Minimization of Unconstrained Nonpolynomial Large-Scale Optimization Problems Using Conjugate Gradient Method Via Exact Line Search

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1 Amerca Joural of Mechacal ad Materals Eeer 207; (): do: 0648/jajmme Mmzato of Ucostraed Nopolyomal Lare-Scale Optmzato Problems Us Cojuate Gradet Method Va Exact Le Search Adam Ajmot, Oah Davd Owumu 2 Departmet of Mathematcs, Uversty of Ilor, Ilor, Nera 2 Departmet of Mathematcs, Federal Uversty Wuar, Wuar, Nera Emal address: adam4best@malcom (A Ajmot), oahdavd200@malcom (O D Owumu) o cte ths artcle: Adam Ajmot, Oah Davd Owumu Mmzato of Ucostraed Nopolyomal Lare-Scale Optmzato Problems Us Cojuate Gradet Method Va Exact Le Search Amerca Joural of Mechacal ad Materals Eeer Vol, No, 207, pp 0-4 do: 0648/jajmme Receved: February 28, 207; Accepted: March 22, 207; Publshed: Aprl 7, 207 Abstract: he olear cojuate radet method s a effectve teratve method for solv lare-scale optmzato problems us the teratve scheme x (+) x () + d () where: x (+) s the ew teratve pot, x () s the curret teratve pot, s the step-sze ad d () s the descet drecto I ths research wor, we employed the techque of exact le search to compute the step-sze the teratve scheme metoed above he le search techque ave ood results whe appled to some o-polyomal ucostraed optmzato problems Keywords: Iteratve Pot, No Polyomal, Ucostraed Optmzato, Cojuate Gradet Method, Descet Drecto, Exact Le Search, Iteratve Scheme Itroducto We wsh to cosder recet developmets lare-scale ucostraed optmzato We would le to state however that small-scale optmzato remas a actve area of research, ad that advaces ths feld ofte traslate to ew alorthms for lare-scale problems he problem uder cosderato s: m f( x), x R () where f s a smooth fucto of varables We assume that s lare, say > 500, ad we deote the radet of f by A mportat recet developmet has bee the appearace of effectve tools for automatcally comput the dervatves ad for detect partally separable structures hese prorams are already hav a sfcat mpact the practce of optmzato ad the des of alorthms, ad ther fluece s certa to row wth tme We devote much atteto to ths topc because uderstad the characterstcs of the objectve fucto s crucal lare-scale optmzato Lare-Scale Problems We cosder the problem of determ the value of a vector of decso varables x R that mmzes a objectve fucto f : R R, whe x s requred to belo to a feasble set S R, e we cosder the problem m x s f( x ) (2) I ths case we say the problem equato (2) s ucostraed f S R ad whe s a lare umber, the the problem equato (2) s called a lare-scale problem 2 Lare-Scale Ucostraed Optmzato Problems Lare-scale ucostraed optmzato problems do arse may sectors such as : () Eery (optmal plat operatos, reveue ad rs maaemet, ad stratec prc) () Face (opto prc, demad optmzato ad olear least-squares data ftt) () elecommucatos (des ad maaemet of

2 Amerca Joural of Mechacal ad Materals Eeer 207; (): 0-4 dstrbuto etwors) (v) Maret (quattatve maret) (v) Ecoomcs (computatoal ecoomcs) (v) Maufactur (CAD, optmal computer chp layout, ad trasportato etwor) (v) M, etc 3 Methods of Soluto Several alorthms have bee developed for soluto of lare-scale problems by dfferet type of methods he methods are Newto's method, quas-newto method, cojuate radet method, etc 3 Newto Method hs s the most powerful alorthm for solv lare-scale olear optmzato problems It ormally requres the fewest umber of fucto evaluatos, s very ood at llcodto, ad s capable of v the most accurate aswers It may ot always requre the least comput tme, ths depeds o the characterstcs of the problem ad the mplemetato of the Newto terato but t represets the most relable method for solv lare-scale problems Newto's method s well ow ad wdely used to fd zeroes of dfferetable fuctos Suffcetly close to a zero, t coveres very rapdly but the coverece s ot uaratee because t requres lare comput storae space 32 Quas-Newto Method here have bee varous attempts to exted the quas- Newto updat to the lare-scale case, ad two of them have proved to be very successful dfferet cotexts he frst dea cossts of explot the structure of partally separable fuctos by updat approxmatos to the Hessas of the terval fuctos hs ves rse to a powerful alorthm of wde applcablty, ad whose oly drawbac s the eed to fully specfy a partally separable represetato of the fucto he secod approach s that of lmted memory updat whch oly a few vectors are ept to represet the quas-newto approxmato to the Hessa hey are ot as robust ad as rapdly coveret as partally separable quas-newto methods, but are probably much more wdely used 33 Cojuate Gradet Method It has bee show that ay mmzato method that maes use of the cojuate drectos s quadratcally coveret hs property of quadratc coverece s very useful because t esures that the method wll mmze a quadratc fucto steps or less Sce ay eeral fucto ca be approxmated reasoably well by a quadratc ear the optmum pot, ay quadratcally coveret method s expected to fd the optmum pot a fte umber of teratos We chose ths partcular method for the mmzato of ucostraed o polyomal lare-scale optmzato problems So, we wll tae ths out to aother secto of ths presetato order to be able to descrbe more features about the method 2 Cojuate Gradet Method he cojuate radet method s a computatoal procedure for solv the ucostraed problem equato () Ad the method has the follow characterstcs: It solves quadratc problems of varables steps It requres relatvely small crease computer tme per terato ad requres very low memory space It has a well wored-out theory All descet methods requre the evaluato of the fucto ad the radet We ow ve the follow defto of the radet of a fucto he partal dervatves of a fucto f, wth respect to each of -varables are collectvely called the radet of the fucto ad deoted by f e, f( x) f ( x) f ( x) f( x),,, x x2 x he radet s a -compoet vectors ad t has a very mportat property that the radet vector represets the drecto of steepest ascet I respect of ths property, the eatve of the radet vector deotes the drecto of steepest descet 2 Cojuate Gradet Method for No-Quadratc Fuctos Whe the cojuate radet method s used for mmz o-quadratc fuctos, the method s ow as o-quadratc CGM he o-quadratc CGM eerates the () mmz sequece { x } start wth tal pot x (0) us the teratve formula (+ ) () () (3) x x + d, K 0,, 2, (4) () where K s the step-leth ad d s a search drecto determed by ad d I whch ( ) x + ( + ) ( ) ( + ) ( ) d, 0 +, ( ) + s the radet of f at ( ) ( + ) ( + ) ( ) x +, e, (5) f x (6) s a scalar ow as the parameter of the method he dfferet formula for dcate dfferet CGMs he follow are the famlar s avalable Hestees-Stefel [9]

3 2 Adam Ajmot ad Oah Davd Owumu: Mmzato of Ucostraed Nopolyomal Lare-Scale Optmzato Problems Us Cojuate Gradet Method Va Exact Le Search HS 2 Pola-Rbere-Polya [7] 3 Fletcher-Reeves [4] 4 Lu-Storey [0] Where: PRP FR LS ( + ) y ( ) d y ( + ) y ( ) ( ) ( + ) ( + ) ( ) ( ) ( + ) y d + ( ) ( ) ( + ) ( ) 22 Alorthm for No-Quadratc CGM y (7) he steps volved the mplemetato of a oquadratc CGM are as stated below: (0) Step : Gve x R ad > 0 (small), set 0 If ( ) <, the stop, otherwse o to Step 2 ( ) ( ) Step 2: Fd arm f( x + d ), > 0 Step 3: Update the varable x () ad () accord to the equato (4) ad (6) respectvely If ( ) < the stop, otherwse o to Step 4 Step 4: Determe for the selected method ad d () us the equato (5) Step 5: Set +, ad o to Step 2 3 Exact Le Search hs s the process of determ the value of the stepleth alo the drecto d () wth the oal of esur coverece wthout deterorat the rate of coverece I ths secto, we cosder the processes volved fd the step-leth Gve that f(x) s the objectve fucto to be mmzed us the CGM, the problem of fd arrows dow to fd the value whch mmze ( + ) ( ) ( ) ( ) ( ) f x f x + d f( ) (8) for fxed values of x () ad d () Sce f(x (+) ) becomes a fucto of oly oe varable, the method of fd volves oe-dmesoal mmzato techque I every le search, the am s to determe the value of the step-leth > 0 as stated above alo the drecto d () wth the am of esur co-verece wthout deterorat the rate of coverece he frst possblty of ths s to set wth ( ) ( ) a f x + d ar m ( ) (9) e, s the value of > 0 that mmzes the fucto ( ) f alo the drecto d obtaed by solv the equato the equato (9) ca be f ( ) ( ) f x + d ( ) 0 (0) he techque employed (0) yelds a exact value ad s referred to as a exact le search Exact le search ca be drectly appled to o-polyomal objectve fuctos o expaso us aylor's seres he the stepleth ca be obtaed from the equato (0) by fd the real roots whch satsfes the equato (9) hs research wor exames the computatoal process volved mplemet the exact le search for solv ucostraed optmzato problems volv o-polyomal objectve fuctos oly 4 Computatoal Detals 4 aylor's Seres aylor's seres expaso for a multvarable fucto f(x, x 2,, x ) s f( a) f ( a) f ( a) f( x) f( a) + ( x a ) + ( x a ) + ( x a ) 2 3 x 2! x 3! x 42 Alorthm for Exact Le Search Alorthm for exact le search as used ths wor are stated below as follows: Step : Gve a o-polyomal objectve fucto f(x), expad aylor's seres ad trucate the seres after a umber of terms Step 2: For the trucated f(x), substtute x wth x + d to et f() e, f() f(x + d) ad wrte as a polyomal of Step 3: Compute the frst-order dervatve of f(x + d) wth f( x) respect to ad equate to zero e, 0 Step 4: Solve for the real root such that > 0 43 Computatoal Examples he follow o-polyomal fuctos obtaed from Adre [] are used as computatoal examples 43 Rayda 2 Fucto f( x) [exp( x ) x ], x [,,,] (0)

4 Amerca Joural of Mechacal ad Materals Eeer 207; (): I trucated form for quartc: 432 Daoal 3 Fucto x x x f( x) f( x) [exp( x ) s( x )], x [,,,] I trucated form for quartc: 433 Cose Fucto (CUE) (0) x x x f( x) 2 (0) ( ) [cos( ) + )], [,,,] f x x x x rucated form for quartc: 434 Rayda Fucto 2 4 x x f( x) + + 2! 24! (0) f( x) [exp( x) x], x [,,,] 0 rucated form for quartc: 435 Daoal 6 Fucto x x x f ( x) 0 ( x) (0) ( ) [ ( ], [,,,] f x e x x rucated form for cubc: 44 Numercal Results 2 3 x x f( x) 2x + + 2! 3! he CGM Alorthm (22) was mplemeted us MALAB [R2009a] codes o a HP laptop 620 wth processor Petum(R) Dual-Core CPU GHz ad RAM 200GB to solve the computatoal examples above Ad the results obtaed are tabulated the tables below us the follow otatos: : (dmeso) IR: (umber of teratos) f : (Optmal value of objectve fucto f ) : (orm of the optmal radet ) Ext: (proram executo tme) CE: (computatoal examples) At: (averae executo tme per computato each method) able Numercal Result wth PR CE PR (Dm) Itr f Ext e003 27e e004 30e e003 42e e004 2e At CE PR (Dm) Itr f Ext e003 9e e004 7e e00 43e e00 60e CE N HS able 2 Numercal Result wth HS (Dm) Itr f Ext e003 27e e004 30e e003 42e e004 2e e003 9e e004 7e e00 43e e00 60e e003 3e e004 8e CE LS able 3 Numercal Result wth LS (Dm) Itr f Ext e003 27e e004 30e e003 42e e004 2e e003 9e e004 7e e00 43e e00 79e e003 3e e004 8e Remars O Computatoal Results I alorthm (22), the orm of s defed as, At 0057 At 0055 < < If all compoets of have the same value, the; < < he hhest value for used all the methods to solve all the problems s 0000 ad the tolerace used 0 6 Substtut for ad the equato above, we have 0 0 < Now f 0 ad 0 e , < hus, 0 0 from whch we deduce that 0

5 4 Adam Ajmot ad Oah Davd Owumu: Mmzato of Ucostraed Nopolyomal Lare-Scale Optmzato Problems Us Cojuate Gradet Method Va Exact Le Search whch s the requremet for exact coverece he mmzato of the problem us all the methods s 6 all satsfed us the tolerace 0 All the methods ave the same value for each of the problem, ths shows accuracy the methods used Also cosder the executo tme for the soluto of all the problems ables -3, we coclude that all the methods covere very fast ad there s cosstecy all the methods Fally, jud by averae executo tme per computato each method, the method LS has the fastest rate of coverece compare to the rest of the cojuate radet methods cosdered based o the problems solved 5 Cocluso Attempt to optmze some lare scale ucostraed fuctos that are o-polyomal ature usually pose a bt of challee hs usual challee optmz ths set of fuctos s what formed the choce of adopt the exact le search techque coupled wth the cojuate radet method (CGM) mmz the lare scale o-polyomal fuctos dscussed ths research wor Moreover, dur the optmzato process of the sad set of fuctos cosdered, the study reveals that adopt the le search techque ves ood results whe appled to some of the o-polyomal ucostraed optmzato problems cosdered ad thus could be recommeded as oe of the ood methods of hadl smlar optmzato stuatos Acowledemets he authors wat to apprecate the maaemet ad edtoral team of the Amerca Joural of Mechacal ad Materals Eeer for ther costructve crtcsms that led to the mprovemet of the study May God bless you all Refereces [] Adre, N (2008) Ucostraed optmzato text fuctos Upub-lshed mauscrpt Research Isttute of Iformatcs Bucharest, Romaa [2] Al, M Lecture o Nolear ucostraed optmzato School of Computato ad Appled Mathematcs, Uversty of Wtwatersad, Johaesbur, South Afrca [3] Bambola, O M, Al, M ad Nwaeze E (200) A effcet ad coveret cojuate radet method for ucostraed olear optmzato (submtted) [4] Fletcher, R ad Reeves, C M (964) Fucto mmzato by co-juate radet Computer Joural Vol 7, No 2, pp [5] Da, Y ad Yua, Y (2000) A olear cojuate radet wth a stro lobal coverece propertes: SIAM Joural o Optmzato Vol 0, pp [6] Fletcher, R (997) Practcal method of optmzato, secod edto Joh Wley, New Yor [7] Pola, E ad Rbere, G (969) Note sur la coverece de drectos cojuees Rev Fracase Iformat Recherche operatolle, 3e Aee 6, pp [8] Polya, B (969) he cojuate radet extreme problems USSR comp Math Math phys 94-2 [9] Hestees, M R ad Stefel, E (952) Method of cojuate radet for solv lear equatos J Res Nat Bur Stad, pp 49 [0] Lu, Y ad Storey, C (992) Effcet eeralzed cojuate radet alorthms Joural of Optmzato heory ad Applcatos Vol 69, pp [] Rao, S S (980) Optmzato theory ad applcatos, secod ed-to, Wley Easter Ltd, New Delh [2] Getr, G ad rod, S (2000) O lare-scale ucostraed optmzato problems ad hher order methods Uversty of Bere, Departmet of Iformatcs, Hh echoloy Cetre N-5020 Bere, Norway