Abbas Y. Al-Bayati - Basim A. Hassan - University of Mosul
Abbas Y. Al-Bayati - Basim A. Hassan - University of Mosul Abbas Y. Al-Bayati - Basim A. Hassan - University of Mosul
Abbas Y. Al-Bayati - Basim A. Hassan and Sawsan S. Ismaelour implementation differs in that we use the scaled quadratic model instead of thequadratic itself.CG-algorithm 1. and the generalized VM-algorithm 2. The objective here is toshow that, using Al-Bayati’s self-scaling VM-update (12), the sequence of thegenerating points is the same in the generalized CG-algorithm 1.Before making a few more observations we shall outline briefly the proposedstrategy for the interleaved generalized CG-VM method Al-Bayati [2].• Algorithm 3:Let f be a non linear scaling of the quadratic function f; given x 1 and a matrixH 1 = I ; set G = ; i = 1 and t = 1 initially, k is the iteration index.*1g 1*Step 1: Set dt = - H igt(13)Step 2: For k = t , t+1 , t+2 , … iterate withx = x + λ dk+ 1 k k k,Hy*Tβk 1 i kk= (g +T ,dkykd*k+ 1= - Higk+1+ βkdk,)T *ykgk+1βk= ,Td ykkw =TdigTd gi*ii+12ggwg- g*= *k + 1 k k+ 1 2 k1g(x λ d )k+= 1 2k−k k*y = gk+1- gk2Here i is the index of the matrix updated only at restart steps and k is the indexof iteration and the algorithm is not converged, until a restart is indicated.Step 3: If a restart is indicated, namely that the Powell [9] restarting criterion issatisfied, i.e.*T *Tk+ 1 k k+ 1 k+1|g g | ‡ 0.2 |g g | , (14)Then reset t to the current k , update H i by:T(H y v ) 2 a H yH Hi t tv [(tT) v - (i ti + 1=i−+2) ]bit.bbtb = vTwhere at= ytHiytand t t tStep 4: Replace i by i + 1 and repeat from (13).3. Hybrid Conjugate Gradient MethodTytDespite the numerical superiority of PR-method over FR-method the later hasbetter theoretical properties than the formal see Al-Baali [3]. Under certain conditionsFR-method can be shown to have global convergence with exact line search Powell [10]t16
The Extended CG Method for Non-Quadratic Modelsand also with inexact line search satisfying the strong Wolf-Powell condition. Thisanomaly leads to speculation on the best way to choose β i .Touati – Ahmed and Storey in (12) proposed the following hybrid method:2 i-1Step 1: if l ||g i+1|| £ (2m) , with 1 m2 > >h and λ > 0 go to Step 2. Otherwise, Setβ i = 0 .PRPRStep 2: If β < 0 , Set β = , otherwise go to Step 3.Step 3: IfiPRβiPRi = β i⎛(1/2m)|| g⎜⎝iβ i2≤i-12|| gi||||⎞⎟ , with m =h, set β = . Otherwise Set⎠PRiβ iβ .Here m, h and λ user supplied parameters. This hybrid was shown to beglobally convergence under both exact and inexact line searches and to be quitecompetitive with PR- and FR-methods.4. New hybrid algorithm (Algorithm 4):*Step1 : Set dt = - H igt, i = 1, t = 1, k is the index of iterations.Step 2: For k = t , t+1 , … iterate withxk+ 1= xk+ λkdk,andH*Tβk 1 i kk= (g +T ,dkyky)Where i is the index of the matrix updated only at restart steps.2 k 1Step 3: If l ||g i-1|| £ (2m) + with 1 m2 > >h go to Step 1. Otherwise set β i = 0.PRPRStep 4: If βi< 0 set β i = β i, otherwise go to Step 5.Step 5:IfPR 12 2PRFRBi£ ( m) ||gi-1|| /||gi||with m > η , Set B i= B i, otherwise set B i= B i.2***Step 6: Compute dk+ 1= - Higk+1+ βkdk, where g = k+ 1wkgk+ - g 1 2 kStep 7: If a restart is indicated, namely that the Powell, restarting criterion is satisfied,*T *Ti.e.: |g g | ‡ 0.2 |g g | , then reset t to the current k, update H i by:k-1 kk(H y v )H +k2 aTi t tti t Ti + 1= Hi−vt[( 2 ) vt- ( )]btbbttStep 8: Replace i by i+1 and repeat from (13).5. Conclusions and Numerical results:Several standard test functions were minimized (2 < n < 400) to compare theproposed algorithm with standard Sloboda algorithm which are coded in doubleprecision Fortran 90. The proposed hybrid algorithm needs matrix calculation for400×400, this is the approximately the latest range for this computation of the matrix.The numerical results are obtained on personal Pentium IV Computer. The compete setof results are given in tables 5.1 and 5.2.Hy17
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The Extended CG Method for Non-Quadratic Modelsand also with inexact line search satisfying the strong Wolf-Powell condition. Thisanomaly leads to speculation on the best way to choose β i .Touati – Ahmed and Storey in (12) proposed the following hybrid method:2 i-1Step 1: if l ||g i+1|| £ (2m) , with 1 m2 > >h and λ > 0 go to Step 2. Otherwise, Setβ i = 0 .PRPRStep 2: If β < 0 , Set β = , otherwise go to Step 3.Step 3: IfiPRβiPRi = β i⎛(1/2m)|| g⎜⎝iβ i2≤i-12|| gi||||⎞⎟ , with m =h, set β = . Otherwise Set⎠PRiβ iβ .Here m, h and λ user supplied parameters. This hybrid was shown to beglobally convergence under both exact and inexact line searches and to be quitecompetitive with PR- and FR-methods.4. New hybrid algorithm (<strong>Al</strong>gorithm 4):*Step1 : Set dt = - H igt, i = 1, t = 1, k is the index <strong>of</strong> iterations.Step 2: For k = t , t+1 , … iterate withxk+ 1= xk+ λkdk,andH*Tβk 1 i kk= (g +T ,dkyky)Where i is the index <strong>of</strong> the matrix updated only at restart steps.2 k 1Step 3: If l ||g i-1|| £ (2m) + with 1 m2 > >h go to Step 1. Otherwise set β i = 0.PRPRStep 4: If βi< 0 set β i = β i, otherwise go to Step 5.Step 5:IfPR 12 2PRFRBi£ ( m) ||gi-1|| /||gi||with m > η , Set B i= B i, otherwise set B i= B i.2***Step 6: Compute dk+ 1= - Higk+1+ βkdk, where g = k+ 1wkgk+ - g 1 2 kStep 7: If a restart is indicated, namely that the Powell, restarting criterion is satisfied,*T *Ti.e.: |g g | ‡ 0.2 |g g | , then reset t to the current k, update H i by:k-1 kk(H y v )H +k2 aTi t tti t Ti + 1= Hi−vt[( 2 ) vt- ( )]btbbttStep 8: Replace i by i+1 and repeat from (13).5. Conclusions and Numerical results:Several standard test functions were minimized (2 < n < 400) to compare theproposed algorithm with standard Sloboda algorithm which are coded in doubleprecision Fortran 90. The proposed hybrid algorithm needs matrix calculation for400×400, this is the approximately the latest range for this computation <strong>of</strong> the matrix.The numerical results are obtained on personal Pentium IV Computer. The compete set<strong>of</strong> results are given in tables 5.1 and 5.2.Hy17
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