattso [3] ad Lag, Wag [3] ad others [],[]-[7]. All there eural etworks ca be classed to the ollowg three types: () The gradet- based odels () The pealty ucto based odels (3) The proecto based odels Nolear ractoal prograg does ot belog to cove optzato probles ad how to costruct a good perorace eural etwork odel to solve ths optzato proble becoes a challege ow sce. Fractoal prograg s a olear prograg ethod that has kow creasg eposure recetly ad ts portace solvg cocrete probles s steadly creasg. Also o-lear optzato odels descrbe practcal probles uch better tha the lear optzato does. The ractoal prograg probles are partcularly useul the soluto o ecooc probles whch varous actvtes use certa resources varous proportos whle the obectve s to optze a certa dcator, usually the ost avorable retur o allocato rato subect to the costrat posed o the avalablty o goods. Pal & Gupta [9] 8 preseted a Goal prograg approach or solvg terval valued ult obectve ractoal prograg probles usg Geetc Algorth. Zhag & Feg [7] developed Neuro dyac aalyss or a class o olear ractoal prograg. We ad Wu [5] solved a cotuous- te lear ractoal prograg probles by usg the paraetrc ethod Neural crcut desg techques ad related characterstcs aalyss s ow becog a typcally challegg udertakg. Motvated by ths dea, ths paper s orgazed as ollows. I ths secto, we orulate the ult obectg o-lear ractoal prograg proble ad ts dualty. I secto 3, soe llustratve eaples are preseted. I secto 4, Neural odel s prarly desged or optzato wth costrats codto. The ethodology s based o the lagrage ultpler wth saddle pot whch satses the optalty ad cocluso part secto 5 are cted. Secto - : We cosder the ollowg probles: (P) a ( ( h ( ) X p Subect to g, X R where, h, =,, p are real valued uctos deed o X, each h s strctly postve ad g = (g, g g ), where each g s a real valued ucto deed o. IJSER

2 Iteratoal Joural o Scetc & Egeerg Research, Volue, Issue, Deceber- ISSN To develop the optalty codtos, cosder the ollowg aulary proble: (Pe) a e h Subect to g (,u) = X p, X R a ( ( e h () p u uber e ad or all X ad ur. g The lagraga dual o (P) s deed as ollows: a X a eh u g u o p For a ed e ad or each X & ur or ed real Deto : I there ests X, u R u such that or all (, u), u, u X ad or all u R, u, the, u saddle pot o proble (Pe). s called a Deto : A par u wth X, u R, s sad to satsy the optalty codtos or the proble (pe) ad oly the ollowg our codtos are satsed (a) zes (, u ) (b) u g ( ) = (c) g ( ) (d) u Deto 3: A pot u R s sad to be a optal ultpler o proble (Pe) ad oly there ests a X such that, u satses optalty codto o deto. Ma Results: I, u satses the optalty codtos or (Pe) wth e a / h, the p, u also satses the optalty codtos or (P) Result : For ay uu, X, (, u) a X X p a Proo: For ay u U, X e h u g p / h (, u) = a e h u g X X p a eh ug p a / eh sce g ( ad u or all. p a IJSER p / h, h or all Hece the cocluso ollows: Result : I u s a optal ultpler or (Pe) the or every optal soluto o (Pe),, u satses the optalty codtos or (Pe). Proo : Sce u s a optal ultpler or (Pe), there ests X such that t satses () zes o (, u ) () u g () g (v) u Fro (), () ad (v) codtos t ollows that u g( ), Now or ed e, p a ( ) -eh ( ) a X p p a a p ( ) -eh eh ( eh u g =, u L, u = a e h u. g p p a e h Sce g, u, t ollows that, u, u X Hece, u satses the optalty codtos or (Pe). Result 3 : A par, u satses the optalty codtos () to (v) o deto ad oly t satses the ollowg codtos () s a optal soluto (Pe) () u s a optal soluto o D () a ( ( ) e h ( )) = (, u) p Proo: Now, u satses the optalty codtos o () to (v) o deto. The or ay ur, u, u g u a e h u g ed e p a a p =, u X, or all p Fro optalty codtos, e h, sce u g e h u g

3 Iteratoal Joural o Scetc & Egeerg Research, Volue, Issue, Deceber- 3 ISSN 9-558, u, u X Result 4 : The ollowg stateets are equvalet () Hece ro equato () ad () t ollows that (a), u s a saddle pot o (, u), u, u, u Ths ples that, u s a saddle pot. Sce, u s a saddle pot o, u, the, u s a optal soluto o ( P e ). Ths proves the result-3 codtos (). To establsh codtos (), (, u) a X X p X a p ( eh ( ( ) eh ( ) a ( ) eh ( ) p a p ug( u g ( ) ( ) eh ( ) a, u, u u g ( ) ( eh ( u g( X p, u a ( eh ( X p X, u u g( u s a optal soluto o (D). Ths establshes the proo or (). To prove (), we cosder ro equato () that, u, u X or, a ( ) eh ( ) p X, u Sce, u g ( ),,..., a p u g ( ) ( ) eh ( ), u X To prove or coverse part, we cosder that t satses the codtos ro () to (). The, a X p a p a p ( eh ( ( ) eh ( ) u u g( g ( ( ) eh ( ), u Hece all the relatos are equal. zes X, u ad u g( ) The optaalty codtos () ad (v) ollow or the easblty o ad u IJSER (b) Codtos o Result hold (c) Codto () to (v) o deto - hold. We observe that (a) (b) (c) (a) Result 5: Suppose that (P) has a optal soluto ad that s stable where e a / h P e p Thus (D) has a optal soluto ad the optal values o (P) ad (D) are equal. Secto 3: Deto 4 : A easble soluto o P e s sad to be a ecet soluto o (Pe) there does ot est ay easble soluto o (Pe) such that ed e. ( eh ( ( ) eh ( ) or, p & or... ad ( eh ( ( ) eh ( ) or soe ad or ed e I s a ecet soluto o (Pe) the soluto o (P) s a ecet Deto 5: A easble soluto o P e s sad to be properly ecet soluto o P e t s a ecet soluto o P ad there ests a scalar M > such that or soe ad e or soe easble, ( e h ( ( ) eh ( ), ( ) eh ( ) ( e h ( ( e h ( eh M such that ( ) ( ) ( ) ( e h eh ) I s a properly ecet soluto o a properly ecet soluto (P). Eaple : F ( eh(, eh p subect to (,, g ad X, ( ( 3 h ( h( g ( g( The easble rego s, or ed real uber. or soe. t us take, ( e h ( ( ) e h ( ) ( ) ( 3) P the s e ad e= Now we ca prove that s a properly ecet soluto o ( F p ).

4 Iteratoal Joural o Scetc & Egeerg Research, Volue, Issue, Deceber- 4 ISSN Secto 4: Neural Model The euros the etwork ca be classed to two classes: varable euros ad Lagraga euros, wth regard to ther role searchg or the soluto. I the dyac process o the eural etwork, Lagraga euros lead the traectory tot the easble rego whle varable euros decrease the Lagraga ucto L(,). The decrease o the Lagraga ucto ca be vered ro the act that alog the traectory o the etwork dl(, ) L(, ) d dt dt cos tat d dt t,h,g be pseudocove uctos. Hece h s pseudocove.e. ( eh ( s pseudocove. Sce, g s pseudocove ad y that ples t ( e h ( y g( s pseudocove. t t Also ( ) ( ) e h ( ) y g( )., ths satses varatoal Iequalty proble over. Hece the ew proecto eural etwork odel s t P ( ) e h ( ) y g( where, X d h, P : s a proecto operator ad g( s closed cove. Secto 5: Cocluso: The paper proposes a ew proecto eural etwork odel ad theoretcally guarateed to solve varatoal equalty probles. The ultobectve a olear ractoal prograg s deed ad ts optalty s derved by usg ts Lagraga dualty. The equlbru pots o the proposed eural etwork odel are oud to correspod to the Karush Kuh Trcker pot assocated wth the olear ractoal prograg proble. Reereces [.] M.S. Bazaraa, H.D. Sheral, C.M. Shetty Nolear Prograg Theory ad Algorths d ed., New York. Joh Wley 993. [.] S. Zhag ad A.G. Costatdes, Lagrage Prograg Neural Networks, IEEE Tras.o Crcuts ad Systes II, Aalog ad Dgtal Sgal Processg. Vol. 39, No. 7, pp July 99. [3.] X.B Lag ad J. Wag, A recurret eural etwork or olear optzato wth a cotuously deretable obectve ucto ad boud costrats IEEE Tras, Neural etwork, vol-ii, o -6 PP. 5 6, Nov [4.] Y. Huag, Lagrage- type Neural etworks or olear prograg probles wth Iequalty costrats, IEEE coerece o decso ad cotrol, PP--5, Dec. 5. [5.] D.W. Tak ad J.J. Hopeld Sple eural optzato etworks ; A A/D coverter, sgal decso crcut ad lear Prograg crcut, IEEE Tras, crcuts systs,vol.cas-33,pp May 986. [6.] A. Jelea A paraetrc study or solvg Nolear ractoal probles A St. Uv Ovdus costata, vol (),PP.87-9, 3. [7.] Q.Zhag ad J.Feg et.al. Neurodyac Aalyss or a class o olear ractoal prograg IEEE Iteratoal coerece o coputatoal Itellgece & Securty 8. [8.] Y.a ad J. Wag A Geeral ethologyor desgg Globally coverget optzato Neural Networks. IEEE trasactos o Neural Networks, Vol 9, No-6,PP , Nov 998. [9.] B.B. Pal & S. Gupta A Goal prograg approach or solvg Iterval valued ultobectve ractoal prograg probles usg Geetc Algorth, IEEE Iteratoal coerece o Idustral ad Iorato Systes, PP: 8-, Dec 8. [.] Y.L, Y.A. Che ad H.S. Ah. Fractoal order teractve learg cotrol I proceedgs o 9 ICROS SICE. [.] D. Kderlehrer, G. Stapcha, A troducto to varatoal Iequaltes ad ther Applcato Acadec, New York (98). [.] R. Fourer, D.M Gay et. al., A odelg laguage or Matheatcal prograg, Scetc press 993,94. [3.] H.Y. Beso, et. al. Iteror pot ethods or ocove olear prograg, Jag ad Coparatve Nuercal testg Operatos Research ad Facal Egeerg, Prceto Uversty, Aug 8,. [4.] A.M Geor, Proper ececy ad theory o vector azato, J. Math, Aalyss ad Applcato Vol-,No- 3 PP: 68-63, 968. [5.] C.F We & H.C Wu A paraetrc ethod or solvg cotuous- te lear ractoal prograg probles IEEE teratoal coerece o coputatoal scece ad optzato.. [6.] S.G Nersesov ad W.M. Hadded. O the Stablty ad cotrol o Nolear Dyacal systes va vector Lyapuov Fuctos IEEE trasactos o Autoatc cotrol vol 5, No-, Feb. 6. [7.] O.L. Magasara. Nolear prograg New York, Mc Graw Hll 969. [8.] J.J. Hopeld. Neuros wth graded respose have collectve coputatoal propertes lke those o two state euro Proc. Natl, Acad. Sc Vol 8, PP , 984. [9.] T.E Ster Theory o No lear Networks ad systes, New York, Addso Wesley, 965. [.] M.P. Keedy ad L.O. Chua, Uyg the Tak ad Hoeld ar Prograg crcut ad the caocal IJSER

5 Iteratoal Joural o Scetc & Egeerg Research, Volue, Issue, Deceber- 5 ISSN No-Lar Prograg crcut o chua ad L,IEEE crcut syst; Vol CAS-34,No- February [.] M.P. Keedy ad L.O.Chua, Neural Networks or olear prograg. IEEE Tras. crcut syst., Vol 35,No. 5,May 988. [.] A.Rodrguez-Vazquez, R.Do guez-castro, et.al. Nolear swtched-capactor eural etworks or optzato probles. IEEE Tras, crcuts systes vol 37, No.3,PP ,Mar 99. [3.] A.Bouzerdow ad T.R.Pattso, Neural etwork or quadratc optzato wth boud costrats, IEEE Tras.Neural Networks.Vol.4,No.-,PP.93-34,993. [4.] Y.S. Xa, Further results o global covergece ad stablty o globally proected dyacal systes, J. Optzato Theory ad Applcato,Vol,No-3PP , 4. IJSER