Research Article | Open Access

Volume 2011 |Article ID 463087 | https://doi.org/10.1155/2011/463087

Liu Jin-kui, Zou Li-min, Song Xiao-qian, "Global Convergence of a Nonlinear Conjugate Gradient Method", Mathematical Problems in Engineering, vol. 2011, Article ID 463087, 22 pages, 2011. https://doi.org/10.1155/2011/463087

# Global Convergence of a Nonlinear Conjugate Gradient Method

Revised21 May 2011
Accepted04 Jun 2011
Published21 Jul 2011

#### Abstract

A modified PRP nonlinear conjugate gradient method to solve unconstrained optimization problems is proposed. The important property of the proposed method is that the sufficient descent property is guaranteed independent of any line search. By the use of the Wolfe line search, the global convergence of the proposed method is established for nonconvex minimization. Numerical results show that the proposed method is effective and promising by comparing with the VPRP, CG-DESCENT, and DL+ methods.

#### 1. Introduction

The nonlinear conjugate gradient method is one of the most efficient methods in solving unconstrained optimization problems. It comprises a class of unconstrained optimization algorithms which is characterized by low memory requirements and simplicity.

Consider the unconstrained optimization problem where is continuously differentiable, and its gradient is available.

The iterates of the conjugate gradient method for solving (1.1) are given by where stepsize is positive and computed by certain line search, and the search direction is defined by where , and is a scalar. Some well-known conjugate gradient methods include Polak-RibiΓ¨re-Polyak (PRP) method [1, 2], Hestenes-Stiefel (HS) method [3], Hager-Zhang (HZ) method [4], and Dai-Liao (DL) method [5]. The parameters of these methods are specified as follows: where is the Euclidean norm and . We know that if is a strictly convex quadratic function, the above methods are equivalent in the case that an exact line search is used. If is nonconvex, their behaviors may be further different.

In the past few years, the PRP method has been regarded as the most efficient conjugate gradient method in practical computation. One remarkable property of the PRP method is that it essentially performs a restart if a bad direction occurs (see [6]). Powell [7] constructed an example which showed that the PRP method can cycle infinitely without approaching any stationary point even if an exact line search is used. This counterexample also indicates that the PRP method has a drawback that it may not globally be convergent when the objective function is nonconvex. Powell [8] suggested that the parameter is negative in the PRP method and defined as Gilbert and Nocedal [9] considered Powellβs suggestion and proved the global convergence of the modified PRP method for nonconvex functions under the appropriate line search. In addition, there are many researches on convergence properties of the PRP method (see [10β12]).

In recent years, much effort has been investigated to create new methods, which not only possess global convergence properties for general functions but also are superior to original methods from the computation point of view. For example, Yu et al. [13] proposed a new nonlinear conjugate gradient method in which the parameter is defined on the basic of such as where (in this paper, we call this method as VPRP method). And they proved the global convergence of the VPRP method with the Wolfe line search. Hager and Zhang [4] discussed the global convergence of the HZ method for strong convex functions under the Wolfe line search and Goldstein line search. In order to prove the global convergence for general functions, Hager and Zhang modified the parameter as where The corresponding method of (1.7) is the famous CG-DESCENT method.

Dai and Liao [5] proposed a new conjugate condition, that is, Under the new conjugate condition, they proved global convergence of the DL conjugate gradient method for uniformly convex functions. According to Powellβs suggestion, Dai and Liao gave a modified parameter The corresponding method of (1.10) is the famous DL+ method. Under the strong Wolfe line search, they researched the global convergence of the DL+ method for general functions. Zhang et al. [14] proposed a modified DL conjugate gradient method and proved its global convergence. Moreover, some researchers have been studying a new type of method called the spectral conjugate gradient method (see [15β17]).

This paper is organized as follows: in the next section, we propose a modified PRP method and prove its sufficient descent property. In Section 3, the global convergence of the method with the Wolfe line search is given. In Section 4, numerical results are reported. We have a conclusion in the last section.

#### 2. Modified PRP Method

In this section, we propose a modified PRP conjugate gradient method in which the parameter is defined on the basic of as follows: in which . We introduce the modified PRP method as follows.

##### 2.1. Modified PRP (MPRP) Method

Step 1. Set , , and , if , then stop.

Step 2. Compute by some inexact line search.

Step 3. Let , , if , then stop.

Step 4. Compute by (2.1), and generate by (1.3).

Step 5. Set , and go to Step 2.

In the convergence analyses and implementations of conjugate gradient methods, one often requires the inexact line search to satisfy the Wolfe line search or the strong Wolfe line search. The Wolfe line search is to find such that where . The strong Wolfe line search consists of (2.2) and the following strengthened version of (2.3):

Moreover, in most references, we can see that the sufficient descent condition is always given which plays a vital role in guaranteeing the global convergence properties of conjugate gradient methods. But, in this paper, can satisfy (2.5) without any line search.

Theorem 2.1. Consider any method (1.2)-(1.3), where . If for all , then

Proof. Multiplying (1.3) by , we get If , from (2.7), we know that the conclusion (2.6) holds. If , the proof is divided into two cases in the following.
Firstly, if , then from (2.1) and (2.7), one has
Secondly, if , then from (2.7), we also have From the above, the conclusion (2.6) holds under any line search.

#### 3. Global Convergences of the Modified PRP Method

In order to prove the global convergence of the modified PRP method, we assume that the objective function satisfies the following assumption.

Assumption H
(i) The level set is bounded, that is, there exists a positive constant such that for all , .
(ii) In a neighborhood of , is continuously differentiable and its gradient is Lipchitz continuous, namely, there exists a constant such that Under these assumptions on , there exists a constant such that
The conclusion of the following lemma, often called the Zoutendijk condition, is used to prove the global convergence properties of nonlinear conjugate gradient methods. It was originally given by Zoutendijk [18].

Lemma 3.1. Suppose that, Assumption H holds. Consider any iteration of (1.2)-(1.3), where satisfies for and satisfies the Wolfe line search, then

Lemma 3.2. Suppose that Assumption H holds. Consider the method (1.2)-(1.3), where , and satisfies the Wolfe line search and (2.6). If there exists a constant , such that then one has where .

Proof. From (2.1) and (3.4), we get By (2.6) and (3.6), we know that for each .
Define the quantities By (1.3), one has Since is unit vector, we get From and the above equation, one has By (2.1), (3.4), and (3.6), one has From (3.3), (2.6), (3.4), and (3.11), one has so By (3.10) and the above inequality, one has

Lemma 3.3. Suppose that Assumption H holds. If (3.4) holds, then has property (*), that is, (1)there exists a constant , such that ,(2)there exists a constant , such that .

Proof. From Assumption (ii), we know that (3.2) holds. By (2.1), (3.2), and (3.4), one has Define . If , then from (2.1), (3.1), (3.2), and (3.4), one has

Lemma 3.4 (see [19]). Suppose that Assumption H holds. Let and be generated by (1.2)-(1.3), in which satisfies the Wolfe line search and (2.6). If has the property (*) and (3.4) holds, then there exits , for any and , for all , such that where , denotes the number of the .

Theorem 3.5. Suppose that Assumption H holds. Let and be generated by (1.2)-(1.3), in which satisfies the Wolfe line search and (2.6), , then one has

Proof. To obtain this result, we proceed by contradiction. Suppose that (3.18) does not hold, which means that there exists such that so, we know that Lemmas 3.2 and 3.4 hold.
We also define , then for all , one has where , that is, From Assumption H, we know that there exists a constant such that From (3.21) and the above inequality, one has Let be a positive integer and where has been defined in Lemma 3.4. From Lemma 3.2, we know that there exists such that From the Cauchy-Schwartz inequality and (3.24), , one has By Lemma 3.4, we know that there exists such that It follows from (3.23), (3.25), and (3.26) that From (3.27), one has , which is a contradiction with the definition of . Hence, which completes the proof.

#### 4. Numerical Results

In this section, we compare the modified PRP conjugate gradient method, denoted the MPRP method, to VPRP method, CG-DESCENT method, and DL+ method under the strong Wolfe line search about problems [20] with the given initial points and dimensions. The parameters are chosen as follows: , , , , and . If is satisfied, we will stop the program. The program will be also stopped if the number of iteration is more than ten thousands. All codes were written in Matlab 7.0 and run on a PC with 2.0βGHz CPU processor and 512βMB memory and Windows XP operation system.

The numerical results of our tests with respect to the MPRP method, VPRP method, CG-DESCENT method, and DL+ method are reported in Tables 1, 2, 3, 4, respectively. In the tables, the column βProblemβ represents the problemβs name in [20], and βCPU,β βNI,β βNF,β and βNGβ denote the CPU time in seconds, the number of iterations, function evaluations, gradient evaluations, respectively. βDimβ denotes the dimension of the tested problem. If the limit of iteration was exceeded, the run was stopped, and this is indicated by NaN.

 Problem Dim NI NF NG CPU ROSE 2 24 109 90 0.3651 FROTH 2 11 80 61 0.0594 BADSCP 2 26 227 210 0.2000 BADSCB 2 11 89 79 0.1085 BEALE 2 21 75 59 0.1449 HELIX 3 25 76 61 0.1754 BRAD 3 20 73 61 0.1380 GAUSS 3 3 8 6 0.0164 MEYER 3 1 1 1 0.0063 GULF 3 1 2 2 0.0173 BOX 3 1 1 1 0.0574 SING 4 67 263 228 0.5000 WOOD 4 33 150 117 0.2421 KOWOSB 4 57 222 195 0.4000 BD 4 26 127 96 0.1995 OSB1 5 1 1 1 0.0157 BIGGS 6 121 449 396 1.0000 OSB2 11 341 900 811 1.3000 JENSAM 6 12 49 32 0.0900 7 13 56 35 0.1872 8 11 53 30 0.1678 9 12 65 38 0.1160 10 26 133 94 0.2604 11 NaN NaN NaN NaN VARDIM 3 4 40 26 0.0135 5 6 57 38 0.0296 6 5 65 43 0.0270 8 7 72 47 0.0327 9 7 78 50 0.0647 10 7 81 52 0.0646 12 7 90 58 0.0647 15 8 92 60 0.0948 WATSON 5 59 200 167 0.2000 6 387 1281 1134 1.4000 7 1768 5834 5191 6.0000 8 3934 13373 11920 14.0000 10 4319 15102 13451 17.0000 12 1892 6762 6007 9.0000 15 1527 5552 4933 7.0000 20 3001 11308 10107 19.0000 PEN2 5 111 439 393 0.4000 10 185 845 752 1.5000 15 154 774 679 0.5000 20 178 989 889 0.6000 30 123 610 534 0.4000 40 147 700 617 0.5000 50 152 744 651 1.2000 60 163 813 720 0.7000 PEN1 5 30 151 125 0.2742 10 88 415 357 0.9000 20 32 155 124 0.3349 30 73 350 290 0.7000 50 72 346 285 0.3000 100 29 189 147 0.2458 200 28 198 152 0.4759 300 27 201 150 1.0464 TRIG 10 41 92 82 0.3817 20 56 136 127 0.5634 50 49 106 103 0.1949 100 61 137 127 0.3857 200 56 116 114 2.0205 300 52 106 101 11.6394 400 57 116 114 44.9734 500 53 109 108 89.8125 ROSEX 100 26 123 103 0.2323 200 26 123 103 0.2583 300 26 123 103 0.3078 400 26 123 103 0.4697 500 26 123 103 0.6781 1000 26 123 103 2.4474 1500 26 123 103 5.3979 2000 26 123 103 9.9364 SINGX 100 78 320 283 0.8000 200 79 335 293 0.8000 300 73 308 269 0.8000 400 89 367 324 1.6000 500 91 374 330 2.2000 1000 93 385 342 8.0000 1500 82 347 306 15.8000 2000 80 341 299 28.4000 BV 200 1813 4326 4063 9.0000 300 636 1501 1418 5.4000 400 226 516 487 2.7000 500 188 420 398 3.2000 600 86 190 184 1.9000 1000 21 40 37 0.9963 1500 11 20 19 1.0900 2000 2 6 5 0.5456 IE 200 6 13 7 0.3063 300 6 13 7 0.6698 400 6 13 7 1.1916 500 6 13 7 1.8511 600 6 13 7 2.6615 1000 6 13 7 7.3635 1500 6 13 7 16.6397 2000 6 13 7 29.4927 TRID 200 35 81 74 0.3327 300 36 83 75 0.3587 400 37 83 75 0.3731 500 35 78 73 0.4935 600 36 80 76 0.6862 1000 35 79 75 1.7180 1500 36 84 79 4.0501 2000 37 85 79 7.5866
 Problem Dim NI NF NG CPU ROSE 2 24 111 85 0.1585 FROTH 2 12 78 59 0.0698 BADSCP 2 99 394 336 0.4000 BADSCB 2 13 30 19 0.0448 BEALE 2 12 48 35 0.0402 HELIX 3 74 203 175 0.5000 BRAD 3 30 100 80 0.2027 GAUSS 3 3 7 4 0.0085 MEYER 3 1 1 1 0.0058 GULF 3 1 2 2 0.0052 BOX 3 1 1 1 0.0561 SING 4 101 341 289 0.7000 WOOD 4 174 482 417 0.6000 KOWOSB 4 71 234 203 0.3000 BD 4 42 161 125 0.2451 OSB1 5 1 1 1 0.0063 BIGGS 6 113 375 330 0.8000 OSB2 11 264 667 603 1.5000 JENSAM 6 9 33 17 0.0696 7 11 39 17 0.1137 8 10 42 19 0.0883 9 17 90 57 0.1850 10 17 124 84 0.1435 11 6 76 46 0.1111 VARDIM 3 4 40 26 0.0290 5 6 57 38 0.0203 6 5 65 43 0.0273 8 7 72 47 0.036 9 7 78 50 0.0715 10 7 81 52 0.0458 12 7 90 58 0.0672 15 8 92 60 0.0622 WATSON 5 193 535 473 1.4000 6 342 1002 890 2.2000 7 1451 4157 3678 5.0000 8 6720 19530 17300 20.0000 10 NaN NaN NaN NaN 12 3507 10432 9234 13.0000 15 5271 15817 14006 24.0000 20 NaN NaN NaN NaN PEN2 5 129 499 434 1.0000 10 90 379 328 0.3000 15 434 1467 1298 2.2000 20 941 2959 2612 3.0000 30 771 2531 2193 3.0000 40 248 978 860 1.6000 50 952 2788 2511 3.0000 60 927 2822 2435 4.0000 PEN1 5 32 151 124 0.3752 10 90 383 324 0.8000 20 28 160 121 0.1119 30 76 334 279 0.3000 50 64 344 280 0.5000 100 23 160 122 0.2212 200 21 174 128 0.4081 300 28 192 143 1.0207 TRIG 10 36 82 71 0.4048 20 56 125 114 0.6413 50 45 93 85 0.3426 100 58 120 113 0.4573 200 64 135 128 2.0248 300 52 102 99 12.4258 400 60 132 125 52.5691 500 59 127 116 101.6207 ROSEX 100 24 111 85 0.2798 200 24 111 85 0.2808 300 24 111 85 0.3136 400 24 111 85 0.4271 500 24 111 85 0.6181 1000 24 111 85 2.1821 1500 24 111 85 4.7644 2000 24 111 85 8.8878 SINGX 100 169 562 483 0.6000 200 199 676 576 1.4000 300 365 1181 1031 3.7000 400 627 2025 1756 9.0000 500 129 431 367 2.5000 1000 229 788 675 16.9000 1500 100 329 280 16.0000 2000 128 428 363 36.0000 BV 200 NaN NaN NaN NaN 300 7278 12990 12989 55.0000 400 3837 6707 6706 42.0000 500 1842 3236 3235 26.0000 600 898 1562 1561 17.6000 1000 133 232 231 6.2000 1500 19 36 35 2.0484 2000 2 6 5 0.5156 IE 200 7 15 8 0.3596 300 7 15 8 0.7986 400 7 15 8 1.4202 500 7 15 8 2.2147 600 7 15 8 3.1811 1000 7 15 8 8.8316 1500 7 15 8 19.7838 2000 7 15 8 34.7553 TRID 200 33 75 71 0.3758 300 35 79 75 0.3654 400 35 78 74 0.3912 500 35 78 74 0.5188 600 37 82 78 0.7388 1000 34 76 72 1.6832 1500 37 86 81 4.1950 2000 37 86 80 7.5255
 Problem Dim NI NF NG CPU ROSE 2 36 132 107 0.2371 FROTH 2 12 64 48 0.0462 BADSCP 2 40 213 189 0.2000 BADSCB 2 16 101 88 0.1212 BEALE 2 11 45 33 0.0405 HELIX 3 66 179 152 0.4397 BRAD 3 47 143 122 0.2292 GAUSS 3 3 10 8 0.0093 MEYER 3 1 1 1 0.0085 GULF 3 1 2 2 0.0068 BOX 3 1 1 1 0.0600 SING 4 62 198 164 0.3000 WOOD 4 103 298 247 0.5000 KOWOSB 4 77 222 192 0.4000 BD 4 53 204 162 0.3000 OSB1 5 1 1 1 0.0084 BIGGS 6 128 395 341 0.7000 OSB2 11 379 915 827 1.2000 JENSAM 6 NaN NaN NaN NaN 7 12 51 32 0.1114 8 11 50 26 0.0844 9 NaN NaN NaN NaN 10 5 59 34 0.0421 11 21 148 105 0.1749 VARDIM 3 4 40 26 0.0246 5 6 57 38 0.0174 6 5 65 43 0.0266 8 7 72 47 0.0226 9 7 78 50 0.0323 10 7 81 52 0.0495 12 7 90 58 0.0610 15 8 92 60 0.0583 WATSON 5 135 391 336 0.5000 6 421 1186 1043 1.4000 7 1822 5278 4655 6.0000 8 2607 7589 6716 9.0000 10 NaN NaN NaN NaN 12 3370 10111 8930 12.0000 15 5749 17442 15368 27.0000 20 5902 18524 16282 36.0000 PEN2 5 128 485 421 1.1000 10 147 571 486 0.4000 15 663 2262 1996 2.0000 20 734 2452 2181 3.0000 30 810 2438 2264 3.0000 40 1488 4502 3960 5.0000 50 744 2342 2056 2.0000 60 755 2457 2188 3.0000 PEN1 5 39 174 141 0.3298 10 94 404 345 0.8000 20 35 162 127 0.1220 30 76 347 286 0.3000 50 81 375 313 0.4000 100 31 184 141 0.2434 200 25 171 126 0.4106 300 26 186 139 1.0160 TRIG 10 33 74 65 0.2781 20 60 145 132 0.5098 50 43 95 90 0.3620 100 53 116 108 0.4474 200 59 123 118 1.7555 300 51 109 105 12.6411 400 58 121 114 45.0506 500 51 110 105 89.7073 ROSEX 100 36 124 99 0.1230 200 34 125 102 0.1539 300 35 133 107 0.3939 400 31 121 100 0.5789 500 31 129 106 0.8736 1000 37 142 116 3.5602 1500 34 132 106 7.4845 2000 32 128 103 12.9238 SINGX 100 77 227 187 0.7000 200 54 172 141 0.2376 300 99 301 248 1.0000 400 79 245 207 1.3000 500 69 215 180 1.6000 1000 101 322 271 8.6000 1500 66 210 174 12.4000 2000 69 210 173 23.1000 BV 200 NaN NaN NaN NaN 300 NaN NaN NaN NaN 400 4509 8044 8043 63.0000 500 1635 2926 2925 34.0000 600 925 1605 1604 24.5000 1000 247 418 417 16.9000 1500 18 38 37 3.0564 2000 2 6 5 0.6883 IE 200 7 15 8 0.3546 300 7 15 8 0.7927 400 7 15 8 1.4039 500 7 15 8 2.2022 600 7 15 8 3.1297 1000 7 15 8 8.6892 1500 7 15 8 19.5607 2000 7 15 8 34.7423 TRID 200 31 70 58 0.2691 300 33 73 65 0.2857 400 32 72 61 0.4264 500 34 76 71 0.6853 600 35 78 74 0.9484 1000 36 81 77 2.6056 1500 36 84 79 5.8397 2000 35 80 73 9.9491
 P Dim It f_iter Grade_iter CPU ROSE 2 25 110 87 0.1192 FROTH 2 9 65 50 0.0283 BADSCP 2 19 146 133 0.1003 BADSCB 2 11 56 47 0.0434 BEALE 2 11 50 39 0.0355 HELIX 3 50 144 122 0.2616 BARD 3 20 76 64 0.0824 GAUSS 3 2 5 3 0.0079 MEYER 3 1 1 1 0.0087 GULF 3 1 2 2 0.0062 BOX 3 1 1 1 0.0601 SING 4 91 295 250 0.5000 WOOD 4 148 402 349 0.8000 KOWOSB 4 76 236 204 0.4000 BD 4 67 208 177 0.3000 OSB1 5 1 1 1 0.0083 BIGGS 6 59 211 189 0.3000 OSB2 11 279 699 614 1.4000 JENSAM 6 9 35 19 0.0570 7 9 35 17 0.1276 8 NaN NaN NaN NaN 9 15 91 57 0.1148 10 17 136 96 0.2084 11 5 80 50 0.0882 VARDIM 3 4 40 26 0.0153 5 6 57 38 0.0192 6 5 65 43 0.0184 8 7 72 47 0.0219 9 7 78 50 0.0438 10 7 81 52 0.0242 12 7 90 58 0.0442 15 8 92 60 0.0535 WASTON 5 62 208 173 0.2000 6 346 1055 927 1.8000 7 1247 3443 3085 4.0000 8 2034 6104 5408 7.0000 10 5973 17601 15699 21.0000 12 3200 9291 8259 12.0000 15 1865 5402 4790 8.0000 20 6420 18320 16360 30.0000 PEN2 5 83 329 283 0.7000 10 128 519 449 1.0000 15 277 1040 922 0.9000 20 290 1113 981 0.9000 30 767 2231 2040 3.0000 40 675 1929 1706 3.0000 50 617 2223 1962 2.0000 60 NaN NaN NaN NaN PEN1 5 33 165 139 0.2615 10 78 353 298 0.7000 20 24 119 94 0.0842 30 73 367 305 0.3000 50 66 356 293 0.6000 100 22 154 117 0.1957 200 27 178 136 0.4346 300 29 195 147 1.0408 TRIG 10 34 82 71 0.2910 20 59 140 129 0.5161 50 46 94 88 0.3865 100 55 120 111 0.4625 200 56 116 111 1.6476 300 52 110 105 12.3062 400 56 115 113 44.8831 500 54 121 116 97.7043 ROSEX 100 25 110 87 0.0859 200 25 110 87 0.2019 300 25 110 87 0.2697 400 25 110 87 0.4169 500 25 110 87 0.5990 1000 25 110 87 2.2166 1500 25 110 87 4.8426 2000 25 110 87 9.1263 SINGX 100 113 384 329 1.0000 200 117 397 341 0.5000 300 215 735 634 2.1000 400 113 384 329 1.6000 500 110 367 313 2.2000 1000 137 452 388 9.2000 1500 152 494 420 22.1000 2000 110 367 313 30.2000 BV 200 NaN NaN NaN NaN 300 NaN NaN NaN NaN 400 5624 8458 8457 49.0000 500 3314 4854 4853 38.0000 600 1618 2291 2290 25.0000 1000 258 312 311 8.7000 1500 18 30 29 1.7251 2000 2 6 5 0.5129 IE 200 6 13 7 0.3075 300 6 13 7 0.6731 400 6 13 7 1.1939 500 6 13 7 1.8608 600 6 13 7 2.6785 1000 6 13 7 7.3685 1500 6 13 7 16.5794 2000 6 13 7 29.5854 TRID 200 33 75 71 0.2606 300 35 79 75 0.2866 400 36 80 76 0.3653 500 35 78 73 0.4857 600 37 82 78 0.6950 1000 34 76 72 1.6550 1500 37 86 81 4.1380 2000 37 86 80 7.4702

In this paper, we will adopt the performance profiles by Dolan and MorΓ© [21] to compare the MPRP method to the VPRP method, CG-DESCENT method, and DL+ method in the CPU time, the number of iterations, function evaluations, and gradient evaluations performance, respectively (see Figures 1, 2, 3, 4). In figures,

Figures 1β4 show the performance of the four methods relative to CPU time, the number of iterations, the number of function evaluations, and the number of gradient evaluations, respectively. For example, the performance profiles with respect to CPU time means that for each method, we plot the fraction of problems for which the method is within a factor of the best time. The left side of the figure gives the percentage of the test problems for which a method is the fastest; the right side gives the percentage of the test problems that are successfully solved by each of the methods. The top curve is the method that solved of the most problems in a time that was within a factor of the best time.

Obviously, Figure 1 shows that MPRP method outperforms VPRP method, CG-DESCENT method, and DL+ method for the given test problems in the CPU time. Figures 2β4 show that the MPRP method also has the best performance with respect to the number of iterations and function and gradient evaluations since it corresponds to the top curve. So, the MPRP method is computationally efficient.

#### 5. Conclusions

We have proposed a modified PRP method on the basic of the PRP method, which can generate sufficient descent directions with inexact line search. Moreover, we proved that the proposed modified method converge globally for general nonconvex functions. The performance profiles showed that the proposed method is also very efficient.

#### Acknowledgments

The authors wish to express their heartfelt thanks to the referees and Professor Piermarco Cannarsa for their detailed and helpful suggestions for revising the paper. This work was supported by The Nature Science Foundation of Chongqing Education Committee (KJ091104) and Chongqing Three Gorge University (09ZZ-060).

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