A COMPARATIVE STUDY OF THE METHODS OF SOLVING NON-LINEAR PROGRAMMING PROBLEM

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1 DAODIL INTERNATIONAL UNIVERSITY JOURNAL O SCIENCE AND TECHNOLOGY, VOLUME, ISSUE, JANUARY 9 A COMPARATIVE STUDY O THE METHODS O SOLVING NON-LINEAR PROGRAMMING PROBLEM Bmal Chadra Das Departmet of Tetle Egeerg Daffodl Iteratoal Uversty, Dhaka, Bagladesh E-mal: bcdar@yahoocom Abstract: The work preset ths paper s based o a comparatve study of the methods of solvg No-lear programmg (NLP problem We kow that Kuh-Tucker codto method s a effcet method of solvg No-lear programmg problem By usg Kuh-Tucker codtos the quadratc programmg (QP problem reduced to form of Lear programmg(lp problem, so practcally smple type algorthm ca be used to solve the quadratc programmg problem (Wolfe s AlgorthmWe have arraged the materals of ths paper followg way st we dscuss about o-lear programmg problems I secod step we dscuss Kuh- Tucker codto method of solvg NLP problems ally we compare the soluto obtaed by Kuh- Tucker codto method wth other methods or problem so cosder we use MATLAB programmg to graph the costrats for obtag feasble rego Also we plot the obectve fuctos for determg optmum pots ad compare the soluto thus obtaed wth eact solutos Keywords: No-lear programmg, obectve fucto,cove-rego, pvotal elemet, optmal soluto Itroducto or decso makg optmzato plays the cetral role Optmzato s the syoym of the word mamzato/mmzato It meas choose the best I our tme to take ay decso, we use most moder scetfc methods best o computer mplemetatos Moder optmzato theory based o computg ad we ca select the best alteratve value of the obectve fucto The optmzato problems have two maor dvsos Oe s lear programmg problem ad other s o-lear programmg problem [] But the moder game theory, dyamc programmg problem, teger programmg problem also part of the optmzato theory havg wde rage of applcato moder scece, ecoomcs ad maagemet Lear ad o-lear programmg problem optmzes a obectve fucto subect to a class of lear ad olear equalty or equalty codtos called costrats, usually subect to o-egatvty restrctos of the varables It was troduced by [] I the preset work we tred to compare some methods of o-lear programmg problem We kow that, for solvg a o-lear programmg problem varous algorthms ca be used, but oly few of the methods wll be affectve for solvg problems Usually oe of the algorthms have o relatos wth others ad there s o uversal algorthm lke smple method lear programmg for solvg a o-lear programmg problem I the work we wll apply the methods for solvg problem rather tha ts theoretcal descrptos No-lear programmg Lke lear programmg, o-lear programmg s a mathematcal techque for determg the optmal solutos to may busess problems The problem of mamzg or mmzg a gve fucto Z f( Subect to ( g ( b ( b s called to geeral o-lear programmg problem, f the obectve fucto e the fucto f ( or ay oe of the costrats fucto g ( s o-lear ; or both are o-lear for oegatve I a word, the o-lear programmg problem s that of choosg oegatve values of certa varables, so as to mamze or mmze a gve o-lear fucto subect to a gve set of lear or o-lear equalty costrats; or mamze or mmze a lear

2 DAODIL INTERNATIONAL UNIVERSITY JOURNAL O SCIENCE AND TECHNOLOGY, VOLUME, ISSUE, JANUARY 9 9 fucto subect to a gve set of o-lear equalty The problem ( ca be re wrtte as: Mamze Z f (,,,, Subect to g (,,,, b ( b g (,,,, b ( b ( g, m(,,, b m ( b m,, Ay vector satsfyg the costrats & o-egatvty restrctos wll be called a feasble soluto for the problem Geometrcally, each of the - o egatvty restrctos,,, defes a half- space of o-egatve values & the tersecto of all such half-spaces s the oegatve orthat, a subset of Euclda - space I E the o-egatve orthat s the o-egatve frst quadrat Each of the - equalty costrats ad t was troduced by [] g b ( b,,,m ( ( - also defes a set of pots Euclda - space ad the tersecto of these m-sets wth o-egatve orthat s called as opportuty set e E : g( b ; ( X { } - So, geometrcally a o-lear programmg problem s that of fdg a pot or a set of pots the opportuty set at whch the hghest cotour of the obectve fucto s attaed We kow that the optmum soluto of lear programmg problem does occur at a etreme pot of the cove opportuty set However case of o-lear programmg problem the soluto ca est at the boudary or the teror of the opportuty set (Weerstrass theorem : If the o-empty set X s compact (e closed ad bouded ad obectve fucto ( s cotuous o X the ( has a global mamum ether X the teror or o the boudary of X [5] Kuh-Tucker Theory The mpetus for geeralzatos of ths theory s cotaed the materal preseted where - we observed that uder certa codtos, a pot at whch f( takes o a relatve mamum for pots satsfyg g b,,,, m s a saddle pot (, of the Lagraga fucto (,The materal preseted was orgally developed By HW Kuh ad AWTucker [5] The theorem has bee of fudametal mportace developg a umercal procedure for solvg quadratc programmg problems A fucto (,, -beg a -compoet ad a m-compoet vector, s sad to have a saddle pot at [, ] f (, (, (, (5 holds for all a -eghborhood of If (5 holds for all ad, the (, s sad have a saddle pot the large o a global saddle pot at (, We shall fd t coveet to specalze the defto of a saddle pot to cases where certa compoets of ad are restrcted to be o-egatve, others are to be o-postve, ad a thrd category s to be urestrcted sg (,,,, s (,, s +,, t ( 6 (,, t +,,,,, s;, s +,, t; urestrcted, t +, ; ( 7 (,,,, ; ( (,,,, u (,, u +,, v ( 9 (,, v +,, m,,, u;, u +,, v + urestrcted, v,, m ( (,,,, m (

3 DAS: A COMPARATIVE STUDY O THE METHODS O SOLVING NON-LINEAR PROGRAMMING PROBLEM Equatos (6 through ( represet a set of ecessary codtos whch [, ] must satsfy f (, has a saddle pot at,, W, provded that [ ] for [ ] / c Now the suffcet codtos ca be stated as follows Let [, ] be a pot satsfyg (6 through ( The f there ests a - eghborhood about [, ] such that for pots [, ] W, ths eghborhood, +, - (, ( ( ( ( (, (, + (, ( - ( the (, has a saddle pot at [, ] for [, ] W If ( ad ( hold for all W,, W t follows that has a global saddle pot at [, ] for [, ] W rom the above we ca deduce Kuh- Tucker s codtos as follows: Let us cosder the o-lear programmg problem as: Mamze Z f ( subect to g( b To obta Kuh-Tucker ecessary codtos, let us covert the equalty costras of the above problem to equalty costrats by addg a vector of m slack (surplus varables: Ma Z f ( subect to g( + s b T or s b g where s s, s,, sm Now the Lagraga fucto for the problem takes the form, L(, f ( + { b g( } / Assumed that f, g c, frst order ecessary codtos ca be obtaed by takg frst order dervatves wth respect to, of the Lagraga fucto f g f g ad b g ( ( b g ( Rewrtg the Lagraga form we get ( ( (, f ( + ( b g (,,, m Now f ( f takes costrats local optmum at t s ecessary that a vector ests such that, f g ( ( ( ( T [ (, ] [ ( f ( g ( ] (, b g ( f g ( b b g ( f g ( b b g ( f g ( b [ (, ] [ b g ( g ] Comparso of solutos by Kuh- Tucker codto ad others I ths part we wat to show that varous NLP problems ca be solved by dfferet methods Our am s to show the affectve ess of the methods cosdered: Eample: Let us cosder the problem Mamze Z + Subect to the costrats: +, rst we wat to solve above problem by graphcal soluto method The gve problem ca be rewrtg as: Mamze Z ( + + Subect to the costrats +, We observe that our obectve fucto s a parabola wth verte at (, -/ ad costrats are lear To solve the problem graphcally, frst we costract the graph of the costrat the frst quadrat sce ad by cosderg the equato to equato + ad t was troduced by [6] Each pot has co-ordates of the tupe (, ad coversely every ordered par (, or real umbers determes a pot the plae Thus our search for the umber par

4 DAODIL INTERNATIONAL UNIVERSITY JOURNAL O SCIENCE AND TECHNOLOGY, VOLUME, ISSUE, JANUARY 9 (, s restrcted to the pots of the frst quadrat oly 5 5 B 7-6 Optmum g g g Optmum soluto by graphcal method We get the cove rego OAB as opportuty set Sce our search s for such a par (, whch gves a mamum value of + ad les the cove rego The desre pot wll be that pot of the rego at whch a sde of the cove rego s taget to the parabola or ths proceed as follows: Dfferetatg the equato of the parabola, we get d + d d d ( 5 d Now dfferetatg the equato of the costrat, we get d d + d (6 d Solvg equato (5 ad (6, we get 5 ad Ths shows that the parabola z ( + + has a taget to t the le + at the 5 pot, Hece we get the mamum value of the obectve fucto at ths pot Therefore, Z ma at, 6 Let us solve the above problem by usg [7] Kuh-Tucker Codtos The Lagraga fucto of the gve problem s A (,, + + ( By Kuh-Tucker codtos, we obta ( a ( b, ( c + ( + ( ( d ( wth Now there arse the followg cases: Case ( : Let, ths case we get from ad whch s mpossble ad ths soluto s to be dscarded ad t was troduced by [] Case (: Let I ths case we get from ( + (7 Also from + ad If we take, the If we cosder the Now puttg the value of (, we get 5 5 (,,,, for ths soluto satsfed / satsfed / / satsfed / + + satsfed 5 satsfed Thus all the Kuh-Tucker ecessary codtos are satsfed at the pot 5,

5 DAS: A COMPARATIVE STUDY O THE METHODS O SOLVING NON-LINEAR PROGRAMMING PROBLEM Hece the optmum (mamum soluto to the gve NLP problem s Z + ma at, 6 Let us solve the problem by Beale s method Mamze f ( + Subect to the costrats: +, Itroducg a slack varables s, the costrat becomes + + s, sce there s oly oe costrat, let s be a basc varable Thus we have by [9] ( s, (, B NB wth s Epressg the basc B ad the obectve fucto terms of o-basc NB, we have s - - ad f + - We evaluated the partal dervatves of f wrto o-basc varables at NB, we get f ( NB NB f NB sce both the partal dervatves are postve, the curret soluto ca be mproved As f gves the most postve value, wll eter the bass Now, to determe the leavg basc varable, we compute the ratos: ho ko m, m, hk kk m, sce the mmum occurs for, s wll leave the bass ad t was troduced by [] Thus epressg the ew basc varable, as well as the obectve fucto f terms of the ew o-basc varables ( ad s we have: s s ad f s we, aga, evaluate the partal dervates of f w r to the o-basc varables: f f s NB NB sce the partal dervatves are ot all egatve, the curret soluto s ot optmal, clearly, wll eter the bass or the et Crtero, we compute the ratos m, /, / sce the mmum of these ratos correspod to, o-basc varables ca be removed Thus we troduce a free varable, u (urestrcted, as a addtoal o-basc varable, defed by f u Note that ow the bass has two basc varables ad (ust etered That s, we have NB ( s, u ad B (, Epressg the basc B terms of o-basc NB, we have, u 5 ad ( + u s The obectve fucto, epressg terms of NB s, 5 f u + + u s u 97 s u 6 Now, s NB u δf ; u δu NB u f f sce for all NB ad, u u

6 DAODIL INTERNATIONAL UNIVERSITY JOURNAL O SCIENCE AND TECHNOLOGY, VOLUME, ISSUE, JANUARY 9 the curret soluto s optmal Hece the optmal basc feasble soluto to the gve problem s: 5 97,, Z 6 Let us solve the gve problem by [] usg Wolfe s algorthm Ma Z + subect to the costrats +, rst we covert the equalty costrats to equalty by troducg slack varables s + + s ( Therefore, the Lagraga fucto of the problem s L(, + + ( - T Here D, A (, A, C, b ( We kow by Wolfe s algorthm: T DX A + V C v + v + v + v + v ( 9 v ( Itroducg two artfcal varables a ad a equatos ( 5 ad ( 6, we get + v + a ( v + a ( The quadratc programmg problem s equvalet to M Z a + a e Ma( Z a a subect to the costrats + + s + v + a v + a,,, v, v, s, a, a wth v, v, s The soluto of the problem ca be show the followg modfed smple tableau: It was Itroduced by [] Here we get three costrats equatos wth eght ukow So, for basc soluto of the system always three varables wll take o-zero values ad others are zero We wll fd out o zero values for, & or s I our case tal basc soluto s a, a, s z Our et goal s to mprove the value of Z I tableau, we put costrats system as: Table Costrats system v v s a a c - - Now takg as leadg varables The we get m (/, / s / So, secod elemet of the st colum s the pvotal elemet ad the correspodg colum s the pvotal colum So, eters bass Reducg the pvotal elemet to uty by dvdg all elemets of the pvotal row by, we get the table Table Reducg frst pvotal elemet to uty v v s a a c / -/ - / Reducg zero all elemets of the pvotal colum ecept pvotal oe, we get the table Table Reducg zero all elemets of st pvotal colum ecept pvotal oe v v s a a c -/ / / -/ - -/ / Now takg as our et leadg varable The we get st elemet of the secod colum s our et pvotal elemet Reducg t to uty by dvdg all elemets of the pvotal ow by ad et takg as our et leadg varable The we get m s So, s our et pvotal, / elemet Reducg the pvotal elemet to uty, we get, the tableau-5 Table Reducg rd pvotal elemet to uty v v s a a c -/ / / -/ -/ / -/ / / / /

DAS: A COMPARATIVE STUDY O THE METHODS O SOLVING NON-LINEAR PROGRAMMING PROBLEM Makg zero all elemets of the pvotal colum ecept pvotal oe, we get the table 5 Table 5 Optmal soluto v v s a a c rom T-5

7 DAS: A COMPARATIVE STUDY O THE METHODS O SOLVING NON-LINEAR PROGRAMMING PROBLEM Makg zero all elemets of the pvotal colum ecept pvotal oe, we get the table 5 Table 5 Optmal soluto v v s a a c rom T-5 we obta the optmal soluto as 5,, Thus for the optmal soluto for the gve QP problem s Ma Z / -/ 5 + at -/ / -/ 5 (,, whch s same as we obta by Kuh-Tuker codto method or all kds of o-lear programmg problem, we ca show that the optmal soluto by Kuh-tucker codto s same as ay other method we cosdered Therefore the soluto obtaed by graphcal soluto method, Kuh-Tucker codtos ad Wolfe s method are same 5 Cocluso To obta a optmal soluto to the olear programmg problem, we observe that Kuh-Tucker codtos are more useful tha ay other methods of solvg NLP problem Because a NLP problem partcular problems are solve by partcular method There s o uversalsm the methods of solvg NLP problem But we have show that by Kuh-Tucker codtos all kds of NLP problems ca be solved The NLP problem volvg two varables ca easly solve by graphcal soluto method, but the / -/ / -/ / 5/ / / problem volvg more varables caot solve by graphcal soluto method Besdes, for all NLP problems, graphcal soluto method do ot gves always-optmal soluto Oly quadratc programmg problems ca solve by Wolfe s method Therefore, from the above dscusso, we ca say that kuh-tucker codtos are the best method for solvg oly NLP problem Refereces [] Greg, D M: Optmzato Lougma- Group Uted, New York (9 [] Gupta, P K Ma Moha: Lear programmg ad theory of Games Sulta Chad & sos, New Delh, Ida [] Hohedalke, B Vo: Smplcal Decomposto No-lear Programmg procedure [] G Hadley: No-lear ad dyamc programmg [5] M S Bazaraa & C M Shetty: No-lear programmg theory ad algorthms [6] Mttal Seth: Lear Programmg Pragat Prakasha, Meerut Ida 997 [7] Adade J: O the Kuh-Tucker Theory, No- Lear Programmg, J Adade (Ed 967b [] Hldreth C: A Quadratc Programmg procedure Naval Research Logstcs Quarterly [9] D MHmmeldlau: Appled No-lear programmg [] G R Walsh: Methods of optmzato Jo Wley ad sos ltd, 975, Rev 95 Bmal Chadra Das has completed M Sc pure Mathematcs ad B Sc (Hos Mathematcs from Chttagog Uversty Now he s workg as a Lecturer uder the Departmet of Tetle Egeerg Daffodl Iteratoal Uversty Hs area of research s Operato Research

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