# The Global Convergence of a Modified BFGS Method under Inexact Line Search for Nonconvex Functions

Authors: Pengyuan Li; Junyu Lu; Haishan Feng
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

Among the quasi-Newton algorithms, the BFGS method is often discussed by related scholars. However, in the case of inexact Wolfe line searches or even exact line search, the global convergence of the BFGS method for nonconvex functions is not still proven. Based on the aforementioned issues, we propose a new quasi-Newton algorithm to obtain a better convergence property; it is designed according to the following essentials: (1) a modified BFGS formula is designed to guarantee thatBk+1 inherits the positive definiteness of Bk; (2) a modified weak Wolfe–Powell line search is recommended; (3) a parabola, which is considered as the projection plane to avoid using the invalid direction, is proposed, and the next point xk+1 is designed by a projection technique; (4) to obtain the global convergence of the proposed algorithm more easily, the projection point is used at all the next iteration points instead of the current modified BFGS update formula; and (5) the global convergence of the given algorithm is established under suitable conditions. Numerical results show that the proposed algorithm is efficient.


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## Body

## 1. Introduction

Consider(1)minfx|x∈ℜn,where f:ℜn⟶ℜ and f∈C2. The multitudinous algorithms for (1) often use the following iterative formula:(2)xk+1=xk+sk,k=0,1,2…,where xk is the current point, sk=xk+1−xk=αkdk, αk is a step size, and dk is a search direction at xk. There exist many algorithms for (1) [1–9]. Davidon [10] pointed out that the quasi-Newton method is one of the most effective methods for solving nonlinear optimization problems. The idea of the quasi-Newton method is to use the first derivative to establish an approximate Hessian matrix in many iterations, and the approximation is updated by a low-rank matrix in each iteration. The primary quasi-Newton equation is as follows:(3)Bk+1sk=yk,yk=gk+1−gk.The search directiondk of the quasi-Newton method is generated by the following equation:(4)d0=−H0g0,dk=−Hkgk,k≥1,where H0 is any given n×n symmetric positive-definite matrix, Hk=Bk−1, the Hessian approximation matrix Bk is the quasi-Newton update matrix, and gk=gxk is the gradient of fx at xk. The BFGS (Broyden [11], Fletcher [12], Goldfarb [13], and Shanno [14]) method is one of the quasi-Newton line search methods and has great numerical stability. The famous BFGS update formula is(5)Bk+1=Bk−BkskskTBkskTBksk+ykykTskTyk,which is effective for solving (1) [15–18]. Powell [19] first proved that the BFGS method possesses global convergence for convex functions under Wolfe line search. Some global convergence results for the BFGS method for convex minimization problems can be found in [19–26]. However, Dai [16] proposed a counterexample to illustrate that the standard BFGS method may not be applicable to nonconvex functions with Wolfe line search, and Mascarenhas [27] demonstrated the nonconvergence of the standard BFGS method even with exact line search. To verify the global convergence of the BFGS method for general functions, some modified BFGS methods [28–31] have been also presented for nonconvex minimization problems. Aiming to obtain a better approximation of the objective function Hessian matrix, Wei et al. [32] proposed a new BFGS method, whose formula is(6)Bk+1=Bk−BkskskTBkskTBksk+yk∗yk∗TskTyk∗,where yk∗=yk+Ck/sk2sk and Ck=2fxk−fxk+αkdk+gxk+αkdk+gxkTsk, and the corresponding quasi-Newton equation is as follows:(7)Bk+1sk=yk∗.For convex functions, convergence analysis of the new BFGS algorithm was given for weak Wolfe–Powell line search:(8)fxk+αkdk≤fk+δαkgkTdk,gxk+αkdkTdk≥σgkTdk,where δ∈0,1/2 and σ∈δ,1.Motivated by the above formula and other observations, Yuan and Wei [33] defined a modified quasi-Newton equation as follows:(9)Bk+1sk=ykm,where ykm=yk+maxCk,0/sk2sk and(10)Bk+1=Bk−BkskskTBkskTBksk+ykmykmTskTykm.It is obvious that ifCk>0 holds, then the quasi-Newton method is the method (7); otherwise, it is the standard BFGS method. Therefore, when Ck>0 holds, the modified quasi-Newton method (10) and the quasi-Newton method (6) have the same approximation of the Hessian matrix. Inspired by their views, we will demonstrate the global convergence of the modified BFGS (MBFGS) method (10) for nonconvex functions with the modified weak Wolfe–Powell (MWWP) line search [34], whose form is as follows:(11)fxk+αkdk≤fk+δαkgkTdk+αkmin−δ1gkTdk,δαk2dk2,(12)gxk+αkdkTdk≥σgkTdk+min−δ1gkTdk,δαkdk2,where δ∈0,1/2,δ1∈δ/2,δ, and σ∈δ,1. The parameter δ1 is different from that in paper [34].This article is organized as follows: Section2 introduces the motivation and states the given technique and algorithm. In Section 3, we prove the global convergence of the modified BFGS method with MWWP line search under some reasonable conditions. Section 4 reports the results of the numerical experiments to show the performance of the algorithms. The last section presents the conclusion. Throughout the article, fxk and fxk+1 are replaced by fk and fk+1, and gxk and gxk+1 are replaced by gk and gk+1. · denotes the Euclidean norm.


## 2. Motivation and Algorithm

The global convergence of the BFGS algorithm has been established for the uniformly convex functions which have many advantages. It is worth considering whether we can use these properties of uniformly convex functions in the BFGS algorithm to obtain global convergence. This idea motivates us to propose a projection technique to acquire better convergence properties of the BFGS algorithm. Given a new numerical formula for (1):(13)zk=xk+αkdk,where zk is the next point generated by the classical BFGS formula. Moreover, a parabolic form is given as follows:(14)x|gzkTzk−x+λzk−x2=0,where λ>2 is a constant. It is not difficult to see that Qkx=gzkTzk−x+λzk−x2 can be considered as the first two terms of the expansion of a quadratic function at zk, whose Hessian matrix is a diagonal matrix with eigenvalue −2λ. Therefore, the BFGS method is globally convergent. By projecting xk onto (14), we obtain the next step xk+1:(15)xk+1=xk+gzkTzk−xk+λzk−xk2gzk−gxk2gzk−gxk.The idea of the projection can also be found in [6, 8, 35]. Based on the above discussions, the modified algorithm is given in Algorithm 1.Algorithm 1: Algorithm MBFGS.
Step 0: given a pointx0∈ℜn, constants ε∈0,1,δ∈0,1/2,σ∈δ,1,δ1∈δ/2,δ,λ∈2,+∞, and an n×n symmetric positive-definite matrix B0, set k=0.Step 1: stop ifgk≤ε.Step 2: obtain a search directiondk by solvingBkdk+gk=0Step 3: calculateαk using the inequalities (11) and (12).Step 4: setwk=xk+αkdk.Step 5: if−δ1gkTdk>δαkdk2 holds, then let xk+1=wk, yk=gxk+1−gxk, and go to Step 7; otherwise, go to Step 6.Step 6: letxk+1 be defined by (15), ykτ=gwk−gxk, and yk:=ykτ.Step 7: updateBk+1 using formula (10).Step 8: letk=k+1, and go to Step 1.Remark 1.
(i)
xk+1 is the defined projection point in Step 6, and vector ykτ is the same as vector yk in Step 5, where the projection point xk+1 does not work in (10) but does in the next iteration.(ii)
If−δ1gkTdk>δαkdk2 holds in Step 5, then the global convergence of the algorithm can be obtained by the modified weak Wolfe–Powell line search, (11) and (12). If not, we can ensure the global convergence of the algorithm using the projection method (15).


## 3. Convergence Analysis

In this section, we concentrate on the global convergence of the modified projection BFGS algorithm. The following assumptions are required.Assumption 1.
(i)
The level setΓ=x∈ℜn|fx≤fx0 is bounded.(ii)
The functionfx is twice continuously differentiable and bounded from below, and its gradient function gx is Lipschitz continuous, that is,(16)gx−gy≤Lx−y,x,y∈ℜn,holds, where L>0 is a constant.Assumption 1(ii) indicates that the relation(17)Gξsk≤Lsk,Gξ=∇2fξ,ξ∈xk,xk+1,holds.Theorem 1.
Suppose that Assumption1 and gkTdk≤0 hold. Then, there exists a constant α satisfying (11) and (12), where α∈pm,pn, and pn>pm>0 are constants.Proof.
The detailed proof of the rationality of the line search is given in paper [35].Lemma 1.
Let Assumption1 and gkTdk≤0 hold. If the sequence xk is generated by Algorithm 1, then we have(18)skTykm≥βαkdk2,where β>0 is a constant.Proof.
According to Lemma1 of paper [35], the following relations are reasonable:(19)skTyk≥δαkdk2,if−δ1gkTdk>δαkdk2,(20)skTyk≥β1αkdk2,if−δ1gkTdk≤δαkdk2,where parameterβ1=λ−δσ−δ1/δ1>0. Combining (19) and (20), we obtain the following formula with β=minδ,β1:(21)skTyk≥βαkdk2.Using the definition ofykm, we obtain(22)skTykm=skTyk+maxCk,0≥skTyk.Therefore,skTykm≥βαkdk2. The proof is complete.Lemma 2.
If the sequenceBk is generated by Algorithm 1 and Assumption 1 holds, then the matrix Bk is positive definite for all k.Proof.
According to (18), the relation skTykm>0 is valid. Thus, the proof is complete.Lemma 3.
If the sequenceαk,dk,xk,gk is generated by Algorithm 1 and Assumption 1 holds, then(23)∑k=0∞gkTdk2dk2<∞.Proof.
According (11) and Assumption 1(ii), the following formula obviously exists(24)∑k=0∞−αkgkTdk<∞.Combining (12) with (16), we obtain(25)αkLdk2≥gk+1−gkTdk≥−1−σgkTdk+min−δ1gkTdk,δαkdk2≥−1−σgkTdk.Thus,(26)αk≥−1−σgkTdkLdk2.Substituting the above inequality into (24), we have (23). The proof is complete.Lemma 4.
Let Assumption1 and the inequality gk+1Tdk≤−δgkTdk hold. If there exist constants ρ1≥ρ2>0, then the following hold:(27)Bksk≤ρ1sk,(28)skTBksk≥ρ2sk2,for at least t/2 values of k∈1,t for any positive integer t.Proof.
The proof will be completed using the following two cases:Case 1. If −δ1gkTdk>δαkdk2 is true, then yk=gzk−gxk=gxk+1−gxk and sk=zk−xk=xk+1−xk=αkdk. By (16) and Assumption 1,(29)yk=gxk+1−gxk≤Lxk+1−xk=Lαkdk.Combining the above inequality with (19), we obtain(30)skTyksk2≥δandyk2skTyk≤L2δ.Assumption1(ii) and the definition of Ck imply that(31)Ck≤yksk+Lsk2.By (31), (19), and the definition of ykm, we have(32)ykm≤yk+maxCk,0sksk2≤2yk+Lδyk=2+Lδyk.Relations (22), (30), and (32) indicate that(33)skTykmsk2≥δandykm2skTykm≤2+Lδ2L2δ.Case 2. −δ1gkTdk≤δαkdk2 holds. From Step 6 of Algorithm 1, we obtain(34)yk2=ykτ2=gwk−gxk2≤L2αkdk2.Together with (32), we have(35)ykm2≤2L+L2δ2αkdk2.Integrating (35) with (20), we obtain(36)ykm2skTykm≤1β12L+L2δ2.Using the definition ofsk, we have(37)sk=xk+1−xk=gzkTzk−xk+λzk−xk2gzk−gxk2gzk−gxk≤gzkTzk−xk+λzk−xk2gzk−gxk≤−δαkgkTdk+λzk−xk2gzk−gxk≤δ/δ1+λαkdk2gzk−gxk,where the second inequality follows fromgk+1Tdk≤−δgkTdk, and the last inequality follows from (16). Combining the above formula with (20), we obtain(38)skTykmsk2≥β1δ12L2δ+λδ12.All in all, the following formula always holds:(39)skTykmsk2≥ω1andykm2skTykm≤ω2,whereω1=minδ,β1δ12L2/δ+λδ12 and ω2=max2+L/δ2L2/δ,1/β12L+L2/δ2. Similar to the proof of Theorem 1 in [36], we obtain (27) and (28). The proof is complete.Theorem 2.
If the conditions of Lemma4 hold, then we obtain(40)limk⟶∞infgk=0.Proof.
By (23), we can obtain(41)gkTdk2dk2⟶0,k⟶∞.Then, using Algorithm1, we have(42)dkTBkdk2dk2⟶0,k⟶∞.The relationship between (27) and (28) indicates that(43)0≤ρ22dk2≤dkTBkdk2dk2⟶0,k⟶∞,which means thatdk⟶0 holds for k⟶∞. Combining Lemma 4 and gk=−Bkdk, we obtain(44)ρ2dk≤gk≤ρ1dk.This implies that (40) holds. The proof is complete.


## 4. Numerical Results

In this section, we perform some numerical experiments to test Algorithm1 with the modified weak Wolfe–Powell line search and compare its performance with that of the normal BFGS method. We call Algorithm 1 as MBFGS.


### 4.1. General Unconstrained Optimization Problems

Tested problems: the problems are obtained from [37, 38]. There are 74 test questions in total, which are listed in Table 1.Dimensionality: problem instances with 300, 900, and 2700 variables are considered.Himmelblau stop rule [39]: if fxk>e1, then set Ter1=fxk−fxk+1, or let Ter1=fxk−fxk+1/fxk. If gx<ε or Ter1<e2 holds, then the program is stopped, where e1=e2=10−5 and ε=10−6.Parameters: in Algorithm1, δ=0.2, δ1=0.15, σ=0.85, λ=5, and B0=I, which is the unit matrix.Experiment environment: all programs are written in MATLAB R2014a and run on a PC with an Inter(R) Core(TM) i5-4210U CPU @ 1.70 GHz, 8.00 GB of RAM, and the Windows 10 operating system.Symbol representation: some definitions of the notation used in Tables1 and 2 are as follows:No: the test problem number.CPUTime: the CPUtime in seconds.NI: the number of iterations.NFG: the total number of function and gradient evaluations.Image description: Figures1–3 show the profiles of CPUTime, NI, and NFG. It is easy to see from these figures that the MBFGS method possesses the best performance since its performance curves of CPUTime, NI, and NFG are better than those of the BFGS method. In addition, numerical results of the total CPUTime, NI, and NFG of the modified BFGS method are lower than those of the BFGS method.Table 1
The test problems.
No.Test problem1Extended Freudenstein and Roth function2Extended trigonometric function3Extended Rosenbrock function4Extended White and Holst function5Extended Beale function6Extended penalty function7Perturbed quadratic function8Raydan 1 function9Raydan 2 function10Diagonal 1 function11Diagonal 2 function12Diagonal 3 function13Hager function14Generalized tridiagonal-1 function15Extended tridiagonal-1 function16Extended three exponential terms function17Generalized tridiagonal-2 function18Diagonal 4 function19Diagonal 5 function20Extended Himmelblau function21Generalized PSC1 function22Extended PSC1 function23Extended Powell function24Extended block diagonal BD1 function25Extended Maratos function26Extended Cliff function27Quadratic diagonal perturbed function28Extended wood function29Extended Hiebert function30Quadratic function QF1 function31Extended quadratic penalty QP1 function32Extended quadratic penalty QP2 function33A quadratic function QF2 function34Extended EP1 function35Extended tridiagonal-2 function36BDQRTIC function (CUTE)37TRIDIA function (CUTE)38ARWHEAD function (CUTE)39NONDIA function (CUTE)40NONDQUAR function (CUTE)41DQDRTIC function (CUTE)42EG2 function (CUTE)43DIXMAANA function (CUTE)44DIXMAANB function (CUTE)45DIXMAANC function (CUTE)46DIXMAANE function (CUTE)47Partial perturbed quadratic function48Broyden tridiagonal function49Almost perturbed quadratic function50Tridiagonal perturbed quadratic function51EDENSCH function (CUTE)52VARDIM function (CUTE)53STAIRCASE S1 function54LIARWHD function (CUTE)55DIAGONAL 6 function56DIXON3DQ function (CUTE)57DIXMAANF function (CUTE)58DIXMAANG function (CUTE)59DIXMAANH function (CUTE)60DIXMAANI function (CUTE)61DIXMAANJ function (CUTE)62DIXMAANK function (CUTE)63DIXMAANL function (CUTE)64DIXMAAND function (CUTE)65ENGVAL1 function (CUTE)66FLETCHCR function (CUTE)67COSINE function (CUTE)68Extended DENSCHNB function (CUTE)69Extended DENSCHNF function (CUTE)70SINQUAD function (CUTE)71BIGGSB1 function (CUTE)72Partial perturbed quadratic PPQ2 function73Scaled quadratic SQ1 function74Scaled quadratic SQ2 functionTable 2
Numerical results for the total CPUTime, NI, and NFG.
AlgorithmCPUTimeNINFGBFGS62299.859382383357444MBFGS60990.640632235751370Figure 1
Performance profiles of the methods (CPUTime).Figure 2
Performance profiles of these methods (NI).Figure 3
Performance profiles of these methods (NFG).


### 4.2. The Muskingum Model in Engineering Problems

The Muskingum model, whose definition is as follows, is presented in this subsection. The key work is to numerically estimate the model using Algorithm1.Muskingum Model [40]:(45)minfx1,x2,x3=∑i=1n−11−Δt6x1x2Ii+1+1−x2Qi+1x3−1−Δt6x1x2Ii+1−x2Qix3−Δt2Ii−Qi+Δt21−Δt3Ii+1−Qi+12,whose symbolic representation is as follows: x1 is the storage time constant, x2 is the weight coefficient, x3 is an extra parameter, Ii is the observed inflow discharge, Qi is the observed outflow discharge, Δt is the time step at time ti (i=1,2,…,n), and n is the total time.The observed data of the experiment are derived from the process of flood runoff from Chenggouwan and Linqing of Nanyunhe in the Haihe Basin, Tianjin, China. To obtain better numerical results, the initial pointx=0,1,1T and the time step Δt=12h are selected. The specific values of Ii and Qi for the years 1960, 1961, and 1964 are stated in article [41]. The test results are listed in Table 3.Table 3
Results of the three algorithms.
Algorithmx1x2x3BFGS [42]10.81560.98261.0219HIWO [40]13.28130.80010.9933MBFGS11.18321.00000.9996The following three conclusions are apparent from Figures4–6 and Table 3: (1) Combined with the Muskingum model, the MBFGS method has great numerical experiment performance, similar to the BFGS method and the HIWO method, and the efficiency of these three algorithms is fascinating. (2) The final points (x1, x2, and x3) of the MBFGS method are emulative of those of other similar methods. (3) Owing to the final points of these three methods being distinct, the Muskingum model may have more approximation optimum points.Figure 4
Performance of the MBFGS in 1960.Figure 5
Performance of the MBFGS in 1961.Figure 6
Performance of the MBFGS in 1964.


## 4.1. General Unconstrained Optimization Problems

Tested problems: the problems are obtained from [37, 38]. There are 74 test questions in total, which are listed in Table 1.Dimensionality: problem instances with 300, 900, and 2700 variables are considered.Himmelblau stop rule [39]: if fxk>e1, then set Ter1=fxk−fxk+1, or let Ter1=fxk−fxk+1/fxk. If gx<ε or Ter1<e2 holds, then the program is stopped, where e1=e2=10−5 and ε=10−6.Parameters: in Algorithm1, δ=0.2, δ1=0.15, σ=0.85, λ=5, and B0=I, which is the unit matrix.Experiment environment: all programs are written in MATLAB R2014a and run on a PC with an Inter(R) Core(TM) i5-4210U CPU @ 1.70 GHz, 8.00 GB of RAM, and the Windows 10 operating system.Symbol representation: some definitions of the notation used in Tables1 and 2 are as follows:No: the test problem number.CPUTime: the CPUtime in seconds.NI: the number of iterations.NFG: the total number of function and gradient evaluations.Image description: Figures1–3 show the profiles of CPUTime, NI, and NFG. It is easy to see from these figures that the MBFGS method possesses the best performance since its performance curves of CPUTime, NI, and NFG are better than those of the BFGS method. In addition, numerical results of the total CPUTime, NI, and NFG of the modified BFGS method are lower than those of the BFGS method.Table 1
The test problems.
No.Test problem1Extended Freudenstein and Roth function2Extended trigonometric function3Extended Rosenbrock function4Extended White and Holst function5Extended Beale function6Extended penalty function7Perturbed quadratic function8Raydan 1 function9Raydan 2 function10Diagonal 1 function11Diagonal 2 function12Diagonal 3 function13Hager function14Generalized tridiagonal-1 function15Extended tridiagonal-1 function16Extended three exponential terms function17Generalized tridiagonal-2 function18Diagonal 4 function19Diagonal 5 function20Extended Himmelblau function21Generalized PSC1 function22Extended PSC1 function23Extended Powell function24Extended block diagonal BD1 function25Extended Maratos function26Extended Cliff function27Quadratic diagonal perturbed function28Extended wood function29Extended Hiebert function30Quadratic function QF1 function31Extended quadratic penalty QP1 function32Extended quadratic penalty QP2 function33A quadratic function QF2 function34Extended EP1 function35Extended tridiagonal-2 function36BDQRTIC function (CUTE)37TRIDIA function (CUTE)38ARWHEAD function (CUTE)39NONDIA function (CUTE)40NONDQUAR function (CUTE)41DQDRTIC function (CUTE)42EG2 function (CUTE)43DIXMAANA function (CUTE)44DIXMAANB function (CUTE)45DIXMAANC function (CUTE)46DIXMAANE function (CUTE)47Partial perturbed quadratic function48Broyden tridiagonal function49Almost perturbed quadratic function50Tridiagonal perturbed quadratic function51EDENSCH function (CUTE)52VARDIM function (CUTE)53STAIRCASE S1 function54LIARWHD function (CUTE)55DIAGONAL 6 function56DIXON3DQ function (CUTE)57DIXMAANF function (CUTE)58DIXMAANG function (CUTE)59DIXMAANH function (CUTE)60DIXMAANI function (CUTE)61DIXMAANJ function (CUTE)62DIXMAANK function (CUTE)63DIXMAANL function (CUTE)64DIXMAAND function (CUTE)65ENGVAL1 function (CUTE)66FLETCHCR function (CUTE)67COSINE function (CUTE)68Extended DENSCHNB function (CUTE)69Extended DENSCHNF function (CUTE)70SINQUAD function (CUTE)71BIGGSB1 function (CUTE)72Partial perturbed quadratic PPQ2 function73Scaled quadratic SQ1 function74Scaled quadratic SQ2 functionTable 2
Numerical results for the total CPUTime, NI, and NFG.
AlgorithmCPUTimeNINFGBFGS62299.859382383357444MBFGS60990.640632235751370Figure 1
Performance profiles of the methods (CPUTime).Figure 2
Performance profiles of these methods (NI).Figure 3
Performance profiles of these methods (NFG).


## 4.2. The Muskingum Model in Engineering Problems

The Muskingum model, whose definition is as follows, is presented in this subsection. The key work is to numerically estimate the model using Algorithm1.Muskingum Model [40]:(45)minfx1,x2,x3=∑i=1n−11−Δt6x1x2Ii+1+1−x2Qi+1x3−1−Δt6x1x2Ii+1−x2Qix3−Δt2Ii−Qi+Δt21−Δt3Ii+1−Qi+12,whose symbolic representation is as follows: x1 is the storage time constant, x2 is the weight coefficient, x3 is an extra parameter, Ii is the observed inflow discharge, Qi is the observed outflow discharge, Δt is the time step at time ti (i=1,2,…,n), and n is the total time.The observed data of the experiment are derived from the process of flood runoff from Chenggouwan and Linqing of Nanyunhe in the Haihe Basin, Tianjin, China. To obtain better numerical results, the initial pointx=0,1,1T and the time step Δt=12h are selected. The specific values of Ii and Qi for the years 1960, 1961, and 1964 are stated in article [41]. The test results are listed in Table 3.Table 3
Results of the three algorithms.
Algorithmx1x2x3BFGS [42]10.81560.98261.0219HIWO [40]13.28130.80010.9933MBFGS11.18321.00000.9996The following three conclusions are apparent from Figures4–6 and Table 3: (1) Combined with the Muskingum model, the MBFGS method has great numerical experiment performance, similar to the BFGS method and the HIWO method, and the efficiency of these three algorithms is fascinating. (2) The final points (x1, x2, and x3) of the MBFGS method are emulative of those of other similar methods. (3) Owing to the final points of these three methods being distinct, the Muskingum model may have more approximation optimum points.Figure 4
Performance of the MBFGS in 1960.Figure 5
Performance of the MBFGS in 1961.Figure 6
Performance of the MBFGS in 1964.


## 5. Conclusion

This paper gives a modified BFGS method and studies its global convergence under an inexact line search for nonconvex functions. A new algorithm is proposed, which has the following properties: (i) The search direction and its associated step size are accepted if a positive condition holds, and the next iterative point is designed; otherwise, a parabola is introduced, which is regarded as the projection surface to avoid using the failed direction, and the next pointxk+1 is designed by a projection technique. (ii) To easily obtain global convergence of the proposed algorithm, the projection point is used at all the next iteration points instead of the current modified BFGS update formula. The global convergence for nonconvex functions and the numerical results of the proposed algorithm indicated that the given method is competitive with other similar methods. As for future work, we have the following points to consider: (a) Is there a new projection technique suitable for the global convergence of the modified BFGS method? (b) Application of the modified BFGS method (10) to other line search techniques should be discussed. (c) Whether the combination of the projection technique mentioned and a conjugate gradient method, especially the PRP method, has good numerical experimental results is worthy of investigation.



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