A new conjugate gradient method with the new Armijo search based on a modified secant equations

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1 ISSN: Enineerin an echnoloy Vol 5 Iue 7 July 8 A new conjuate raient metho with the new Armijo earch bae on a moifie ecant equation Weijuan Shi Guohua Chen Zhibin Zhu Department of Mathematic & Applie Mathematic Hunan Univerity of Humanitie Science an echnoloy Loui 477 P RChina Department of Mathematic & Applie Mathematic Hunan Univerity of Humanitie Science an echnoloy Loui 477 P RChina School of Mathematic an Computin Science Guilin Univerity of Electronic echnoloy Guilin 544 P R China ABSRAC: It very effective for the conjuate raient metho to olve lare-cale minimization problem In thi paper bae on the moifie ecant equation we propoe a new conjuate raient metho with the moifie linear earch Uner ome proper conition the lobal converence of thi metho i etablihe Armijo type KEYWORDS: uncontraine optimization problem; conjuate raient metho; ecant equation; Armijo-type earch; lobal converence I INRODUCION It i well nown that the conjuate raient metho i an effective metho to olve lare-cale minimization problem ([ ]) he conjuate raient metho ha a wie rane of application in many omain lie control cience enineerin an operation reearch etc he iterative formula of the conjuate raient metho i iven a follow: x x () where enote the tep ize i efine by: () here are many formulae about ee[] for example ome famou formulae are efine a follow: FR PRP HS LS Recently [3] an [4] propoe a new ecant equation Aume that the objective function f i mooth ufficiently we can mae it aylor expanion at point x x Copyriht to IJARSE wwwijaretcom 643

2 ISSN: Enineerin an echnoloy Vol 5 Iue 7 July 8 f f G O 6 G O 4 4 where n 3 f x i j l i j l i j l x x x hi formula can be written a (ee[4]): G y where where (3) f f y 6 3 Y are anaano [5] coniere the followin extene ecant equation : B Z Z y u (4) u Bae on the formula (3) u n R i any vector which atifie u Generally peain bae on the claical Armijo linear earch technique uner ome proper conition many conjuate raient metho poe the ecent property an the lobal converence But the rawbac of the Armijo linear earch i how to chooe the initial tep ize If it i too lare then more function evaluation are neee if it i too mall then the efficiency of relevant alorithm will be ecreae In thi paper firtly we moify the ecant equation(4) an obtain a new ecant equation then preent a new conjuate raient metho an propoe a moifie Armijo linear earch technique which aim at the above rawbac of Armijo linear earch Uner ome appropriate conition the lobal converence i iven for the new conjuate raient metho with the moifie Armijo linear earch We propoe the followin moifie ecant equation B y II NEW CONJUGAE GRADIEN MEHOD () y y y y Bae on the above mentione ecant equation a new formula of i propoe: if if y y t wheret y i preente by () he forthcomin propoition i clearly nown in [34] () Copyriht to IJARSE wwwijaretcom 644

3 ISSN: Enineerin an echnoloy Propoition L 3 Vol 5 Iue 7 July 8 Definition A twice continuouly ifferentiable function f i uniformly convex on the nonempty open convex et if an only if there exit M uch that x y x y M x y x y S In orer to icu the effectivene of the conjuate raient metho () the followin baic aumption are iven n H he objective function f x i continuouly ifferentiable an ha a lower boun on R H he raient x fx of function n f x f y L x y x y B (3) H 3he level et Lx x f x f x the level et Lx x R f x f x f x i Lipchitz continuouly on the open convex et B with ( x i iven) that i there exit a contant L uch that ha a boun that i there exit a contant C uch that Accorin to our moifie ecant equation the followin propoition i obtaine clearly x C x L x (4) Propoition y 3L 4 L M Proof By Definition we have y y y M y Conierin () if the aumption H an H 3 hol an wehave y y y y (5) 3L 3L L y M 3L 4 L M Recently ome metho to obtain the Lipchitz contant L were propoe [] If let y the followin three etimatin formulae were obtaine L y (6) Copyriht to IJARSE wwwijaretcom 645

4 ISSN: Enineerin an echnoloy Vol 5 Iue 7 July 8 y L y L Copyriht to IJARSE wwwijaretcom 646 y In fact any calar which i reater than L can be reare a a Lipchitz contant however it i poible to caue the low converence rate So it i very important to fin the Lipchitz contant which i a mall a poible an i effective for practical computation In theth iteration we tae repectively the Lipchitz contant a: y L max L (9) y ' () L max L min M y y () L max L ' where L an M i a lare poitive number Correponin to the three Lipchitz contant we call the conjuate metho a A A A 3 repectively Given let l Now bae on [] we preent the followin moifie Armijo linear earch: μ ρ c ε c y 3L θ 4 L t M α i the laretα which belon to l l ρ l ρ atifyin: Definition an while f f x α αμ L i iven in 9 an wheret i mentione in repectively Bae on the moifie Armijo linear earch an the new formula of conjuate raient alorithm Alorithm: n Step Chooe x R Step if Step Let x x α et top otherwie o to Step where i followe by () (7) (8) M i efine in β we propoe the followin moifie βi efine by ()an a i efine by the

5 ISSN: Enineerin an echnoloy moifie Armijo type linear earch Step 3 Let : o bac to tep Vol 5 Iue 7 July 8 IIIGLOBAL CONVERGENCE OF HE ALGORIHM Lemma 3Suppoe that H an H hol an the new conjuate raient metho with the moifie Armijo type M uch that m L M linear earch enerate an infinite equence x then there exit the contant m an Lemma 3Suppoe that H an H hol the new conjuate raient metho with the new Armijo type α where c θ linear earch enerate an infinite equence y ε 3L 4 L t we have M c x Proof By the Cauchy Schwarz inequality we have c y a thereby we now that x then for y y y y t y t y o 3L 4L t M = Copyriht to IJARSE wwwijaretcom 647

6 ISSN: Enineerin an echnoloy c y Vol 5 Iue 7 July 8 y t y y y thu c that i c he proof i complete Lemma 33Suppoe that H an H hol the new conjuate raient metho with the new Armijo type where linear earch enerate an infinite equence x then m i efine in Lemma 3 Proof When For we have or c he proof i complete Lemma34Suppoe that H an H hol then the moifie Armijo type linear earch i wellefine c t y y y c Proof When c c y y c we have that Copyriht to IJARSE wwwijaretcom 648

7 When ISSN: Enineerin an echnoloy c y f f x Vol 5 Iue 7 July 8 for we have the followin inequality: Uin the Mean Value heorem on the left-han ie of the above inequality there exit a calar t uch that x t that i x t By the conition H accorin to the Cauchy - Schwarz inequality an Lemma 3 it hol that L x t x t c ie c L L c c c c Sothere exit min uch that the moifie Armijo L c he proof i complete type linear earch i well-efine heorem 3Suppoe that H an H hol the new conjuate raient metho with the new Armijo type x hen lim linear earch enerate an infinite equence Proof Let inf if then f f x c By the conition H we have lim By the contrary uppoe that K uch that o it hol that hen there exit an infinite ubet Copyriht to IJARSE wwwijaretcom 649

8 ISSN: Enineerin an echnoloy Vol 5 Iue 7 July 8 lim K x By Lemma 3 an Lemma 34 we now that y c c ' ' From 3 there exit uch that l Let it i obviou that f f x By the proof of Lemma 34 we have that L c hen by Lemma 33 it hol that c (3) a l K c ' c K L L Which contraict with 3 he proof i complete IVNUMERICAL EXPERIMENS In thi ection we carry out ome numerical experiment Our alorithm ha been tete on ome problem a follow where x i the initial point an x i the final point Example f x = x + x + x + x 7 Example f x = x + x 5 Example 3 f x = x + x + 4 x 4 x + + x x + We et the parameter δ = 5ρ = 5c = 75 an L = in the numerical experimenthe numerical reult are iven in able able: NO ep x x time() e e e e e e Copyriht to IJARSE wwwijaretcom 643

9 ISSN: Enineerin an echnoloy Vol 5 Iue 7 July 8 VCONCLUSION able how the performance of the alorithm about relative to the iteration It i eay to ee that for above problem the alorithm i efficient In particular when the preciion i not very trict reult for each problem are baically correct an with le number of time of iteration REFERENCES []Z J Shi J Shen Converence of Liu-Storey conjuate raientmethoeuropean Journalof Operational Reearch 8 (7) []B K Saman G Reza M A Nezam wo new conjuate raient metho bae on moifie ecant equation Journal of Computational an ApplieMathematic 34 () [3]X D Chen J Sun Global converence of a two-parameter family of conjuate raientmetho without line earch Journal of Computational an Applie Mathematic 46() [4]Y H Dai Y Yuan Converence propertie of the FletcherCReeve metho IMA Journalof Numerical Analyi 6 (996) [5]Y H Dai Y Yuan Converence propertie of the conjuate ecent metho Avancein Mathematic 5 (996) [6]Y H Dai Y Yuan A nonlinear conjuate raient metho with a tron lobal conver-ence property SIAM Journal of Optimization (999) 77-8 [7]R Fletcher Practical Metho of Optimization econeuncontraine Optimization voli Wiley New Yor 987 R Fletcher C Reeve Function minimization by conjuate raient Computer Journal7 (964) [8]J Sun J P Zhan Converence of conjuate raient metho without line earch Annalof Operation Reearch 3 () 6-73 [9]Sun X Q Yan X D Chen Quaratic cot flow an the conjuate raient methoeuropean Journal of Operational Reearch 64 (5) 4-4 []Z J Shi J Shen Converence of ecent metho without line earch Applie Mathemat-ic an Computation 67 (5) 94-7 []Z J Shi J Shen Step-ize etimation for uncontraine optimization metho Computational an Applie Mathematic 4 (5) []J Z Zhan N Y Den L H Chen New quai-newton equation an relate metho foruncontraine optimization Journal of Optimization heory an Application (999)47-67 [3]J Z Zhan C X Xu Propertie an numerical performance of quai-newton methowith moifie quai-newton equation Journal of Computational an Applie Mathemat-ic 37 () [4]H Yabe M aano Global converence propertie of nonlinear conjuate raient metho with moifie ecant conition Computational Optimization an Application 8(4) 3-5 Copyriht to IJARSE wwwijaretcom 643