A modified quadratic hybridization of Polak-Ribière-Polyak and Fletcher-Reeves conjugate gradient method for unconstrained optimization problems

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1 An International Journal of Optimization Control: Theories & Applications ISSN: eissn: Vol.7, No.2, pp (207) RESEARCH ARTICLE A modified quadratic hybridization of Pola-Ribière-Polya Fletcher-Reeves conjugate gradient method for unconstrained optimization problems Sindhu Narayanan, P. Kaelo * M.V. Thuto Department of Mathematics, University of Botswana, Private Bag UB00704, Gaborone, Botswana sindhu.op@gmail.com, aelop@mopipi.ub.bw, thutomv@mopipi.ub.bw ARTICLE INFO Article History: Received 5 December 206 Accepted 23 February 207 Available 5 July 207 Keywords: Hybridization Conjugate gradient Wolfe line search conditions Global convergence AMS Classification 200: 90C06, 90C30, 65K05 ABSTRACT This article presents a modified quadratic hybridization of the Pola Ribière Polya Fletcher Reeves conjugate gradient method for solving unconstrained optimization problems. Global convergence, with the strong Wolfe line search conditions, of the proposed quadratic hybrid conjugate gradient method is established. The new method is tested on a number of benchmar problems that have been extensively used in the literature numerical results show the competitiveness of the new hybrid method.. Introduction Nonlinear conjugate gradient method is a very powerful technique for solving large scale unconstrained optimization problems min{f(x) : x R n }, () where f : R n R is a continuously differentiable function. It has advantages over Newton quasi-newton methods in that it only needs the first order derivative hence less storage capacity is needed. It is also relatively simple to program. Given an initial guess x 0 R n, the nonlinear conjugategradientmethodgeneratesasequence{x } for problem () as x + = x +α d, = 0,,2,..., (2) where α is a step length which is determined by a line search d is a descent direction of f at x generated as { g, if = 0, d = (3) g +β d, if, *Corresponding Author 77 where g = f(x ) is the gradient of f at x β is a parameter. Conjugate gradient methods differ in their way of definingtheparameterβ. Overtheyears, several choices of β, which give rise to different conjugate gradient methods, have been proposed. The most famous formulas for β are Fletcher-Reeves (FR) method [20] β FR = g 2 g 2, Pola-Ribiére-Polya (PRP) method [32, 33] β PRP = gt y g 2, Dai-Yuan (DY) method [2] β DY = g 2 d T y, the Hestenes-Stiefel (HS) method [8, 23]

2 78 S. Narayanan, P. Kaelo, M.V. Thuto / IJOCTA, Vol.7, No.2, pp (207) β HS = gt y d T y, where y = g g denotes the Euclidean norm of vectors. These were the first scalars β for nonlinear conjugate gradient methods to be proposed. Since then, other parameters β have been proposed in the literature (see for example [,2,4 6,4,5,7,9,22,28,35,39,40] references therein). From the literature, it is well nown that FR DY methods have strong convergence properties. However, they may not perform well in practice. Ontheotherside, PRPHSmethods are nown to perform better numerically but may not converge in general. Given this, researchers try to devise some new methods, which have the advantages of these two inds of methods. This has been done mostly by combining two or more β parameters in the same conjugate gradient method to come up with hybrid methods. Thus, hybrids try to combine attractive features of different algorithms. For example, Touati-Ahmed Storey [36] proposed this hybrid method β TS = max { 0,min(β FR,β PRP ) } to tae advantage of the attractive convergence properties of β FR numerical performance of β PRP. Many other hybrids have been proposed by parametrically combining different parameters β. In Dai Yuan [], for instance, a one-parameter family of conjugate gradient methods is proposed as g 2 β = λ g 2 +( λ )d T y, where the parameter λ is such that λ [0,]. Liu Li [28] proposes a convex combination of β LS β DY to get where β LS β = ( γ )β LS +γ β DY, = gt y d T g is the Liu-Storey (LS) [26] parameter γ [0,]. Other hybrid conjugate gradient methods can be found in [2,4 8,3,2,22,24,25,27,29,35,38,4]. The step length α is often chosen to satisfy certain line search conditions. It is very important in the convergence analysis implementation of conjugate gradient methods. The line search in the conjugate gradient methods is often based on the wea Wolfe conditions f(x +α d ) f(x )+µα g T d (4) g(x +α d ) T d σg T d, (5) or the stronger version of the Wolfe line search conditions f(x +α d ) f(x )+µα g T d (6) g(x +α d ) T d σg T d, (7) where 0 < µ < σ <. More information on these line search methods other line search methods can be found in the literature [9,4,25,3, 34,37,39,4]. In this paper, we suggest another approach to get a new hybrid nonlinear conjugate gradient method. The rest of the paper is organised as follows. In section 2, we present the proposed method. In Section 3 we prove that the proposed algorithm (method) globally converges. Section 4 presents some numerical experiments conclusion is given in Section A new hybrid conjugate gradient method We now present our proposed hybrid conjugate gradient method. The hybrid method we propose is motivated by the wor of Babaie-Kafai [4,5] Mo, Gu Wei [29]. Babaie-Kafai [4,5] suggested a quadratic hybridization of β FR β PRP where method of the form β HQ± = β + (θ ) = ( θ 2 )βprp β + (θ± ), θ± [,], β PRP+, θ ± C, β FR, θ ± <, β FR, θ ± >, +θ β FR, (8) θ [,],

3 A modified quadratic hybridization of PRP FR conjugate gradient method θ ± = βfr ± (β FR ) 2 4β PRP (β HS 2β PRP is the solution of the quadratic equation θ 2 βprp θ β FR +β HS β PRP ) β PRP = 0. Thus, the author suggested two methods β HQ+ β HQ. The parameter β PRP+ = max{0,β PRP } is a hybrid parameter that was suggested by Gilbert Nocedal [2] to improve on the convergence properties of β PRP. In Mo, Gu Wei [29], the authors suggest a β defined by β = βprp + 2gT g g 2, (9) which then modifies the Touati-Ahmed Storey method [36] to give β = max { 0,min(β FR,β PRP,β )}. This method by Mo et al. [29] was shown to be very competitive with the other hybrids in the literature it was shown to perform much better than the original β PRP. Now, motivated by this suggestion (9) from [29] the wor of Babaie-Kafai [4,5], in this wor we modify Babaie-Kafai s method by introducing β S as where β + (θ ), θ [,], max{0,β β S = }, θ C, β FR, θ <, β FR, θ >, θ = βfr (β FR ) 2 4β (βhs β ) 2β (0) β + (θ ) = ( θ 2 )(max{0,β })+θ β FR, θ [,], () where β is as defined in (9), then define g, = 0, d = (2) g +β S d,. This leads to our hybrid conjugate gradient method presented below. Algorithm. New Hybrid β S Conjugate Gradient Method Step Give initial guess x 0 R n, the parameters 0 < µ < σ < ǫ > 0. Step 2 Set d 0 = g 0 = 0. If g 0 < ǫ, stop. Step 3 Compute α using the strong Wolfe conditions (6) (7). Step 4 Set x + = x +α d, = +. Step 5 If g < ǫ, stop. Step 6 Compute β using (0 ). Step 7 Compute d = g + β S d, go to Step Global convergence of the proposed method The global convergence analysis in this section follows that of Babaie-Kafai [4, 5]. To analyze the global convergence property of our hybrid method, the following assumptions are required. These assumptions have been used extensively in the literature for the global convergence analysis of conjugate gradient methods. Assumption. Let the level set Ω = {x R n : f(x) f(x 0 )}, where x 0 is the initial guess, be bounded. That is, there exists a positive constant B such that x B, x Ω. (3) Assumption 2. In some neighbourhood N of Ω, the function f is continuously differentiable its gradient, g(x) = f(x), is Lipschitz continuous, that is, there exists a constant L > 0 such that g(x) g(y) L x y for all x,y N. These assumptions imply that there exists a positive constant ˆγ such that g(x) ˆγ. (4)

4 80 S. Narayanan, P. Kaelo, M.V. Thuto / IJOCTA, Vol.7, No.2, pp (207) Also, under Assumptions 2, the following lemma can be established. Lemma (Zoutendij lemma). Consider any iteration of the form x + = x + α d, where d is a descent direction α satisfies the wea Wolfe conditions (4) (5). Suppose Assumptions 2 hold, then cos 2 θ g 2 <. =0 It follows from Lemma the sufficient descent condition with the Wolfe line search that =0 g 4 <. (5) d 2 Lemma 2. Suppose that Assumptions 2 hold. Consider any conjugate gradient method in the form of x + = x +α d, = 0,,2,, (2) in which, for all 0, the search direction d is a descent direction the step length α is determined to satisfy the Wolfe conditions. If =0 =, (6) d 2 then the method converges in the sense that lim inf g = 0. (7) Lemma 3. Suppose that Assumptions 2 hold. Consider any conjugate gradient method in the form of x + = x +α d, = 0,,2,, (2), with the conjugate gradient parameter β + (θ ) defined by (), in which the step length α is determined to satisfy the strong Wolfe conditions (6) (7). Also assume that the descent condition d T g < 0, 0 (8) holds there exists a positive constant ξ such that θ ξα, 0. (9) then d 0 g γ, 0, (20) u + u 2 <, (2) =0 where u = d d. Proof. Firstly, note that the descent condition (8) guarantees that d 0. So, u is welldefined. Moreover, from (20) Lemma 2, we have =0 <, (22) d 2 since otherwise (7) holds contradicting (20). Now, we divide β + (θ ) into two parts as β () = ( θ 2 )max(0,β ), β(2) = θ β FR,, for all 0, we define r + = v + d +, δ + = β () d d +, where v + = g + +β (2) d. Therefore, from (2) we obtain that u + = r + +δ + u. (23) Since u = u + =, from (23) we can write r + = u + δ + u = δ + u + u. (24) Because θ [,], we have δ + 0. Using the condition δ + 0, the triangle inequality (24), we get u + u (+δ + )u + (+δ + )u u + δ + u + δ + u + u = 2 r +. (25) If, for a positive constant γ, we have Also, from (3), (4), (9) (20) we have

5 A modified quadratic hybridization of PRP FR conjugate gradient method... 8 v + = g + +β (2) d = g + +θ g + 2 g 2 d g + + θ g + 2 g 2 d ˆγ + ξα ˆ d = ˆγ + ξˆγ2 x + x ˆγ + ξˆγ2 ( x + + x ) ˆγ + 2Bξˆγ2. (26) Theorem. Suppose that Assumptions 2 hold. Consider any conjugate gradient method in the form of x + = x +α d, = 0,,2,, (2), with the conjugate gradient parameter β + (θ ) defined by (), in which the step length α is determined to satisfy the strong Wolfe conditions (6) (7). If the search directions satisfy the descent condition (8) there exists a positive constant η such that θ η α d, 0, (30) Now, from (22), (25), (26) we have u + u 2 4 r + 2 =0 = 4 =0 =0 v + 2 d + 2 ( ) 2 4 ˆγ + 2Bˆγ2 ξ =0 ( ) 2 4 ˆγ + 2Bˆγ2 ξ γ 4 d + 2 =0 g 4 d + 2 then the method converges in the sense that lim inf g = 0. Proof. Because of the descent condition strong Wolfe conditions, we have proven that the sequence {x } 0 is a subset of the level set Ω. Also, sincealltheassumptionsof Lemma 2hold, the inequality (2) holds. Now, to prove the convergence, it is enough to show that the method has property (*). Since θ [,], from (), (4), (20) we have < (27) β + (θ ) = ( θ ) 2 (max{0,β })+θ β FR ( θ ) 2 βprp + 2gT + g g 2 + θ β FR We now define the following property, called property (*). Definition. [0] Consider any conjugate gradient method in the form of x + = x +α d, = 0,,2,, (2). Suppose that for a positive constant γ the inequality (20) holds. Under this assumption, we say that the method has property (*) if only if there exist constants b > λ > 0 such that for all 0, β b, (28) α d λ β b. (29) β PRP + 2gT + g +β FR g 2 g + ( g + + g ) g 2 2ˆγ2 + 2ˆγ2 + ˆγ2 = 5ˆγ g + g + g + 2 g 2 g 2 (3) Moreover, from Assumption 2 equations (), (4), (20), (30) we get β + (θ ) = ( θ ) 2 (max{0,β })+θ β FR ( θ ) 2 βprp + 2gT + g g 2 + θ β FR β PRP + 2gT + g + θ g 2 β FR g + g + g g g + g + θ g 2 g + 2 g 2

6 82 S. Narayanan, P. Kaelo, M.V. Thuto / IJOCTA, Vol.7, No.2, pp (207) Lˆγ x + x + 2ˆγ2 + ηˆγ2 α d Lˆγ α d + 2ηˆγ2 α d + ηˆγ2 α d = Lˆγ+3ηˆγ2 α d. So, from (3) (32), if we let b = 5ˆγ2 λ = b(lˆγ +3ηˆ ), (32) then (28) (29) hold consequently, the method has property (*). 4. Numerical Experiments We now present numerical experiments obtained byourmethodonsometestproblemschosenfrom Morè, et al. [30] Andrei [3] to analyse its efficiency effectiveness. A number of these test problems are widely used in the literature for testing unconstrained optimization methods. We present these test problems in Table, where the columns Prob Dim, respectively, represent the name dimension of the test problem, the dimensions of the problems range from 2 to We compare our proposed new hybrid conjugate gradient method (β S ) with the quadratic hybridization β HQ of Babaie-Kafai [4, 5] the method β by Mo, Gu Wei [29]. In [4], βhq was shown to be the better hybridization as comparedtoβ HQ+, hence our comparison will only focus on β HQ. For all the methods, we considered the stopping condition to be ǫ = 0 5, that is, the algorithms (methods) were stopped once the condition g < 0 5 was satisfied, or the maximum number of iterations of 5000 was reached. For the line search, the strong Wolfe conditions (6) (7) were used to find the step length α, with µ = σ = 0.6. All the methods were coded in MATLAB R205b numerical results are compared based on number of gradient evaluations, function evaluations CPU time. In Table, we present the number of functions evaluations(n F E) gradient evaluations (NGE) obtained for the methods β HQ, β S β, where the best results for each problem are indicated in bold. We observe from the table that, overall, the incorporation of β in the quadratic hybridization has a positive effect on β HQ, even though for some problems it is worse off. We also compare the methods using the performance profiles tool suggested by Dolan Moré [6] which, over the years, has been used extensively to judge the performance of different methods on a given set of test problems. The tool evaluates then compares the performance of the set of methods S on a set P of test problems. That is, using the ratio r p,s = t p,s min{t p,s : s S}, wheret p,s is(function, gradient, CPUtime)evaluations required to solve p by method s, the overall performance profile function is ρ s (τ) = n p size{p : p n p,log(r p,s ) τ}, where n p is the total number of problems in P τ 0. In case the method s fails to solve problem p, the ratio r p,s is set to some sufficiently large number. The function ρ s (τ) is then plotted against τ to give the performance profile. Notice that the function ρ s (τ) taes the values ρ s (τ) [0,] so the inequality ρ s (τ ) < ρ t (τ ) shows that the method t outperforms the method s at τ. We now present the plots of these performance profiles on function evaluations, gradient evaluations CPU time as figures. The function evaluations performance profile is presented in Figure, gradient evaluations in Figure 2 CPU time in Figure 3. It is clear from the figures that replacing β PRP by β in the quadratic hybridization β HQ has a positive effect. From the figures, we observe that β is the best method overall. As for β S βhq, we see that in Figures 2, for τ 0.5, β HQ is slightly better than β S. However, infigure3, theplotshowsthatinterms ofcputime, thereisnotmuchdifferencebetween β S βhq with the methods being much competitive. Overall the figures show the influence of β on the quadratic hybridization over the use of β PRP.

7 A modified quadratic hybridization of PRP FR conjugate gradient method Table. Results of test problems. Prob Dim β HQ β S β NFE NGE NFE NGE NFE NGE Rosenbroc Freud Roth Beale Helical valley Bard Gaussian Box Powell Singular Wood Biggs EXP Osborne Broyden tridiagonal Ext. TET Gen. White & Holst Ext. Penalty Ext. Maratos Gen. Rosenbroc Fletcher Ext. Rosenbroc Ext. Powell singular Raydan Ext. Beale Ext. Himmelblau Ext. DENSCHNB Ext. DENSCHNF Ext. Freud & Roth Ext. White & Holst Ext. Wood NONSCOMP Quartic β HQ- 0.9 β HQ β S β * β S β * ρ s (τ) 0.6 ρ s (τ) τ Figure. Function evaluations profile τ Figure 2. Gradient evaluations profile.

8 84 S. Narayanan, P. Kaelo, M.V. Thuto / IJOCTA, Vol.7, No.2, pp (207) ρ s (τ) τ 5. Conclusion Figure 3. CPU time profile. In this article, a modified quadratic hybridization of Pola Ribière Polya Fletcher Reeves conjugate gradient method (β S ) was presented. Its global convergence under the strong Wolfe line search conditions was also established. The β S method presented was tested on a number of unconstrained problems that have been extensively used in the literature compared to the original quadratic hybridization of Pola Ribière Polya Fletcher Reeves conjugate gradient method β HQ. The numerical results show that this proposed modification has a positive effect on the performance of β HQ. However, the numerical results from this study show that further research to improve the efficiency effectiveness of β HQ other conjugate gradient hybrids is still needed. A number of hybrid conjugate gradient methods have been proposed in the literature but there are many problems that are currently not properly hled by these methods, hence the need for more research in this field. Acnowledgments The authors than the anonymous Referees the Editors whose comments suggestions largely improved this wor. References [] Al-Baali, M., Narushima, Y. Yabe, H., A family of three term conjugate gradient methods with sufficient descent property for unconstrained optimization, Comput. Optim. Appl., 60, 89 0 (205). [2] Andrei, N., A new three-term conjugate gradient algorithm for unconstrained optimization, Numer. Algorithms, 68, (205). [3] Andrei, N., An unconstrained optimization test functions collection, Adv. Model. Optim., 0(), 47 6 (2008). β HQβ S β * [4] Babaie-Kafai, S., A quadratic hybridization of Pola- Ribiere-Polya Fletcher-Reeves conjugate gradient methods, J Optim. Theory Appl., 54(3), (202). [5] Babaie-Kafai, S., A Note on the global convergence of the quadratic hybridization of Pola-Ribiere-Polya Fletcher-Reeves conjugate gradient methods, J Optim. Theory Appl., 57(), (203). [6] Babaie-Kafai, S., Two modified scaled nonlinear conjugate gradient methods, J Comput. Appl. Math., 26, (204). [7] Babaie-Kafai, S., Fatemi, M., Mahdavi-Amiri, N., Two effective hybrid conjugate gradient algorithms based on modified BFGS updates, Numer. Algorithms, 58, (20). [8] Babaie-Kafai, S. Ghanbari, R., A descent extension of the Pola-Ribière-Polya conjugate gradient method, Comp. Math. Appl., 68, (204). [9] Dai, Y.-H., Conjugate gradient methods with Armijotype line searches, Acta Math. Appl. Sin. Engl. Ser., 8(), (2002). [0] Dai, Y.-H. Liao, L.Z., New conjugacy conditions related nonlinear conjugate gradient methods, Appl. Math. Optim., 43, 87 0 (200). [] Dai, Y.-H. Yuan, Y., A class of globally convergent conjugate gradient methods, Sci. China Ser. A., 46(2), (2003). [2] Dai, Y.-H. Yuan, Y., A nonlinear conjugate gradient method with strong global convergence property, SIAM J. Optim., 0(), (999). [3] Dai, Y.-H., Yuan, Y., An efficient hybrid conjugate gradient method for unconstrained optimization, Ann. Oper. Res., 03, (200). [4] Dai, Z. Wen, F., Another improved Wei-Yao-Liu nonlinear conjugate gradient method with sufficient descent property, Appl. Math. Comput., 28: (202). [5] Dai, Z.-F. Tian, B.-S., Global convergence of some modified PRP nonlinear conjugate gradient methods, Optim. Lett., 5, (20). [6] Dolan, E.D. Moré, J.J., Benchmaring optimization software with profile performance profiles, Math. Program., 9(2), (2002). [7] Dong, X.L., Liu, H. He, Y., A self-adjusting conjugate gradient method with sufficient descent condition conjugacy condition, J Optim. Theory Appl., 65, (205). [8] Dong, X.L., Liu, H.W., He, Y.B. Yang, X.M., A modified Hestenes-Stiefel conjugate gradient method with sufficient descent condition conjugacy condition, J Comput. Appl. Math., 28, (205). [9] Dong, Y., A practical PR+ conjugate gradient method only using gradient, Appl. Math. Comput., 29(4), (202). [20] Fletcher, R. Reeves, C., Function minimization by conjugate gradients, Comp. J., 7, (964). [2] Gilbert, J.C., Nocedal, J., Global convergence properties of conjugate gradient methods for optimization, SIAM J. Optim., 2(), 2 42 (992). [22] Hager, W.W., Zhang, H., A survey of nonlinear conjugate gradient methods, Pac. J. Optim., 2(), (2006). [23] Hestenes, M.R., Stiefel, E., Methods for conjugate gradients for solving linear systems, J Res. Natl. Bur. St., 49, (952).

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