Solution of Linear Programming Problems using a Neural Network with Non-Linear Feedback

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1 RADIOENGINEERING, VOL 21, NO 4, DECEMBER Solutio of Liear Prograig Probles usig a Neural Network with No-Liear Feedback Syed Atiqur RAHMAN, Mohd Saar ANSARI, Athar Ali MOINUDDIN Departet of Electroics Egieerig, AMU, Aligarh, Idia atiqau@gailco, dsaar@gailco, aaoi@gailco Abstract This paper presets a recurret eural circuit for solvig liear prograig probles The objective is to iiize a liear cost fuctio subject to liear costraits The proposed circuit eploys o-liear feedback, i the for of uipolar coparators, to itroduce trascedetal ters i the eergy fuctio esurig fast covergece to the solutio The proof of validity of the eergy fuctio is also provided The hardware coplexity of the proposed circuit copares favorably with other proposed circuits for the sae task PSPICE siulatio results are preseted for a chose optiizatio proble ad are foud to agree with the algebraic solutio Hardware test results for a 2-variable proble further serve to stregthe the proposed theory Keywords Liear prograig, dyaical systes, eural etworks, feedback systes, o-liear feedback 1 Itroductio Matheatical prograig, i geeral, is cocered with the deteriatio of a iiu or a axiu of a fuctio of several variables, which are required to satisfy a uber of costraits Such solutios are sought i diverse fields icludig egieerig, operatios research, aageet sciece, coputer sciece, uerical aalysis, ad ecooics [1, [2 A geeral atheatical prograig proble ca be stated as [2: Miiize f (x) (1) subject to g i (x) 0 (i 1,2,,), (2) h j (x) 0 ( j 1,2,, p), (3) x S (4) where x (x 1,x 2,,x ) T is the vector of ukow decisio variables, ad f, g i (i 1,2,,), h j ( j 1,2,, p) are the real-valued fuctios of the real variables x 1,x 2,,x I this forulatio, the fuctio f is called the objective fuctio, ad iequalities (2), equatios (3) ad the set restrictios (4) are referred to as the costraits It ay be etioed that although the atheatical prograig proble (MPP) has bee stated as a iiizatio proble i the descriptio above, the sae ay readily be coverted ito a axiizatio proble without ay loss of geerality, by usig the idetity give i (5) ax f (x) i [ f (x) (5) As a special case, if all the fuctios appearig i the MPP are liear i the decisio variables x, the proble is referred to as a liear prograig proble (LPP) Such LPPs have bee ivestigated extesively over the past decades, i view of their fudaetal roles arisig i a wide variety of egieerig ad scietific applicatios, such as patter recogitio [3, sigal processig [4, hua oveet aalysis [5, robotic cotrol [6, ad data regressio [7 Other real life applicatios iclude portfolio optiizatio [8, crew schedulig [9, aufacturig ad trasportatio [10, telecouicatios [11, ad the Travelig Salesa Proble (TSP) [12 Traditioal ethods for solvig liear prograig probles typically ivolve a iterative process, but log coputatioal tie liits their usage A alterative approach to solutio of this proble is to exploit the artificial eural etworks (ANN s) which ca be cosidered as a aalog coputer relyig o a highly siplified odel of euros [13 ANN s have bee applied to several classes of costraied optiizatio probles ad have show proise for solvig such probles ore effectively For exaple, the Hopfield eural etwork has prove to be a powerful tool for solvig soe of the optiizatio probles Tak ad Hopfield first proposed a eural etwork for solvig atheatical prograig probles, where a liear prograig proble (LPP) was apped ito a closed-loop etwork [14 A brief overview of the various eural etwork based approaches which have bee proposed over the past is preseted i the ext sectio I this paper, a hardware solutio to the liear prograig proble is preseted The proposed architecture uses o-liear feedback which leads to a ew eergy fuctio that ivolves trascedetal ters This trascedetal eergy fuctio is fudaetally differet fro the stadard quadratic for associated with Hopfield etwork ad its

2 1172 S A RAHMAN, M S ANSARI, A A MOINUDDIN, SOLUTION OF LINEAR PROGRAMMING PROBLEMS USING variats To solve a LPP i variables with costraits, the circuit requires opaps, uipolar coparators ad ( + ) resistors thereby causig the hardware coplexity of the proposed etwork to copare favorably with the existig hardware ipleetatios It ay be etioed that a siilar approach of usig o-liear syaptic itercoectios betwee euros has also bee eployed to solve systes of siultaeous liear equatios [15 ad quadratic prograig [16 A iitial result o the solutio of a liear prograig proble i two variables usig this approach appears i [17 The reaider of this paper is arraged as follows A brief review of relevat techical literature o the solutio of LPP usig eural etwork based ethods is preseted i Sectio 2 Sectio 3 outlies the atheatical forulatio of the basic proble ad details of the proposed etwork Sectio 4 cotais explaatio of the eergy fuctio ad the proof of its validity Sectio 5 cotais the circuit ipleetatio of the proposed etwork for a set of saple proble i four variables PSPICE siulatio results of the proposed circuit are also preseted Results of breadboard ipleetatio of the proposed circuit for a 2-variable proble are cotaied i Sectio 6 Issues that are expected to arise i actual oolithic ipleetatios are discussed i Sectio 7 Cocludig rearks are preseted i Sectio 8 2 Existig Neural Networks for LPP LPP has received cosiderable research attetio fro the eural etworks couity The first solutio of the liear prograig proble was proposed by Tak ad Hopfield wherei they used the cotiuous-tie Hopfield etwork [14 Fro the coputatioal aspect, the operatio of Hopfield etwork for a optiizatio proble, like the LPP, aages a dyaic syste characterized by a eergy fuctio, which is the cobiatio of the objective fuctio ad the costraits of the origial proble [18 Over the years, the pealty fuctio approach has becoe a popular techique for solvig optiizatio probles Keedy & Chua proposed a iproved versio of Tak & Hopfield s etwork for LPP i which a iexact pealty fuctio was cosidered [19 The requireet of settig a large uber of paraeters was a ajor drawback of Keedy & Chua s LPP etwork [4 Rodriguez-Vazquez et al used a differet pealty ethod to trasfor the give LPP ito a ucostraied optiizatio proble [20 Although Rodriguez-Vazquez et al later poited out that their etwork had o equilibriu poit i the classical sese [21, ivestigatios by La et al proved that the etwork ca ideed coverge to a optial solutio of the give proble fro ay arbitrary iitial coditio [22 Maa & Shablatt eployed a two-phase eural etwork architecture for solvig LPPs [23 Chog et al aalyzed a class of eural etwork odels for the solutio of LPPs by dyaic gradiet approaches based o exact o-differetiable pealty fuctios [24 They also developed a aalytical tool aied at helpig the syste coverge to a solutio withi a fiite tie I a approach differet fro the pealty fuctio ethods, Zhu, Zhag ad Costatiides proposed a Lagrage ethod for solvig LPPs through Hopfield etworks [25 Xia ad Wag used bouded variables to costruct a ew eural etwork approach to solve LPP with o pealty paraeters They suggested that the equilibriu poit is the sae as the exact solutio whe the prial ad dual probles are solved siultaeously [26 More recetly, Malek & Yari proposed two ew ethods for solvig the LPP ad preseted optial solutios with efficiet covergece withi a fiite tie [27 Lastly, Ghasabi-Oskoei, Malek ad Ahadi have preseted a recurret eural etwork odel for solvig LPP based o a dyaical syste usig arbitrary iitial coditios The ethod does ot require aalog ultipliers thereby reducig the syste coplexity [28 3 Proposed Circuit Let the first-order fuctio to be iiized be F [ c 1 c 2 c subject to the followig liear costraits a 11 a 12 a 1 a 21 a 22 a 2 a 1 a 2 a V 1 V 2 V V 1 V 2 V b 1 b 2 b where V 1,V 2,,V are the variables, ad a i j, c j ad b i (i 1,2,,; j 1,2,,) are costats The proposed eural-etwork based circuit to iiize the fuctio give i (6) i accordace with the costraits of (7) is preseted i Fig 1 As ca be see fro Fig 1, idividual iequalities fro the set of costraits are passed through oliear syapses which are realized usig uipolar coparators The outputs of the coparators are fed to euros havig weighted iputs The euros are realized by usig opaps ad the weights are ipleeted usig resistaces R pi ad C pi are the iput resistace ad capacitace of the opap that is used to eulate the fuctioality of a euro These parasitic copoets are icluded to odel the dyaic ature of the opap If the uipolar coparator is operated with a sigle +V supply, while the opap realizig the euroal fuctioality beig biased with ± V, the obtaied trasfer characteristics of the uipolar voltage coparator are preseted i Fig 2 ad ca be atheatically odelled by (8) As explaied i the ext sectio, such uipolar coparator characteristics are utilized to obtai a eergy fuctio which acts to brig the euroal states to the feasible regio (6) (7)

3 RADIOENGINEERING, VOL 21, NO 4, DECEMBER Fig 1 i th euro of the proposed feedback eural etwork circuit to solve a liear prograig proble i variables with liear costraits uipolar coparators to the iput of the i-th euro As it is show later i this sectio, the values of these resistaces are govered by the etries i the coefficiet atrix of (7) Resistace R i causes ters correspodig to the liear fuctio to be iiized, to appear i the eergy fuctio It ay be etioed that the cocept of a Lyapuov or eergy fuctio beig associated with gradiet-type eural etworks was first eployed by Hopfield i the stability aalysis of the so-called Hopfield Neural Network (HNN) [14 The coputatioal eergy for dyaical systes like the HNN ad the No-Liear Syapse Neual Network preseted i this paper, decreases cotiuously i tie with the etwork covergig to a iiu i state space The evolutio of the syste is i the geeral directio of the egative gradiet of the eergy fuctio Typically, the etwork eergy fuctio is ade equivalet to a certai objective fuctio that eeds to be iiized The search for a eergy iiu perfored by the etwork correspod to the search for the solutio of a optiizatio proble As explaied i Sectio 4, the eergy fuctio associated with the o-liear feedback eural circuit of Fig 1, for iiizig a liear objective fuctio subject to liear costraits, is give by + V 2β E l cosh β c i V i + V 2 a i j V j [ (a i j V j b i ) (12) Fig 2 Trasfer characteristics of the uipolar coparator x 1 2 V [ tah β (V i V j ) + 1 (8) Usig (8), the output of the i-th uipolar coparator i Fig 1 ca be give by (9) where β is the ope-loop gai of the coparator (practically very high), ± V are the output voltage levels of the coparator ad V 1, V 2,, V are the euro outputs x i 1 2 V [tah β (a i1 V 1 + a i2 V a i V b i ) + 1 Applyig ode equatios for ode A i Fig 1, the equatio of otio of the i-th euro ca be give as [ du i x j C pi + c i u i (10) R i R ieqv R c ji where R ieqv is the parallel equivalet of all resistaces coected at ode A i Fig 1 ad is give by (11) [ (11) R ieqv R i R pi R c ji where u i is the iteral state of the i-th euro, R c1i, R c2i,, R ci are the weight resistaces coectig the outputs of the (9) This expressio of the eergy fuctio ca be writte i a slightly differet (but ore illuiatig) for as E c i V i + (P 1 + P P ) (13) where the first ter is the sae as the first-order fuctio to be iiized, as give i (6), ad P 1, P 2,, P are the pealty ters The i-th pealty ter ca be give as P i V 2 a i j V j + V 2β l cosh β [ (a i j V j b i ) [ (a i j V j b i ) (14) Obtaiig a partial differetial of the cobied pealty ter, P( P 1 + P P ) with respect to V i we have P V a ji + V a ji tah β (15) V i 2 2 which ay be siplified to P V i a ji x j (16) Usig the above relatios to fid the derivative of the eergy fuctio E with respect to V i we have

4 1174 S A RAHMAN, M S ANSARI, A A MOINUDDIN, SOLUTION OF LINEAR PROGRAMMING PROBLEMS USING which i tur yields [ E c i V i + P (17) V i V i V i E c i + V i a ji x j (18) Also, if E is the eergy fuctio, it ust satisfy the followig coditio [29: E du i KC pi (19) V i where K is a costat of proportioality ad has the diesios of resistace Usig (9), (10) ad (18) i (19) results i (for the i th euro) R c ji K/a ji ; ( j 1,2,,) (20) de Usig (19) i (23) we get de E dv i V i E dv i du i V i du i (23) ( ) 2 dui dv i KC pi (24) du i The trasfer characteristics of the output opap used to ipleet the euros i Fig 1 ipleets the activatio fuctio of the euro ad ca be writte as V i f (u i ) (25) where V i deotes the output of the opap ad u i correspods to the iteral state at the ivertig terial The fuctio f is typically a saturatig, ootoically decreasig oe, as show i Fig 4, ad therefore [15, A siilar copariso of the reaiig partial fractios for the reaiig euros yields the followig: R c ji K/a ji ; 4 Eergy Fuctio ( j 1,2,,), (i 1,2,,), (21) R i K ; (i 1,2,,) (22) This sectio deals with the explaatio of idividual ters i the eergy fuctio expressio give i (12) The last ter is trascedetal i ature ad a idicative plot showig the cobied effect of the last two ters is preseted i Fig 3 As ca be see, oe side of the eergy ladscape is flat whilst the other has a slope directed to brig the syste state towards the side of the flat slope Durig the actual operatio of the proposed LPP solvig circuit, the coparators reai effective oly whe the euroal output states reai outside the feasible regio ad durig this coditio, these uipolar coparators work to brig (ad restrict) the euro output voltages to the feasible regio Oce that is achieved, first ter i (12) takes over ad works to iiize the give fuctio Fig 3 Cobied effect of last two ters i (12) The validity of the eergy fuctio of (12) ca be proved as follows The tie derivative of the eergy fuctio is give by Fig 4 Trasfer characteristics of the opap used to realize the euros dv i 0 (26) du i thereby resultig i de 0 (27) with the equality beig valid for du i 0 (28) Equatio (27) shows that the eergy fuctio ca ever icrease with tie which is oe of the coditios for a valid eergy fuctio The secod criterio ie the eergy fuctio ust have a lower boud is also satisfied for the circuit of Fig 1 wherei it ay be see that V 1, V 2,, V are all bouded (as they are the outputs of opaps, as give i (25)) aoutig to E, as give i (12), havig a fiite lower boud 5 Siulatio Results This sectio deals with the applicatio of the proposed etwork to the task of iiizig the objective fuctio V 1 + 2V 2 V 3 + 3V 4 (29) subject to V 1 V 2 +V 3 4, V 1 +V 2 + 2V 4 6, V 2 2V 3 +V 4 2, V 1 + 2V 2 +V 3 2, V 1 0, V 4 0 (30)

5 RADIOENGINEERING, VOL 21, NO 4, DECEMBER Fig 5 Obtaiig uipolar coparator characteristics usig a opap ad a diode Fig 7 Siulatio results for the proposed circuit applied to iiize (29) subject to (30) Fig 6 Trasfer characteristics for opap based uipolar ad bipolar coparators as obtaied fro PSPICE siulatios The values of resistaces actig as the weights o the euros are obtaied fro (21), (22) For the purpose of siulatio, the value of K was chose to be 1 kω Usig K 1 kω i (21), (22) gives R c11 R c12 R c13 R c14 R c21 R c22 R c23 R c24 R c31 R c32 R c33 R c34 R c41 R c42 R c43 R c44 R c51 R c52 R c53 R c54 R c61 R c62 R c63 R c64 1K 1K 1K 1K 1K 05K 1K 05K 1K 1K 05K 1K 1K 1K, (31) R 1 R 2 R 3 R 4 1K (32) For the purpose of PSPICE siulatios, the uipolar voltage coparator was realized usig a diode with a opap based coparator as show i Fig 5 The trasfer characteristics obtaied durig the PSPICE siulatios for opap based bipolar ad uipolar coparators are preseted i Fig 6 fro where it ca be observed that the obtaied uipolar characterisitics are i agreeet with the ideal characteristics of Fig 2 For the purpose of this siulatio, the LMC7101A CMOS opap odel fro the Orcad library i PSPICE was utilised The value of β for this opap was easured to be usig PSPICE siulatio For the egative values arisig i (31), iverted outputs would be required fro the uipolar voltage coparators For the preset siulatio, ivertig aplifiers were eployed at the outputs of the coparators which eed egative weights Routie atheatical aalysis of (29) yields: V 1 0, V 2 10, V 3 6 ad V 4 0 The resultat plots of the euro output voltages as obtaied after PSPICE siulatio are preseted i Fig 7 fro where it ca be see that V(1) 110 µv, V(2) 1028 V, V(3) 617 V ad V(4) 110 µv which are very ear to the algebraic solutio thereby cofirig the validity of the approach For eulatig a ore realistic power-up sceario, rado iitial values i the illi-volt rage were assiged to ode voltages Oe set of iitial ode voltage was: V(1) 2 V, V(2) 7 V, V(3) -5 V ad V(4) 10 V 6 Hardware Test Results Breadboad ipleetatio of the proposed circuit was also carried out Apart fro verificatio of the workig of the proposed circuit, the actual circuit realizatio also served the purpose of testig the covergece of the circuit to the solutio startig fro differet iitial coditios The oise preset i ay electroic circuit acts as a rado iitial coditio for the covergece of the eural circuit Stadard laboratory copoets ie the µa741 opap ad resistaces were used for the purpose A 2 variable proble for the iiizatio of the objective fuctio subject to 2V 1 + 6V 2 (33) V 1 1, V 2 1 (34) was chose for the hardware tests The values of resistaces actig as the weights o the euros are obtaied fro (21), (22) The value of K was chose to be 1 kω Usig K 1 kω i (21), (22) gives R 1 R 2 1 kω, R c11 R c22 1 kω ad R c12 R c21 The voltages applied were b 1 b 2 1 V, c 1 2 V ad c 2 6 V The obtaied values of the euroal voltages were V V ad V V which are i close agreeet with the exact atheatical solutio which is V 1 1 ad V 2 1 a sapshot of the obtaied results is preseted i Fig 8 7 Issues i Actual Ipleetatio This sectio deals with the oolithic ipleetatio issues of the proposed circuit The PSPICE siulatios assued that all operatioal aplifiers (ad diodes) are idetical, ad therefore, it is required to deterie how deviatios fro this assuptio affect the perforace of the etwork

6 1176 S A RAHMAN, M S ANSARI, A A MOINUDDIN, SOLUTION OF LINEAR PROGRAMMING PROBLEMS USING Tab 1 Effect of variatio i resistaces o the obtaied results Fig 8 Hardware test result for the proposed circuit applied to iiize (33) subject to (34) A 10 % tolerace with Gaussia deviatio profile was put o the resistaces used i the circuit to solve (29) The aalysis was carried out for 200 rus ad the Mea Deviatio was foud out to be ad Mea Siga (Stadard Deviatio) was 0013 Offset aalysis was also carried out by icorporatig rado offset voltages (i the rage of 1 V to 10 V) to the opaps The Mea Deviatio i this case was easured to be ad the Mea Siga (Stadard Deviatio) was 0007 As ca be see, the effects of isatches ad offsets o the overall precisio of the fial results are i a acceptable rage Effects of o-idealities i various copoets were further ivestigated i PSPICE by testig the circuit with all resistaces havig the sae percetage deviatio fro their assiged values The resultig assesset of the quality of solutio is preseted i Tab 1 fro where it is evidet that the solutio poit does ot chage uch eve for high deviatios i the resistace values Next, the effect of offset voltages i the opap-based uipolar coparators was explored Offset voltages were applied at the ivertig terials of the coparators ad the results of PSPICE siulatios for the chose LPP are give i Tab 2 As ca be see, the offset voltages of the coparators do ot affect the obtaied solutios to ay appreciable extet However, the error does ted to icrease with icreasig offset voltages Fially, offset voltages for the opaps eulatig the euros were also cosidered Offset voltages were applied at the o-ivertig iputs of the opaps ad the results of PSPICE siulatios were copared with the algebraic solutio as give i Tab 3 As ca be see, the offset voltages of the opaps have little effect o the obtaied solutios Tab 2 Effect of offset voltages of the opap-based uipolar coparators o the solutio quality Tab 3 Effect of offset voltages of the opap-based euros o the solutio quality ight be quite differet Alterative realizatios based o the differetial equatios (10) goverig the syste of euros are beig ivestigated Other approaches to obtai the tah() o-liearity iclude the use of a MOSFET operated i the sub-threshold regio [30 ad the use of Curret Differecig Trascoductace Aplifier (CDTA) to provide the sae oliearity i the curret-ode regie [31 8 Coclusio I this paper, a CMOS copatible approach to solve a liear prograig proble i variables subject to liear costraits, which uses -euros ad -syapses is preseted Each euro requires oe opap ad each syapse is ipleeted usig a uipolar voltage-ode coparator This results i sigificat reductio i hardware over the existig schees The proposed etwork was tested o a saple proble of iiizig a liear fuctio i 4 variables ad the siulatio results cofir the validity of the approach Hardware verificatio for a 2 variable proble further validated the theory proposed I fact, the realizatio of uipolar coparators by the use of opaps ad diodes i the proposed circuit teds to icrease the circuit coplexity The trasistor cout ca be further reduced by utilisig voltage-ode uipolar coparators istead of the opap-diode cobiatio This also suggests that a real, large scale ipleetatio for solvig liear prograig probles with high variable couts Refereces [1 KREYSZIG, E Advaced Egieerig Matheatics 8 th ed New Delhi (Idia): Wiley-Idia, 2006 [2 KAMBO, N S Matheatical Prograig Techiques New Delhi (Idia): Affilated East-West Press Pvt Ltd, 1991

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