Numerical Experiments with the Lagrange Multiplier and Conjugate Gradient Methods (ILMCGM)

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1 Aerca Joural of Appled Matheatcs 4; (6: -6 Publshed ole Jauary 5, 5 ( do: 648/aa465 ISSN: (Prt; ISSN: 33-6X (Ole Nuercal Eperets wth the Larae Multpler ad Couate Gradet Methods (ILMCGM Saso Adebayo Olorusola, etayo Eauel Olaosebka, Kayode Jaes Adebayo Departet of Matheatcal Sceces, Ekt State Uversty, Ado Ekt, Nera Eal address: fuladebayo@yahooco (S A Olorusola, eolaosebka@hotalco ( E Olaosebka, ak77@yahooco (K J Adebayo o cte ths artcle: Saso Adebayo Olorusola, etayo Eauel Olaosebka, Kayode Jaes Adebayo Nuercal Eperets wth the Larae Multpler ad Couate Gradet Methods (ILMCGM Aerca Joural of Appled Matheatcs Vol, No 6, 4, pp -6 do: 648/aa465 Abstract: I ths paper, we bed Larae Multpler Method (LMM Couate Gradet Method (CGM, whch eables Couate Gradet Method (CGM to be eployed for solv costraed optzato probles of ether equalty, equalty costrat or both I the past, Larae Multpler Method has bee used etesvely to solve costraed optzato probles However, wth soe specal features CGM whch akes t uque solv ucostraed optzato probles, we see that ths features we be advataeous to solve costraed optzato probles f we ca add or subtract oe or two ths to the CGM hs, the call for the Nuercal Eperets wth the Larae Multpler Couate Gradet Method (ILMCGM that s aed at tak care of ay costraed optzato probles, ether wth equalty or equalty costrat he authors of ths paper desre that, wth the costructo of the Alorth, oe wll crcuvet the dffcultes uderoe us oly LMM to solve costraed optzato probles ad ts applcato wll further prove the result of the Couate Gradet Method solv ths class of optzato proble We appled the ew alorth to soe costraed optzato probles of two, three ad four varables whch soe of the probles are perta to quadratc fuctos Soe of these fuctos are subect to lear, olear, equalty ad equalty costrats Keywords: Larae Multpler Method, Costraed Optzato Proble, Couate Gradet Method, Nuercal Eperets of the Larae Multpler Couate Gradet Method Itroducto he eeral optzato proble to be cosdered s of the for descrbed by [] ad [] as: Optze: Subect to: R, h ( f ( ( (,,, h = = ( (,,, = (3 where, a equalty vector equatos of deso, ad ( s a equalty vector of deso, such that the su of the costrats = ( + he fuctos f (, h ( ad ( are dfferetable fuctos Methods for solv ths odel have bee developed, tested ad successfully appled to ay portat probles of scetfc ad ecooc terest However, spte of the prolferato of the ethods, there s o uversal ethod for solv all optzato probles whch calls for applcato of ILMCGA to solve costraed optzato probles Couate Gradet Method I 95, Hestees ad Stefel developed a Couate Gradet Method (CGM alorth for solv alebrac equatos whch was successfully appled to olear equatos wth results reported by Fletcher ad Reeves 964 he CGM alorth for teratvely locat the u of f ( H s descrbed as follows: Step : Guess the frst eleet ϵ H ad copute the rea ebers of the sequece wth the ad of the forulae the steps throuh 6 Step : Copute the descet drecto

2 Saso Adebayo Olorusola et al: Nuercal Eperets wth the Larae Multpler ad Couate Gradet Methods (ILMCGM Step 3: Set Step 4: Copute Step 5: Set p = (4 + = + α p ; where =, H (5, H = + + αgp (6 p + = + + β p ; =, H (7, H Step 6: If = for soe, the, terate the sequece; else set = + ad o to step 3 I the teratve steps throuh 6 above, deotes the descet drecto at h step of the alorth,, s the step leth of the descet sequece ad deotes the radet of at Steps 3, 4 ad 5 of the alorth reveal the crucal role of the lear operator G deter the step leth of the descet sequece ad also eerat a couate drecto of search Doctoral hess of [3] threw lht o the theoretcal applcablty of the CGM, whch was eteded to optal cotrol probles by [4], [5] ad [6]Applcablty of the CGM alorth thus depeds solely o the eplct kowlede of the lear operator, G Geerally, for optzato probles, G s readly detered ad such eoys the beauty of the CGM as a coputatoal schee sce the CGM ehbts quadratc coverece ad requres oly a lttle ore coputato per terato Larae Multplers Method I atheatcal optzato, the ethod of Larae ultplers (aed after Joseph Lous Larae provdes a stratey for fd the au/u of a fucto subect to costrats he Larae ultpler ethod was bascally troduced to solve optzato probles wth equalty costrats of the for ( ad ( I solv ths, a ew varable, λ, called the Larae ultpler troduced to apped the costrat ( to the obectve fucto ( to ve a ew fucto: ( λ ( L X, = f X + λ h ( X (8 = the equvalet atr for of (4 s of the for: ( λ ( L X, = f X + λ h( X (9 (8 ad (9 are referred to as Laraa fuctos where λ s a vector of Larae ultplers I fd the u of the fucto f(, eerally we ca set the partal dervatves of (8 or (9 to zero such as: L ( X, λ =, =,,, ( L ( X, λ =, =,,, λ ( where X, λ ( ad ( are the u soluto ad the set of assocated Larae ultplers of (8 or (9 Also, ( ad ( are referred to as Kuh-ucker ecessary codtos for a local u of (8 or (9 whle the secod dervatves of the fucto f( ve as: L ( X, λ =, =,,, L ( X, λ =, =,,, λ ( (3 are referred to as suffcet codtos for a local u of (8 or (9 3 Larae Multplers Method Alorth I order to aze or ze the fucto ( subect to the costrat (, the follow procedures are take: Step : frst create the Larae Fucto ( λ ( L X, = f X + λ h ( X (4 = Step : Copute the partal dervatves wth respect to ad the Larae ultpler of the fucto (4 Step 3: Set each of the partal dervatves of (4 equal to zero to et: L = X L = λ (5 (6 us (4 proceed to solve for X ter of Now substtute the solutos for X so that (5 s ters of oly he, solve for λ ad use ths value to fd the optal values X Ibedded Larae Multpler Couate Gradet Method (ILMCGM Alorth Have vestated the two ethods; we ow draw out the follow steps whch wll be used to solve soe costraed optzato probles he steps are as follows: Step : Equate the costrat to zero ( case of equato s of the for: AX = b Step : Apped the ew equato step (e AX b = to the perforace de us Larae Multpler to for Laraa or Aueted Laraa fucto

3 Aerca Joural of Appled Matheatcs 4; (6: -6 3 L, = f + λ( AX b ] [e ( λ ( Step 3: Guess the tal eleets, λ > Step 4: Copute the tal radet,, as well as the tal descet drecto, p = Step 5: Copute the Hessa Matr, H, step Step 6: Set = + α p, whereα = +,,,, p Hp = Step 7: Update the radet us: = + α Hp, = +,,, Step 8: Update the descet drecto us: p p where = + β, β = + + +,,,, = Step 9: If = stop, else, set = + ad retur to step 6 NOE: f ( ad L(, λ are the perforace de ad Laraa fucto respectvely whch are dfferetable 3 Coputatoal Procedure of the ILMCGA Alorth Cosder ( ad (, there ests a Larae Multpler whch bed ( to ( to ve a Laraa fucto such as: ( λ ( L X, = f X + λ h ( X (3 Let the tal uess be: ( ( = ( λ( λ( λ = λ ( = (3 (33 Putt (3 ad (33 ( ad (3 respectvely ves the tal fuctos values e ( ad L(, λ Coput the radet of (3 wth respect to ( #,,, % & we have: L( X, λ = f ( X + λ h ( X = L( X, λ = f ( X + λ h ( X = L( X, λ = f ( X + λ h ( X = (34 Putt (3 ad (33 for ad respectvely (34 ves us the tal radet as: L(, λ L(, λ = L(, λ (35 Multply (35 by eatve ves the decet drecto as: p L(, λ L(, λ = = L(, λ Coput the Hessa Matr of (3 us (34 ves: ( λ ( λ (, λ L, L, L L(, λ L(, λ L(, λ H = L(, λ L(, λ L(, λ (36 (37

4 4 Saso Adebayo Olorusola et al: Nuercal Eperets wth the Larae Multpler ad Couate Gradet Methods (ILMCGM ad O traspos (35 ad (36 respectvely, we have: = L L L (, λ (, λ (, λ p = L L L (, λ (, λ (, λ Multply (35 ad (38 ves us a scalar, ' e k = (38 (39 L(, λ L(, λ k = L (, λ L (, λ L(, λ L(, λ k = L(, λ + L(, λ + + L (, λ (3 Slarly, ultply (39, (37 ad (36 ves a scalar, z e z = p Hp ( λ ( λ (, λ L, L, L L(, λ L(, λ L(, λ L(, λ L(, λ z = L(, λ L(, λ L(, λ L(, λ L(, λ L(, λ L(, λ Hp ( λ ( λ (, λ L, L, L L(, λ L(, λ L(, λ L(, λ (, L λ = L(, λ L(, λ L(, λ (, L λ ( λ ( λ ( L, L, L, λ L(, λ + L(, λ + + L(, λ L(, λ L(, λ L(, λ L(, λ + L(, λ + + L(, λ = L(, λ L(, λ L(, λ L(, λ + (, (, L λ + + L λ (3 (3 putt (3 to (3, we have:

5 Aerca Joural of Appled Matheatcs 4; (6: -6 5 ( λ ( λ L, L, L, λ L(, λ + L(, λ + + L(, λ L(, λ L(, λ L(, λ L(, λ + L(, λ + + L(, λ z = L(, λ L(, λ L(, λ L(, λ L(, λ L(, λ L(, λ + (, + + (, L λ L λ ( (33 Wth atr ultplcato, (33 becoes: + - -, - ( 3, , - ( 3, , ( 3,4 3 - ( = - -, - ( 3, : dvd (3 ad (34 e: -, ( 3, , ( 3, , ( 3, , 69 8 = -, , ( 3,4 3 - : -, ( 3,4 3 - :8-8, 69 ; (34 = 5 > >? ( 3,4 3 > >? ( 3,4 > >?8 ( 3, A > >? ( 3,4 3 > B(?3,C3 >? 5A > >? ( 3,4 3 6@ > B(?3,C3 5A > >? >? ( 3,4 3 > B(?3,C3 5A > >?8 >?8 ( 3, = A > >? ( 3,4 3 > B(?3,C3 5A > >? >? ( 3,4 3 6@ > B(?3,C3 >? 5A > >? ( 3,4 3 > B(?3,C3 5A > >?8 >?8 ( 3, A > >?8 ( 3,4 3 > B(?3,C3 5A > >? : >? ( 3,4 3 6@ > B(?3,C3 5A > >?: >? ( 3,4 3 > B(?3,C3 >?:8 5A > >?8 ( 3, ; (35 (35 s the step leth Now set = + α p, = +,,,, 4 Coputatoal Results he follow probles were evaluated us the ILMCGM alorth thus: Proble : Mzef ( = Subect to: = Proble : 3 4 Subect to : ( + ( 3 4 Proble 3: = 4 Mzef ( = Subect to: + 3 = Proble 4: ( Mzef = ( = + Mze f Subect to: + = able able of result for proble, at λ = No of Iteratos 3 4 Fucto values Gradet Nors

6 6 Saso Adebayo Olorusola et al: Nuercal Eperets wth the Larae Multpler ad Couate Gradet Methods (ILMCGM λ able able of result for proble, at =, = the slack varable λ θ = No of teratos Fucto values Gradet Nors able3 able of result for proble 3, at λ = No of Iteratos 3 Fucto values Gradet Nors able 4 able of result for proble 4, at λ = No of teratos Fucto values Gradet Nors Cocluso Coputatoally, the result alorth fro the Larae Multpler Method bedded Couate Gradet Method was tested o soe costraed optzato probles of two, three ad four varables he probles are pertaed to quadratc fuctos Soe of these fuctos are subect to lear ad olear costrats wth vary Larae paraeter,, betwee ad Whle the slack varable paraeter, θ, s Suppose we take the fucto value as the terat crtero, Proble ad 3 wth the uercal results ad whe copare wth the aalytcal results whch are: 5 ad respectvely, t varably establshes the relevace of the ew alorth for solv costraed optzato probles Proble ad 4 decreases ootocally establsh the coverece of the costraed Optzato Probles O us the Gradet Nor as the stopp crtero, the Gradet Nor of Probles,, 3 ad 4 teds to zero whch show the coverece of the probles O us the Gradet Nor as the stopp crtero, the Gradet Nor of Probles,, 3 ad 4 teds to zero whch show the coverece of the probles All these pots to the fact that, the costructed ILMCGM alorth effcetly solve the probles as supposed Refereces [] RAO, S S, (978, Optzato heory ad Applcatos, Wlly ad Sos [] HOMAS, FE, ad DAVID, MH, (, Optzato of Checal Processes, McGraw Hll [3] IGOR, G, SEPHEN, G N ad ARIELA, S, (9, Lear ad Nolear Optzato, Geore Maso Uversty, Farfa, Vra, SIAM, Phladelpha [4] Davd, G Hull, (3, Optal Cotrol heory for Applcatos, Mechacal Eeer Seres, Sprer-Verla, New York, Ic, 75 Ffth Aveue, New York, NY [5] Bersekas, D P, (98, Costraed Optzato ad Larae Multplers Method, Acadec Press, Ic [6] Rockafellar, R, (5, Multpler Method of Hestees ad Powell appled to cove Prora, Joural of Optzato heory ad Applcatos, Vol 4, No 4 [7] rphath S S ad Nareda K S, (97, Costraed Optzato Probles Us Multpler Methods, Joural of Optzato heory ad Applcatos: Vol 9, No