Research article

New iterative approach for the solutions of fractional order inhomogeneous partial differential equations

  • Received: 17 August 2020 Accepted: 29 October 2020 Published: 18 November 2020
  • MSC : 11J20, 32W50, 39B12

  • In this paper, the study of fractional order partial differential equations is made by using the reliable algorithm of the new iterative method (NIM). The fractional derivatives are considered in the Caputo sense whose order belongs to the closed interval [0, 1]. The proposed method is directly extended to study the fractional-order Roseau-Hyman and fractional order inhomogeneous partial differential equations without any transformation to convert the given problem into integer order. The obtained results are compared with those obtained by Variational Iteration Method (VIM), Homotopy Perturbation Method (HPM), Laplace Variational Iteration Method (LVIM) and the Laplace Adominan Decomposition Method (LADM). The results obtained by NIM, show higher accuracy than HPM, LVIM and LADM. The accuracy of the proposed method improves by taking more iterations.

    Citation: Laiq Zada, Rashid Nawaz, Sumbal Ahsan, Kottakkaran Sooppy Nisar, Dumitru Baleanu. New iterative approach for the solutions of fractional order inhomogeneous partial differential equations[J]. AIMS Mathematics, 2021, 6(2): 1348-1365. doi: 10.3934/math.2021084

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  • In this paper, the study of fractional order partial differential equations is made by using the reliable algorithm of the new iterative method (NIM). The fractional derivatives are considered in the Caputo sense whose order belongs to the closed interval [0, 1]. The proposed method is directly extended to study the fractional-order Roseau-Hyman and fractional order inhomogeneous partial differential equations without any transformation to convert the given problem into integer order. The obtained results are compared with those obtained by Variational Iteration Method (VIM), Homotopy Perturbation Method (HPM), Laplace Variational Iteration Method (LVIM) and the Laplace Adominan Decomposition Method (LADM). The results obtained by NIM, show higher accuracy than HPM, LVIM and LADM. The accuracy of the proposed method improves by taking more iterations.


    In this paper, we consider the following bound constrained nonlinear systems of equations:

    F(x)=0,s.t.xΩ, (1.1)

    where F(x)=(F1(x),F2(x),,Fn(x))T, and Fi:RnR is a nonlinear continuously differentiable function whose gradient is available. We denote by F(x)=(F1(x),F2(x),,Fn(x))T the Jacobian matrix of F at a given point x. The set ΩRn is defined as

    Ω:={xRn|lixiui,i=1,2,,n}

    for some given lower and upper bounds satisfying li<ui+ for all i=1,2,,n.

    The bound constrained nonlinear equation of the type (1.1) is an important problem in the practical problems. There are a couple of different mathematical programming problems like Karush-Kuhn-Tucker systems and complementarity problems can be reformulated as the problem (1.1), see [1,2,3,4,5,6,7,8,9]. On the other hand, in many cases, the function Fi(x) is not always defined on the whole space Rn, and one usually puts some suitable bounds on some or all of the variables.

    The Newton type method is one of the most important numerical methods for problem (1.1) and many researchers are interested in this method [7,10,11,12,13,14]. Given a current iterate xkΩ, the Newton method considers the least-squares solutions dk of the following nonlinear constrained equation:

    min12F(xk)+F(xk)d2s.t.xk+dΩ. (1.2)

    We set the next iterate to be xk+1=xk+dk and call (1.2) the constrained Gauss-Newton method.

    Another natural possibility is to consider solving the basic unconstrained Newton equation:

    F(xk)+F(xk)d=0. (1.3)

    Denote the solution of (1.3) by dkN if it exists and then define xk+1 as the projection of xk+dkN onto Ω. This scheme can be called the projected Newton method.

    On the other hand, there are many versions of the Newton type method, such as the constrained Levenberg-Marquardt method [6,15,16] usually used to solve the following subproblems:

    min12F(xk)+F(xk)d2+σd2s.t.xk+dΩ, (1.4)

    where σ is a positive constant.

    Along with constrained versions of the methods in question, one can also consider their projected variants. The projected Levenberg-Marquardt method has been proposed in [15] and its iteration consists of finding the solution dkLM of the unconstrained subproblem

    min12F(xk)+F(xk)d2+σd2, (1.5)

    and then defines the next iterate xk+1 as the projection of xk+dkLM onto Ω.

    The Newton iteration can be costly, since partial derivatives must be computed and the linear system (1.3) must be solved at every iteration. This fact motivates the development of quasi-Newton methods [10,14,17] which are defined as the generalizations of (1.3) given by

    F(xk)+Bkd=0. (1.6)

    In quasi-Newton methods, the matrices Bk are intended to be approximations of F(xk) and be updated by some quasi-Newton formulas. Another well known algorithm is the trust region type algorithm, for example, [3,4,9,18,19,20,21].

    Whether Newton method or quasi-Newton, one has to solve a linear system with full dimension, which will be expensive for large scale problems. To overcome this drawback, the active set methods are developed by many authors [7,12,13,22]. Since only a reduced dimension linear system to be dealt with at each iteration, the active set Newton methods are more efficient than the full Newton method especially for large scale problems.

    To prove global convergence of the method outlined above, one often assumes that the iteration sequence is contained in a bounded set. If li and ui are bounded and the algorithm generates a feasible sequence, the assumption holds naturally. Otherwise, one often makes an assumption that the level set is bounded. For unconstrained nonlinear equations system, M.Solodov designed a Newton method with projection technique, the method can generate a bounded iteration sequence without additional assumption and the global convergence is obtained. Motivated by the idea of M.Solodov [23], in this paper, we extend the method to constrained equations (1.1). By using this active set strategy, we only need to solve a linear system with reduced dimension at each iteration. The algorithm generates a bounded sequence automatically even if li and ui are infinite. We obtain the global convergence and give numerical tests to show the efficiency of the proposed algorithm.

    The paper is organized as follows: In section 2, we describe our algorithm in detail. In section 3, we prove the global convergence of the proposed algorithm. Some numerical tests are shown in Section 4 and a conclusion is given in section 5. Throughout this paper, we use to denote the 2norm and E denotes the identity matrix.

    We now describe our active set quasi-Newton method with projection technique in detail. To describe our algorithm, we introduce the definition of projection operator which is defined as a mapping form Rn to a nonempty closed convex subset Ω:

    PΩ(x)=argmin{yx|yΩ},xRn. (2.1)

    A well-known property of the operator is that it is nonexpensive, namely,

    PΩ(x)PΩ(y)xy,x,yRn. (2.2)

    Given a current iterate xk, let

    δk:=min{δ,c||F(xk)||},

    where δ and c are positive constants such that

    δ12mini=1,2,,n|uili|,

    and define the index sets

    Ak:={i{1,2,...,n}|xkilkδkoruixkiδk},
    Ik:={1,2,...,n}Ak={i|lk+δk<xki<uiδk}.

    The precise statement of our algorithm is as follows:

    Algorithm 2.1: (Active Set-type Quasi-Newton Method)

    (S.0) Choose a positive definite matrix Bk, x0[l,u], choose parameters β(0,1), λ(0,1), δ>0, c>0, ε>0, μk>0, and ρk[0,1), and set k:=0.

    (S.1) If F(xk)ε, stop.

    (S.2) Try to compute a vector dkRn in the following way:

    For iAk, set

    dki=Fi(xk)/(1ρk)μk. (2.3)

    For iIk, set solve the linear system

    (Bk+μkEIk)dki=Fi(xk)+ek, (2.4)

    where

    ekμkρkdki.

    (S.3) Find zk=xk+αkdk, where αk=βmk with mk being the smallest nonnegative integer m such that

    F(xk+βmdk),dkλ(1ρk)μkdk2. (2.5)

    (S.4) Compute

    xk+1=PΩ[xkF(zk),xkzk||F(zk)||2F(zk)]. (2.6)

    (S.5) Update Bk+1, set k:=k+1, go to (S.1).

    Just as mentioned in [13], throughout this paper, we assume that the parameter δ>0 is chosen sufficiently small such that

    δ12mini=1,2,,n|uili|.

    This implies that we cannot have xkiliδk and uixkiδk for the same index iAk.

    Our algorithm is somewhat different from the traditional active set Newton method as described in [13], where the search step dk in (S.2) is computed in the following formulas:

    For iAk, set

    dki={lixkiifxkiliδk,uixkiifuixkiδk. (2.7)

    For iIk, solve the linear system

    F(xk)IkIkdIk=F(xk)IkF(xk)IkAkdAk. (2.8)

    As described in [13], in order to understand the formula for the computation of the components dki for iIk, note that, after a possible permutation of the rows and columns, [13] rewrite the standard (unconstrained) Newton equation F(xk)d=F(xk) as

    (F(xk)IkIkF(xk)IkAkF(xk)AkIkF(xk)AkAk)(dIkdAk)=(F(xk)IkF(xk)Ak) (2.9)

    Here we replace (2.7) by (2.3) and (2.8) by (2.4), the main proposal is to guarantee that the inequality (2.5) holds. On the other hand, we compute dIk by (2.4) instead of (2.8) which can be seen as an inexact Newton method.

    The matrix Bk is updated by the well known rank two secant type formula updated by the well known BFGS formula

    Bk+1=BkBksksTkBksTkBksk+ykyTkyTksk, (2.10)

    where yk=F(xk+1)F(xk) and sk=xk+1xk.

    In the section, we prove the global convergence of Algorithm 2.1, we make the following assumption.

    Assumption:

    (A1) The function F(x) is Lipschitz continuous and monotone, i.e., there exists a positive constant L such that

    F(x)F(y)Lxy (3.1)

    and

    F(x)F(y),(xy)0,x,yΩ. (3.2)

    (A2) The sequence of matrices {Bk} is positive definite and bounded, i.e., there exists a positive constant κ such that Bkκ for all k.

    We first show that the algorithm is feasible, i.e., there exists a positive m such that (2.5) holds.

    Lemma 3.1. The Algorithm 2.1 is well defined.

    Proof. We prove that the inequality (2.5) will hold with a nonnegative integer m. Suppose that for some index k this is not the case, which means, for all integer m, we have

    F(xk+βmdk),dk<λ(1ρk)μkdk2. (3.3)

    We further get

    limmF(xk+βmdk),dk=F(xk),dk=FAk,dAkFIk,dIk=FAk2/(1ρk)μk+(Bk+μkEIk)dIkek,dIk(1ρk)μkdAk2+μkdIk2ekdIk(1ρk)μkdAk2+(1ρk)μkdIk2(1ρk)μkdk2. (3.4)

    Now we take the limit of both sides of (3.4) as m, when (3.4) holds which implies that λ1, which contradicts the choice of λ(0,1). Hence we have that the inequality holds for some integer m, and the whole algorithm is well defined.

    In what follows, we assume that the algorithm generates an infinite iteration sequence. The following result shows that the algorithm generates a bounded sequence automatically and the proof is similar to Lemma 3.2 in [24] and we omit it here.

    Theorem 3.2. Suppose assumptions (A1) and (A2) hold, sequences {xk} and {zk} are generated by Algorithm 2.1, then {xk} and {zk} are both bounded. Furthermore, for any ¯x such that F(¯x)=0, it holds that

    xk+1¯x2xk¯x2xk+1xk2. (3.5)
    limkxkzk=0. (3.6)

    and

    limkxk+1xk=0. (3.7)

    Now we give the global convergence result of the Algorithm 2.1.

    Lemma 3.3. Let {xk} be generated by Algorithm 2.1, assume Assumption (A1) and (A2) hold, and there exists constants 0<ρ_<¯ρ<1, and μ_<¯μ such that ρ_ρk¯ρ, and μ_μk¯μ. Then {xk} converges to some x such that F(x)=0.

    Proof. By the inequality (2.5), we have

    F(zk),xkzk=αkF(zk),dkλ(1ρk)μkαkdk2. (3.8)

    By the definition of dk, we have that

    dAk=FAk/(1ρk)μkFAk/(1¯ρ)μ_. (3.9)

    and

    FIk(Bk+μkEIk)dIkek(1ρk)μkdIk(1¯ρ)μ_dIk. (3.10)

    Combining (3.9) and (3.10), we can assume that there exists a positive constant c1 such that

    F(xk)c1dk. (3.11)

    On the other hand, the definition of dk also gives that

    FAk=(1ρk)μkdAk(1ρ_)¯μdAk. (3.12)

    From (2.4) and Assumption (A2), we have

    FIk(Bk+μkEIk)dIk+ek(κ+μk+ρkμk)dIk[κ+(1+¯ρ)¯μ]dIk. (3.13)

    Combining (3.12) and (3.13), we can assume that there exists a positive constant c2 such that

    F(xk)c2dk. (3.14)

    Now by (3.8), we obtain

    F(zk)xkzkF(zk),xkzkλ(1¯ρ)μ_αkdk2. (3.15)

    By the continuity of F(x), the bound of sequence {zk} and (3.6), we have

    limkαkdk2=0. (3.16)

    We consider the two possible cases:

    lim infkF(xk)=0andlim infkF(xk)>0. (3.17)

    In the first case, the continuity of F and the boundness of {xk} imply that the sequence {xk} has some accumulation point x such that F(x)=0. Since ¯x was an arbitrary solution, we can choose ¯x=x in (3.5). The sequence {xkx} converges and since x is an accumulation point of {xk}, it must be the case that {xk} converges to x.

    Now consider the second case. From (3.14), we have

    lim infkdk>0.

    Hence by (3.16), we have

    lim infkαk=0.

    (The following proof is very similar to the last part in Theorem 2.1 [23], for complement, we list it here.) By the step rule, we have the inequality (2.5) is not valid for the value βmk1, i.e.,

    F(xk+βmk1dk),dk<λ(1ρk)μkdk2 (3.18)

    Let k, we get

    F(x),d<λ(1ρ)μd2, (3.19)

    Here x, d, ρ, μ denote the limits of the corresponding sequence respectively. On the other hand, by (3.4), we get

    F(x),d(1ρ)μd2, (3.20)

    that contradicts the choice for λ(0,1). Hence the case lim infkF(xk) is impossible.

    This completes the proof.

    In this section, we demonstrate the numerical performance of Algorithm 2.1 (AQN) and its computational advantage by comparing with the modified Kanzow [13] ACTN method (denoted as AKP) and the classical Quasi-Newton method with project (denoted as CQN). All presented codes are written in MATLAB2019 and run on a PC with 3.30GHz CPU processor, 4.0GB memory and Windows 8 operation system.

    We consider ten problems with dimension n = 1000, 5000, 10000. We use six different starting points, that is:

    x1=(0.1,0.1,...,0.1)T,x2=(12,122,...,12n)T,x3=(2,2,...,2)T,x4=(1,12,...,1n)T,x5=(1,112,...,11n)T,x6=rand(0,1).

    After several parameter selection experiments, we select the initial parameters that can make the three algorithms have better performance :

    β=0.5,λ=0.6,δ=0.001,c=1,μk=0.5,ε=106,ρk=0.3.

    Set the terminating criterion for the iteration process as ||F(xk)||106. The problems are listed as follows.

    Problem 1. [25]

    Fi(x)=exi1,i=1,2,...,n, (4.1)

    where Ω=Rn+.

    Problem 2. [25]

    F1(x)=ex11,Fi(x)=exi+xi11,i=2,...,n, (4.2)

    where Ω=Rn+.

    Problem 3. [25]

    F1(x)=2x1x2+ex11,Fi(x)=xi1+2xixi+1+exi1,i=2,...,n1,Fn(x)=xn1+2xn+exn1, (4.3)

    where Ω=Rn+.

    Problem 4. [25]

    F1(x)=52x1+x21,Fi(x)=xi1+52xi+xi+11,i=2,...,n1,Fn(x)=xn1+52xn1, (4.4)

    where Ω=Rn+.

    Problem 5. [25]

    Fi(x)=exi+32sin(2xi)1,i=1,2,...,n, (4.5)

    where Ω=Rn+.

    Problem 6. [25]

    F1(x)=x1ecos(h(x1+x2)),Fi(x)=xiecos(h(xi1+xi+xi+1)),i=2,...,n1,Fn(x)=xnecos(h(xn1+xn)), (4.6)

    where h=1n+1 and Ω=Rn+.

    Problem 7. [25]

    Fi(x)=2xisin|xi|,i=1,2,...,n, (4.7)

    where Ω=Rn+.

    Problem 8. [26]

    Fi(x)=22xi1,i=1,2,...,n, (4.8)

    where Ω=Rn+.

    Problem 9. [26]

    Fi(x)=ex2i+3sinxicosxi1,i=1,2,...,n, (4.9)

    where Ω=Rn+.

    Problem 10. [24]

    Fi(x)=xisin(|xi1|),i=1,2,...,n, (4.10)

    where Ω=Rn+.

    Comprehensive results of our numerical experiment are presented in Tables 110. The columns of the presented tables have the following definitions:

    Table 1.  Numerical results for Problem 1.
    IP DIM AQN CQN AKP
    NI NF CPU NORM NI NF CPU NORM NI NF CPU NORM
    X1 1000 30 61 3.656 8.63E-07 22 45 9.419 6.82E-07 22 45 1.365 7.86E-07
    5000 32 65 123.832 7.97E-07 23 47 252.343 7.62E-07 23 47 94.233 8.78E-07
    10000 33 67 707.123 7.25E-07 24 49 1204.500 5.39E-07 24 49 801.382 6.21E-07
    X2 1000 28 57 5.164 9.12E-07 19 39 8.252 7.03E-07 18 37 1.256 8.22E-07
    5000 28 57 143.052 9.12E-07 19 39 199.614 7.03E-07 18 37 83.818 8.22E-07
    10000 28 57 779.363 9.12E-07 19 39 913.557 7.03E-07 18 37 609.232 8.22E-07
    X3 1000 34 69 5.087 7.49E-07 26 53 10.410 5.92E-07 27 55 2.115 5.01E-07
    5000 36 73 161.879 6.92E-07 27 55 288.600 6.62E-07 28 57 128.120 5.60E-07
    10000 36 73 853.569 9.79E-07 27 55 1328.800 9.36E-07 28 57 920.577 7.92E-07
    X4 1000 34 69 7.029 8.51E-07 20 41 7.855 9.44E-07 22 45 2.102 5.97E-07
    5000 35 71 208.186 6.74E-07 20 41 210.414 9.44E-07 22 45 100.805 5.97E-07
    10000 35 71 1078.500 9.74E-07 20 41 966.249 9.44E-07 22 45 721.224 5.97E-07
    X5 1000 34 69 5.434 9.47E-07 24 49 9.660 8.28E-07 25 51 1.683 8.35E-07
    5000 36 73 169.626 8.76E-07 25 51 264.494 9.26E-07 26 53 118.489 9.34E-07
    10000 37 75 907.287 7.96E-07 26 53 1266.100 6.55E-07 27 55 910.632 6.61E-07
    X6 1000 34 69 5.624 9.32E-07 24 49 9.475 8.07E-07 25 51 1.702 8.48E-07
    5000 36 73 169.128 8.70E-07 25 51 264.354 9.25E-07 26 53 119.666 9.25E-07
    10000 37 75 912.983 7.91E-07 26 53 1283.200 6.54E-07 27 55 916.142 6.61E-07

     | Show Table
    DownLoad: CSV
    Table 2.  Numerical results for Problem 2.
    IP DIM AQN CQN AKP
    NI NF CPU NORM NI NF CPU NORM NI NF CPU NORM
    X1 1000 60 121 10.393 9.69E-07 40 81 16.089 9.12E-07 45 91 8.107 7.83E-07
    5000 59 119 304.457 9.61E-07 40 81 427.665 8.75E-07 44 89 198.837 9.90E-07
    10000 59 119 1494.000 8.76E-07 40 81 2015.480 8.62E-07 44 89 1590.250 9.57E-07
    X2 1000 72 145 15.477 8.99E-07 46 93 18.481 9.61E-07 47 95 3.669 9.48E-07
    5000 72 145 439.560 8.99E-07 46 93 503.471 9.61E-07 47 95 216.065 9.48E-07
    10000 72 145 2181.400 8.99E-07 46 93 2308.030 9.61E-07 47 95 1613.963 9.48E-07
    X3 1000 76 153 16.627 8.33E-07 48 97 19.281 7.93E-07 40 81 2.786 8.10E-07
    5000 74 149 451.509 9.91E-07 48 97 519.860 7.65E-07 38 77 176.960 9.97E-07
    10000 74 149 2251.200 9.34E-07 48 97 2434.370 7.55E-07 38 77 1279.398 9.46E-07
    X4 1000 75 151 16.529 9.82E-07 48 97 19.329 8.67E-07 50 101 3.162 9.43E-07
    5000 75 151 472.387 9.78E-07 48 97 519.555 8.67E-07 50 101 229.550 9.43E-07
    10000 75 151 2346.100 9.76E-07 48 97 2408.623 8.67E-07 50 101 1657.715 9.43E-07
    X5 1000 74 149 15.728 9.03E-07 47 95 18.880 8.61E-07 50 101 3.553 8.40E-07
    5000 73 147 448.860 9.98E-07 47 95 509.295 8.29E-07 48 97 218.293 7.60E-07
    10000 73 147 2209.300 8.51E-07 47 95 2358.238 8.17E-07 48 97 1607.768 9.27E-07
    X6 1000 90 181 19.039 8.23E-07 56 113 22.527 8.90E-07 57 115 3.537 9.40E-07
    5000 94 189 563.484 8.70E-07 59 119 636.860 8.24E-07 62 125 285.941 7.93E-07
    10000 95 191 2835.600 1.00E-06 60 121 3044.774 8.69E-07 62 125 2105.600 9.51E-07

     | Show Table
    DownLoad: CSV
    Table 3.  Numerical results for Problem 3.
    IP DIM AQN CQN AKP
    NI NF CPU NORM NI NF CPU NORM NI NF CPU NORM
    X1 1000 93 187 19.750 9.30E-07 81 163 33.507 8.00E-07 44 89 3.338 8.25E-07
    5000 92 185 562.208 9.53E-07 84 169 918.799 8.97E-07 56 93 214.676 8.47E-07
    10000 99 199 3150.498 9.36E-07 87 175 4584.800 7.85e-07 48 97 1611.711 9.30E-07
    X2 1000 76 153 16.189 9.89E-07 61 123 25.470 8.65E-07 36 73 2.281 8.46E-07
    5000 76 153 463.761 9.89E-07 61 123 667.653 8.65E-07 36 73 164.605 8.46E-07
    10000 76 153 2325.900 9.89E-07 61 123 3055.546 8.65E-07 36 73 1190.508 8.46E-07
    X3 1000 121 243 31.734 9.47E-07 94 189 38.858 9.03E-07 57 115 3.676 6.93E-07
    5000 106 213 697.003 8.87E-07 100 201 1093.100 8.52E-07 63 127 288.233 6.82E-07
    10000 112 225 3761.469 9.95E-07 101 203 5055.167 8.92E-07 - - - -
    X4 1000 87 175 19.633 9.94E-07 72 145 29.727 8.19E-07 41 83 3.892 7.88E-07
    5000 88 177 569.847 9.97E-07 72 145 791.423 8.20E-07 41 83 186.841 7.94E-07
    10000 88 177 2817.300 9.81E-07 72 145 3597.463 8.20E-07 41 83 1368.882 7.94E-07
    X5 1000 109 219 25.967 9.32E-07 89 179 37.222 8.30E-07 53 107 3.714 9.84E-07
    5000 116 233 788.855 8.80E-07 93 187 1035.100 8.54E-07 57 115 263.580 8.93E-07
    10000 117 235 3934.500 9.97E-07 95 191 4746.173 7.82E-07 - - - -
    X6 1000 107 215 25.975 9.97E-07 95 191 39.617 7.52E-07 53 107 3.527 9.08E-07
    5000 116 233 783.719 8.42E-07 98 197 1090.700 9.45E-07 55 111 255.271 9.85E-07
    10000 120 241 4148.100 9.15E-07 101 203 5045.600 7.46E-07 58 117 1955.841 7.26E-07

     | Show Table
    DownLoad: CSV
    Table 4.  Numerical results for Problem 4.
    IP DIM AQN CQN AKP
    NI NF CPU NORM NI NF CPU NORM NI NF CPU NORM
    X1 1000 96 193 39.937 9.05E-07 96 193 43.074 9.05E-07 62 125 6.113 8.47E-07
    5000 97 195 1080.500 7.73E-07 97 195 1085.200 7.73E-07 64 129 292.586 8.46E-07
    10000 96 193 4915.800 7.99E-07 96 193 4809.600 7.99E-07 66 133 2193.345 9.51E-07
    X2 1000 96 193 39.898 7.99E-07 94 189 38.846 8.52E-07 76 153 4.878 8.00E-07
    5000 94 189 1049.800 8.95E-07 93 187 1015.018 9.44E-07 - - - -
    10000 94 189 4742.816 9.23E-07 92 185 4602.733 9.80E-07 - - - -
    X3 1000 75 151 31.182 8.40E-07 75 151 30.961 8.40E-07 66 133 4.018 9.87E-07
    5000 73 147 802.568 9.18E-07 73 147 796.533 9.18E-07 73 147 337.588 6.90E-07
    10000 75 151 3808.900 7.44E-07 75 151 3748.947 7.44E-07 75 151 2478.365 8.36E-07
    X4 1000 90 181 37.247 9.77E-07 93 187 38.923 8.42E-07 66 133 4.171 9.87E-07
    5000 93 187 1025.700 8.81E-07 92 185 1005.456 9.24E-07 76 153 348.842 9.56E-07
    10000 93 187 4705.600 9.01E-07 92 185 4622.813 9.17E-07 79 159 2634.719 8.62E-07
    X5 1000 91 183 37.749 8.51E-07 93 187 38.567 8.93E-07 70 141 4.794 9.67E-07
    5000 91 183 1002.900 8.29E-07 92 185 1004.607 9.91E-07 75 151 348.323 9.69E-07
    10000 110 221 5628.200 7.64E-07 94 189 4710.946 7.83E-07 75 151 2492.402 9.22E-07
    X6 1000 142 285 58.851 8.81E-07 143 287 60.753 9.66E-07 79 159 7.068 9.92E-07
    5000 151 303 1669.800 8.98E-07 151 303 1653.700 8.47E-07 85 171 393.990 7.41E-07
    10000 154 309 7856.500 9.98E-07 154 309 7653.971 9.64E-07 86 173 2869.170 8.31E-07

     | Show Table
    DownLoad: CSV
    Table 5.  Numerical results for Problem 5.
    IP DIM AQN CQN AKP
    NI NF CPU NORM NI NF CPU NORM NI NF CPU NORM
    X1 1000 11 23 2.351 7.06E-07 11 23 5.609 7.06E-07 30 61 7.252 7.27E-07
    5000 12 25 69.706 1.55E-07 12 25 67.909 1.55E-07 30 61 138.354 9.03E-07
    10000 12 25 352.637 2.19E-07 12 25 348.586 2.19E-07 35 71 1172.793 6.11E-07
    X2 1000 12 25 2.984 9.73E-07 12 25 2.678 6.04E-07 61 123 4.507 7.65E-07
    5000 13 27 80.610 2.13E-07 13 27 79.346 1.33E-07 - - - -
    10000 13 27 405.754 3.02E-07 13 27 398.015 1.88E-07 - - - -
    X3 1000 12 25 2.017 4.61E-07 12 25 1.936 4.61E-07 - - - -
    5000 13 27 61.161 1.82E-07 13 27 60.556 1.82E-07 - - - -
    10000 13 27 323.124 2.58E-07 13 27 319.935 2.58E-07 - - - -
    X4 1000 16 33 4.417 1.21E-07 13 27 3.303 3.32E-07 49 99 3.172 9.84E-07
    5000 14 29 93.207 1.19E-07 13 27 88.261 7.42E-07 65 131 298.947 9.85E-07
    10000 14 29 454.358 1.67E-07 13 27 397.148 1.87E-07 63 127 2097.393 7.91E-07
    X5 1000 17 35 4.812 2.09E-07 13 27 2.720 1.52E-07 59 119 4.057 7.74E-07
    5000 17 35 133.735 4.68E-07 13 27 79.443 3.40E-07 60 121 277.731 9.35E-07
    10000 17 35 659.338 6.62E-07 13 27 396.954 4.81E-07 62 125 2093.674 7.82E-07
    X6 1000 13 27 2.824 1.63E-07 13 27 2.754 1.51E-07 59 119 4.104 8.52E-07
    5000 17 35 134.016 3.29E-07 13 27 79.329 3.39E-07 60 121 297.163 8.30E-07
    10000 17 35 651.481 6.60E-07 13 27 396.469 4.82E-07 59 119 1976.682 8.91E-07

     | Show Table
    DownLoad: CSV
    Table 6.  Numerical results for Problem 6.
    IP DIM AQN CQN AKP
    NI NF CPU NORM NI NF CPU NORM NI NF CPU NORM
    X1 1000 49 99 20.419 7.13E-07 42 85 19.977 8.74E-07 28 57 7.443 9.06E-07
    5000 50 101 549.836 9.05E-07 43 87 484.930 9.65E-07 30 61 136.093 6.79E-07
    10000 51 103 2623.200 8.85E-07 44 89 2252.977 8.71E-07 30 61 1008.218 9.29E-07
    X2 1000 48 97 20.223 6.99E-07 42 85 17.454 9.11E-07 29 59 1.842 9.19E-07
    5000 51 103 560.601 7.23E-07 44 89 491.333 6.51E-07 32 65 149.412 5.94E-07
    10000 51 103 2607.600 8.64E-07 44 89 2251.287 9.07E-07 35 71 1172.464 5.23E-07
    X3 1000 39 79 16.445 7.27E-07 39 79 16.669 7.27E-07 27 55 1.868 7.22E-07
    5000 40 81 440.412 7.64E-07 40 81 439.550 7.64E-07 27 55 123.541 8.95E-07
    10000 41 83 2082.900 7.31E-07 41 83 2039.955 7.31E-07 28 57 934.110 5.24E-07
    X4 1000 48 97 20.053 7.21E-07 42 85 17.886 9.11E-07 28 57 3.648 9.14E-07
    5000 49 99 540.145 7.74E-07 44 89 490.692 8.50E-07 31 63 148.501 6.77E-07
    10000 51 103 2582.600 7.23E-07 44 89 2264.235 6.61E-07 31 63 1213.002 7.78E-07
    X5 1000 43 87 18.038 7.87E-07 42 85 17.649 9.23E-07 28 57 3.153 5.72E-07
    5000 44 89 483.664 9.58E-07 43 87 489.753 9.44E-07 29 59 151.122 8.58E-07
    10000 45 91 2279.000 9.44E-07 45 91 2273.992 5.95E-07 30 61 1025.754 5.67E-07
    X6 1000 44 89 18.394 9.10E-07 42 85 17.362 9.04E-07 28 57 2.144 9.01E-07
    5000 50 101 550.710 8.75E-07 43 87 491.334 9.04E-07 29 59 132.839 8.87E-07
    10000 51 103 2599.800 8.75E-07 44 89 2265.719 9.95E-07 30 61 1022.843 7.92E-07

     | Show Table
    DownLoad: CSV
    Table 7.  Numerical results for Problem 7.
    IP DIM AQN CQN AKP
    NI NF CPU NORM NI NF CPU NORM NI NF CPU NORM
    X1 1000 30 61 3.984 9.52E-07 22 45 9.309 7.52E-07 22 45 10.004 7.55E-07
    5000 32 65 124.936 8.79E-07 23 47 257.722 8.41E-07 23 47 103.523 8.44E-07
    10000 33 67 696.347 7.99E-07 24 49 1205.700 5.95E-07 24 49 850.861 5.97E-07
    X2 1000 30 61 8.726 8.29E-07 20 41 10.117 5.27E-07 20 41 4.586 5.67E-07
    5000 30 61 180.592 8.29E-07 20 41 225.843 5.27E-07 20 41 95.335 5.67E-07
    10000 30 61 886.356 8.29E-07 20 41 1006.400 5.27E-07 20 41 663.478 5.67E-07
    X3 1000 35 71 5.263 6.87E-07 26 53 10.435 8.45E-07 25 51 1.791 9.09E-07
    5000 36 73 160.033 9.88E-07 27 55 286.293 9.45E-07 27 55 123.108 5.08E-07
    10000 37 75 862.174 8.98E-07 28 57 1366.000 6.68E-07 27 55 894.793 7.19E-07
    X4 1000 34 69 7.383 8.53E-07 21 43 8.777 5.35E-07 21 43 1.328 6.68E-07
    5000 36 73 224.577 7.58E-07 21 43 230.923 5.35E-07 21 43 96.340 6.68E-07
    10000 35 71 886.356 8.29E-07 21 43 1051.442 5.35E-07 21 43 693.062 6.68E-07
    X5 1000 35 71 5.908 7.47E-07 24 49 10.020 9.58E-07 25 51 1.524 5.81E-07
    5000 37 75 180.603 6.89E-07 26 53 284.448 5.36E-07 26 53 119.156 6.50E-07
    10000 37 75 944.657 9.75E-07 26 53 1318.300 7.58E-07 26 53 861.209 9.19E-07
    X6 1000 35 71 5.919 7.63E-07 24 49 10.713 9.75E-07 25 51 6.592 5.83E-07
    5000 37 75 180.124 6.98E-07 26 53 284.553 5.40E-07 26 53 119.424 6.50E-07
    10000 37 75 934.966 9.82E-07 26 53 1310.800 7.60E-07 26 53 913.455 9.17E-07

     | Show Table
    DownLoad: CSV
    Table 8.  Numerical results for Problem 8.
    IP DIM AQN CQN AKP
    NI NF CPU NORM NI NF CPU NORM NI NF CPU NORM
    X1 1000 14 29 5.802 7.75E-07 14 29 5.782 7.75E-07 24 49 1.915 7.92E-07
    5000 15 31 165.321 5.08E-07 15 31 164.148 5.08E-07 27 55 124.365 8.29E-07
    10000 15 31 756.748 7.18E-07 15 31 751.927 7.18E-07 28 57 933.741 6.67E-07
    X2 1000 15 31 7.203 8.97E-07 15 31 6.145 3.16E-07 40 81 2.560 7.75E-07
    5000 16 33 176.418 5.87E-07 15 31 164.099 7.08E-07 - - - -
    10000 16 33 816.974 8.31E-07 16 33 802.254 2.93E-07 - - - -
    X3 1000 16 33 6.622 4.32E-07 16 33 6.548 4.32E-07 27 55 1.773 8.73E-07
    5000 16 33 177.685 9.66E-07 16 33 174.795 9.66E-07 31 63 143.292 6.08E-07
    10000 17 35 862.303 4.00E-07 17 35 851.777 4.00E-07 31 63 1043.391 8.67E-07
    X4 1000 18 37 7.461 3.10E-07 15 31 6.472 3.12E-07 28 57 1.923 8.04E-07
    5000 16 33 178.233 6.05E-07 15 31 163.715 7.05E-07 41 83 189.831 9.66E-07
    10000 16 33 814.770 8.43E-07 15 31 750.481 9.99E-07 42 85 1412.292 7.38E-07
    X5 1000 18 37 7.470 2.97E-07 14 29 5.710 9.89E-07 28 57 2.999 7.11E-07
    5000 18 37 197.939 6.65E-07 15 31 163.855 6.48E-07 29 59 132.820 7.97E-07
    10000 18 37 913.455 9.40E-07 15 31 751.494 9.17E-07 30 61 1030.916 5.65E-07
    X6 1000 18 37 7.583 2.97E-07 15 31 6.453 2.96E-07 28 57 1.812 7.13E-07
    5000 18 37 199.224 4.30E-07 15 31 163.758 6.42E-07 29 59 132.158 8.65E-07
    10000 19 39 962.107 2.93E-07 15 31 756.324 9.11E-07 29 59 966.698 6.08E-07

     | Show Table
    DownLoad: CSV
    Table 9.  Numerical results for Problem 9.
    IP DIM AQN CQN AKP
    NI NF CPU NORM NI NF CPU NORM NI NF CPU NORM
    X1 1000 38 77 3.055 8.24E-07 12 25 4.959 5.05E-07 23 47 3.828 5.40E-07
    5000 40 81 113.409 9.88E-07 13 27 143.222 2.82E-07 24 49 109.381 6.04E-07
    10000 42 85 712.896 7.49E-07 13 27 651.289 3.99E-07 24 49 802.195 8.54E-07
    X2 1000 37 75 4.855 8.16E-07 11 23 4.708 3.37E-07 21 43 1.338 7.99E-07
    5000 37 75 153.719 8.16E-07 11 23 120.945 3.37E-07 21 43 94.816 7.99E-07
    10000 37 75 835.797 8.16E-07 11 23 555.333 3.37E-07 21 43 699.885 7.99E-07
    X4 1000 44 89 6.384 8.49E-07 12 25 6.552 5.24E-07 25 51 1.755 5.99E-07
    5000 45 91 187.263 9.32E-07 12 25 132.641 5.24E-07 25 51 114.659 6.45E-07
    10000 47 95 1195.000 9.22E-07 12 25 607.4983 5.24E-07 25 51 834.238 6.56E-07
    X5 1000 43 87 5.098 9.20E-07 14 29 6.816 3.27E-07 31 63 3.127 6.74E-07
    5000 46 93 171.163 8.08E-07 14 29 156.394 7.32E-07 32 65 146.725 7.64E-07
    10000 47 95 967.804 8.34E-07 15 31 764.984 2.59E-07 33 67 1127.503 5.43E-07
    X6 1000 43 87 5.095 9.54E-07 14 29 7.562 3.29E-07 31 63 2.035 6.66E-07
    5000 46 93 170.719 8.49E-07 14 29 157.247 7.28E-07 32 65 152.282 7.68E-07
    10000 47 95 924.105 8.18E-07 15 31 763.091 2.57E-06 33 67 1103.404 5.36E-07

     | Show Table
    DownLoad: CSV
    Table 10.  Numerical results for Problem 10.
    IP DIM AQN CQN AKP
    NI NF CPU NORM NI NF CPU NORM NI NF CPU NORM
    X1 1000 28 57 11.625 5.43E-07 28 57 11.710 5.43E-07 25 51 2.226 6.68E-07
    5000 29 59 323.043 6.46E-07 29 59 316.952 6.46E-07 26 53 118.148 7.47E-07
    10000 29 59 1476.900 9.13E-07 29 59 1452.863 9.13E-07 27 55 895.844 5.28E-07
    X2 1000 25 51 10.099 8.50E-07 25 51 10.236 8.50E-07 30 61 1.998 5.30E-07
    5000 27 55 290.044 5.38E-07 27 55 285.598 5.38E-07 - - - -
    10000 27 55 1328.900 7.61E-07 27 55 1312.528 7.61E-07 - - - -
    X3 1000 28 57 11.242 7.69E-07 28 57 11.411 7.69E-07 26 53 1.740 9.50E-07
    5000 29 59 311.758 9.14E-07 29 59 310.210 9.14E-07 28 57 126.392 5.31E-07
    10000 30 61 1482.600 6.88E-07 30 61 1469.212 6.88E-07 28 57 931.480 7.51E-07
    X4 1000 25 51 10.069 8.57E-07 25 51 10.064 8.57E-07 28 57 1.864 5.98E-07
    5000 27 55 290.624 5.38E-07 27 55 287.511 5.38E-07 29 59 131.109 7.84E-07
    10000 27 55 1331.600 7.61E-07 27 55 1316.705 7.61E-07 30 61 999.476 6.92E-07
    X5 1000 27 55 11.246 6.81E-07 27 55 11.420 6.81E-07 27 55 1.838 9.05E-07
    5000 28 57 309.842 8.10E-07 28 57 309.207 8.10E-07 29 59 135.589 8.78E-07
    10000 29 59 1472.000 6.10E-07 29 59 1471.858 6.10E-07 30 61 1043.962 6.19E-07
    X6 1000 27 55 11.266 6.71E-07 27 55 11.374 6.79E-07 27 55 2.818 6.93E-07
    5000 28 57 310.428 8.08E-07 28 57 308.040 8.07E-07 29 59 147.317 8.16E-07
    10000 29 59 1460.100 6.10E-07 29 59 1459.546 6.08E-07 30 61 1177.273 5.73E-07

     | Show Table
    DownLoad: CSV

    IP: the initial points.

    DIM: the dimension of the problem.

    NI: the iterative number.

    NF: the iterative number of function evaluation.

    CPU: the CPU time in seconds when the algorithm terminate.

    NORM: the final norm equation.

    We denote result by '' whenever the number of iterations exceeds 500 or the terminating criterion has not been satisfied. Among these results, none of the three methods were able to solve Problem 9 when initial point is x3=(2,2,...,2)T. Therefore, Table 9 does not include the case when the initial point is x3. Meanwhile, in the drawing process, when the result was denoted by '', its NI, NF, CPU and NORM are counted as .

    The performance of the three methods was evaluated using the performance profile which is presented by Dolan and Moré [27]. We comparing three methods with the same problem, dimension and initial point in an experiment, and recoding information of interest such as NI, NF, CPU and NORM.

    We denote the set of problems as P and the set of methods as M. For example, for each problem p and method m, we define

    tp,m=CPUtimerequiredtosolveproblempbymethodm. (4.11)

    Compare the performance on problem p by method m with the best performance by any method on this problem, that is, we use the performance ratio

    rp,m=tp,mmin{tp,m:mM}. (4.12)

    We assume that a parameter Rrp,m for all p,m is chosen, and rp,m=R if and only if method m does not solve problem p. If method m can solve problem p successfully, we obtain an overall assessment of the performance between these methods. It can be described as follows:

    ρm(τ):=1npsize{pP:rp,mτ}

    where np represents the number of elements in set P, then ρm(τ) is the probability for method mM that a performance ratio rp,m is within a factor τR of the best possible ratio. The function ρm is the (cumulative) distribution function for the performance ratio.

    The performance profile ρm:R[0,1] for a method is a nondecreasing, piecewise constant function, continuous from the right at each breakpoint. We are interested in methods with a high probability of solve success, then we need only to compare the values of ρm(τ) for all of the methods and choose the method with the largest, there means that we need to find which method's function ρm first rearch the line ρm(τ)=1. In the same way, we can obtain the performance profile with respect to NI, NF and NORM.

    As can be seen from the information in the Table 110, AQN has more stable solving performance and can solve more problems, such as what AKP cannot solve: x3 and x5 of Problem 3 when n=10000; x2 of Problem 3 when n=5000,10000; x2 of Problem 5 when n=1000,5000,10000; x3 of Problem 5 when n=5000,10000; x2 of Problem 8 when n=5000,10000; x2 of Problem 10 when n=5000,10000. Compared with CQN, from Figure 1 and Figure 2, we can see that AQN reaches the line that ρm(1)=1 before CQN, which demonstrates AQN has a faster solution time(CPU) and the solution results of the final norm equation(NORM) are more accurate. Although from Figure 3 and Figure 4, there shows the iterative number of CQN is less than AQN, in practice, we pay more attention to the advantage of solution time. To sum up, AQN has a more stable and faster solving performance.

    Figure 1.  Performance profiles based on CPU time.
    Figure 2.  Performance profiles based on the final norm equation.
    Figure 3.  Performance profiles based on number of iterations.
    Figure 4.  Performance profiles based on number of function evaluation.

    In this paper, we propose an active set quasi-Newton method for the solution of optimization problem with bound constraints. The implementation of the method uses the quasi-Newton step as a trial step and the project step as the correction step. By using active set technique, we only need to solve a reduced dimension linear equation at each iteration to generate the search direction. We prove that the generated sequence is bounded automatically and obtain the global convergence of the proposed algorithm. Meanwhile, compared with other algorithms, our method has the most stable performance. There are some questions that need studying in the near future. Firstly, it is possible to get the global convergence of the proposed algorithm without the assumption of the positive definite of the matrix Bk. Secondly, how to get the local convergence of the proposed algorithm especially under some weak condition such as the local error bound condition needs further studying.

    All authors declare no conflicts of interest in this paper.



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