Loading [MathJax]/jax/element/mml/optable/GeneralPunctuation.js
Research article

Modified nonmonotonic projection Barzilai-Borwein gradient method for nonnegative matrix factorization

  • Received: 24 November 2023 Revised: 09 April 2024 Accepted: 23 April 2024 Published: 15 July 2024
  • MSC : 15A23, 65F30

  • In this paper, an active set recognition technique is suggested, and then a modified nonmonotonic line search rule is presented to enhance the efficiency of the nonmonotonic line search rule, in which we introduce a new parameter formula to attempt to control the nonmonotonic degree of the line search, and thus improve the chance of discovering the global minimum. By using a modified linear search and an active set recognition technique, a global convergence gradient solution for nonnegative matrix factorization (NMF) based on an alternating nonnegative least squares framework is proposed. We used a Barzilai-Borwein step size and greater step-size tactics to speed up the convergence. Finally, a large number of numerical experiments were carried out on synthetic and image datasets, and the results showed that our presented method was effective in calculating the speed and solution quality.

    Citation: Xiaoping Xu, Jinxuan Liu, Wenbo Li, Yuhan Xu, Fuxiao Li. Modified nonmonotonic projection Barzilai-Borwein gradient method for nonnegative matrix factorization[J]. AIMS Mathematics, 2024, 9(8): 22067-22090. doi: 10.3934/math.20241073

    Related Papers:

    [1] Gafurjan Ibragimov, Omongul Egamberganova, Idham Arif Alias, Shravan Luckraz . On some new results in a pursuit differential game with many pursuers and one evader. AIMS Mathematics, 2023, 8(3): 6581-6589. doi: 10.3934/math.2023332
    [2] Hamidou Tembine . Mean-field-type games. AIMS Mathematics, 2017, 2(4): 706-735. doi: 10.3934/Math.2017.4.706
    [3] Haishu Lu, Xiaoqiu Liu, Rong Li . Upper semicontinuous selections for fuzzy mappings in noncompact WPH-spaces with applications. AIMS Mathematics, 2022, 7(8): 13994-14028. doi: 10.3934/math.2022773
    [4] Jun Moon . The Pontryagin type maximum principle for Caputo fractional optimal control problems with terminal and running state constraints. AIMS Mathematics, 2025, 10(1): 884-920. doi: 10.3934/math.2025042
    [5] Ebrahim. A. Youness, Abd El-Monem. A. Megahed, Elsayed. E. Eladdad, Hanem. F. A. Madkour . Min-max differential game with partial differential equation. AIMS Mathematics, 2022, 7(8): 13777-13789. doi: 10.3934/math.2022759
    [6] Zifu Fan, Kexin Fan, Wei Zhang . Research on the data integration strategy of local governments and enterprises under central government subsidies. AIMS Mathematics, 2022, 7(6): 10143-10164. doi: 10.3934/math.2022564
    [7] Jun Wang, Dan Wang, Yuan Yuan . Research on low-carbon closed-loop supply chain strategy based on differential games-dynamic optimization analysis of new and remanufactured products. AIMS Mathematics, 2024, 9(11): 32076-32101. doi: 10.3934/math.20241540
    [8] Bei Yuan . Mathematical modeling of intellectual capital asymmetric information game in financial enterprises. AIMS Mathematics, 2024, 9(3): 5708-5721. doi: 10.3934/math.2024277
    [9] Jun Moon . A Pontryagin maximum principle for terminal state-constrained optimal control problems of Volterra integral equations with singular kernels. AIMS Mathematics, 2023, 8(10): 22924-22943. doi: 10.3934/math.20231166
    [10] Regina S. Burachik, Bethany I. Caldwell, C. Yalçın Kaya . Douglas–Rachford algorithm for control- and state-constrained optimal control problems. AIMS Mathematics, 2024, 9(6): 13874-13893. doi: 10.3934/math.2024675
  • In this paper, an active set recognition technique is suggested, and then a modified nonmonotonic line search rule is presented to enhance the efficiency of the nonmonotonic line search rule, in which we introduce a new parameter formula to attempt to control the nonmonotonic degree of the line search, and thus improve the chance of discovering the global minimum. By using a modified linear search and an active set recognition technique, a global convergence gradient solution for nonnegative matrix factorization (NMF) based on an alternating nonnegative least squares framework is proposed. We used a Barzilai-Borwein step size and greater step-size tactics to speed up the convergence. Finally, a large number of numerical experiments were carried out on synthetic and image datasets, and the results showed that our presented method was effective in calculating the speed and solution quality.



    In the literature, there are many works that are concern with the study of differential game problem in the Hilbert spaces such as Rn and 2 (see, for example [5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30]). In particular, the space 2, consists of elements of the form α=(α1,α2,), such that k=1α2k<. The inner product and norm are defined as follows:

    α,β=k=1αkβk, ||ϱ||=(k=1ϱ2k)1/2,

    where ,α,β,ϱ2.

    The works cited above can be categorized into three groups. The first group are the woks [1,6,10,12,14,16,17,20,25,26,27,28,29] and [30] concern with finding conditions for completion of pursuit. Whereas, the works [13,16,17,23,27] and [28] are concern with evasion problems. Problems of finding the value of the game and construction of optimal strategies of the players are considered in [5,7,8,9,11,15,18,22] and [24]. The last category is of special interest in this research.

    Among the last category, the problems considered in the papers [1,9,11,18,22] and [24] involve first order differential equations describing the motion of the players. On the other hand, problems involving second order differential equations are contained in [5,7] and [1]. The works [1] and [29] in the first category, are concern with problems in which dynamic equations of the players given as a combination of first and second order differential equations.

    In another point of view, the references can be grouped into two. Those works with pursuer's and evader's dynamic equations are given as differential equations of the same order in one group. These works include [5,6,7,8,9,10,11,12,13,15,17,18] and [22,23,24,25,26,27,28,29,30]. The other group consists of the works [1] and [29] in which pursuer's and evader's dynamic equations are given as differential equations of different orders.

    In [7], Ibragimov and Salimi investigated a differential game with infinitely many inertial players with integral constraints on the control functions of the players. Players' motion described by second order differential equations in the Hilbert space 2. Duration of the game is fixed. Payoff functional is the infimum of the distance between the evader and the pursuers when the game is terminated. The pursuer's goal is to minimize the payoff, and the evader's goal is to maximize it. Under certain condition, the value of the game is found and the optimal strategies of players are constructed. Subsequently, this result was improved and reported in the paper [15].

    Ibragimov and Kuchkarov in [8] examined pursuit-evasion differential game of countably many pursuers and one evader. Players move according to first order equations. Control of the pursuers and evader are subject to integral restrictions. The duration of the game is fixed and the payoff functional is the infimum of the distance between the evader and the pursuers when the game is terminated. The pursuer's goal is to minimize the payoff, and the evader's goal is to maximize it. Optimal strategies of the players are constructed, and value of the game is found.

    Inertial Pursuit-evasion game with a finite or countable number of pursuers and evader in Hilbert space 2 was studied by Ibragimov et al. in [13]. Dynamic of the players is described by second order differential equations in a Hilbert space. Control of the pursuers and evader are subject to integral restrictions. The period of the game is fixed. They formulated the value of the game and identified explicitly optimal strategies of the players. They assumed that there is no relation between the control resource of any pursuer and that of the evader.

    Salimi, M. and Ferrara, M. in [24] studied differential game in which a finite or countable number of pursuers pursue a single evader. Game is described by an infinite system of differential equation of first order in Hilbert space. The control function of the players satisfy the integral constraints. The period of the game is pre-defined. The farness between the evader and the closest pursuer when the game is finished is the payoff function of the game. They introduce the value of the game and identify optimal strategies of the pursuers.

    The game of boy and crocodile is studied by Siddiqova et. al. in [29]. The boy(evader) moves according to first order differential equation with its control subject to geometric constraint and is not allowed to leave a closed ball in Rn during the game. The pursuer(crocodile) moves according to second order differential equation with integral constraint on its control function. Sufficient conditions for completion of pursuit from all initial positions are obtained.

    A game problem in which pursuers move according to first order differential equation and evader moves according to second order differential equation is studied in [1]. Control function of pursuers and evader are subject to integral and geometric constraints respectively. Theorems each of which provides sufficient conditions for completion of pursuit are stated and proved.

    There are several other pursuit-evasion scenarios involving multiple pursuers and one evader studied in the literature, with dynamic equations not necessarily in the aforementioned groups. For instance; scenarios involving attackers, defenders and one evader are considered in [3,4], scenarios where the multiple pursuers and evader are constrained within a bounded domain are analyzed in [2,19,21,31,32].

    In view of the literature presented above and the references therein, there is no study that belongs to the third category and deals with the problem in which dynamic equations of the players are given as combination of first and second order differential equations. That is, problem of finding values of the game and construction of optimal strategies of the players with players having dissimilar laws of motion. Therefore, the need to contribute to the literature in that direction.

    In the present paper, we study pursuit-evasion differential game problem in a Hilbert space 2, in which motion of the pursuers and evader described by first and second order differential equations respectively. Control functions of both the pursuers and evader are subject to integral constraints.

    In the space 2, we define a ball with center at a and radius r by B(a,r)={x2:||xa||r}, and a sphere with center at a and radius r by S(a,r)={x2:||xa||=r}. Control function of the ith pursuer Pi, is the function ui()2, with Borel measurable coordinates uk:[0,θ]R1. In a similar way, we define control function v() of the evader E.

    Suppose that motion of the players is described by the equations:

    {Pi:˙xi=ui(t),  xi(0)=xi0,iI,E:¨y= v(t),  ˙y(0)=y1, y(0)=y0, (2.1)

    where xi,xi0,ui,y,y0,y1,v2, ui=(ui1,ui2,) and v=(v1,v2,) are control functions of the ith pursuer Pi and evader E respectively. It is required that the control functions ui() and v() satisfy the inequalities

    θ0||ui(t)||2dtρ2i, (2.2)
    θ0||v(t)||2dtσ2, (2.3)

    where ρi, iI={1,2,} and σ are given positive numbers. The stoppage time of the game is fixed and is denoted by θ. In what follows in the paper, we shall refer to the control of the ith pursuer ui() satisfying the inequality (2.2) and that of the evader v() satisfying the inequality (2.3) as admissible control of the ith pursuer and admissible control of the evader respectively.

    Definition 2.1. A function Ui(t,xi,y,v(t)), Ui:[0,)×2×2×22, such that the system

    {˙xi=Ui(t,x,y,v(t)),    xi(0)=xi0,¨y= v(t),   ˙y(0)=y1,   y(0)=y0 (2.4)

    has a unique solution (xi(), y()) with continuous components xi(), y() in 2, for an arbitrary admissible control v=v(t), 0tθ, of the evader E is called strategy of the pursuer Pi. A strategy Ui is said to be admissible if each control formed by this strategy is admissible.

    Definition 2.2. Strategies ˉUi of the pursuers Pi are said to be best (optimal) if

    infU1,,Un,Γ1(U1,,Un,)=Γ1(ˉU1,ˉU2,,ˉUm,) (2.5)

    where

    Γ1(U1,U2,,Um,,):=supv()infiI||xi(θ)y(θ)||, (2.6)

    and Ui are admissible strategies of the pursuers Pi and v() is an admissible control of the evader E.

    Definition 2.3. A function V(t,xi,,xm,,y), V:[0,)×2×,,×2×,,×22, such that the system

    {˙xi=ui,    xi(0)=xi0,¨y= V(t,x1,,xm,y),   ˙y(0)=y1,    y(0)=y0 (2.7)

    has a unique solution (x1(),,xm(),,y()) with continuous components x1(),x2(),,y() in 2, for an arbitrary admissible control ui=ui(t), 0tθ, of the pursuers Pi, is called strategy of the evader E. A strategy V is said to be admissible if each control formed by this strategy is admissible.

    Definition 2.4. Strategy ˉV of the evader E is said to be best (optimal) if

    supVΓ2(V)=Γ2(ˉV) (2.8)

    where

    Γ2(V):=infu1(),,um(),infiI||xi(θ)y(θ)||, (2.9)

    and ui() are admissible control of the pursuers Pi and V is an admissible strategy of the evader E.

    In the paper [9], it is reported that the game has the value ϕ if

    Γ1(¯U1,¯U2,,¯Un,)=ϕ=Γ2(ˉV). (2.10)

    Research question: What is the game value for the game problem (2.1)- (2.3)?

    If the pursuer Pi and evader E choose their admissible controls ui(t)=(ui1(t),ui2(t),) and v(t)=(v1(t),v2(t),) respectively, then by (2.1) their corresponding motion is given by

    xi(t)=(xi1(t),xi2(t),xik(t),),   y(t)=(y1(t),y2(t),,yk(t),). (2.11)

    where

    xik(t)=xi0k+t0uik(s)ds, yk(t)=y0k+ty1k+t0s0vk(r)drds.

    It is clear that x() and y() belongs to the space of continuous functions in the norm of 2 for 0tθ, where the component xk(t) and yk(t) are absolutely continuous.

    Proposition 2.5. The attainability domain of

    (a) the pursuers Pi from the initial state xi0 at time t0=0 is the ball HPi(xi0,ρiθ).

    (b) the evader E from the initial state y0 at time t0=0 is the ball HE(y0,σ(θ33)12).

    Proof. For the proof of (a), using (2.11) we have

    ||xi(θ)xi0||=||xi0+θ0ui(s)dsxi0||=||θ0ui(s)ds||θ0||ui(s)||ds(θ012ds)12(θ0||ui(s)||2ds)12ρiθ.

    Let ˉxHPi(xi0,ρiθ). If the pursuer Pi uses the control

    ui(t)=ˉxxi0θ.

    Then we have

    xi(θ)=xi0+θ0ui(t)dt=xi0+θ0ˉxxi0θdt=xi0+ˉxxi0=ˉx

    The prove of (b) is similar, for the evader's control v(t)=3(θt)θ3(ˉyy0), 0tθ.

    In this section, we consider a game with a single pursuer and a single evader by dropping the index i in the game problem (2.1)-(2.3). In view of this, we have the solutions of the dynamic equations of the players (2.1) are given by

    P:xi(θ)=xi0+θ0ui(t)dt, (3.1)
    E:y(θ)=y0+θy1+θ0t0v(s)dsdt=y0+θ0(θt)v(t). (3.2)

    where y0=y0+θy1.

    We define the set

    Ω={z2:2y0x0,zθ(ρ22σ2θ2)+||y0||2||x0||2}.

    The goal of the pursuer P is to ensure the equality x(θ)=y(θ) and that of evader E is the opposite. The problem is to construct a strategy for the pursuer that ensure the equality x(θ)=y(θ) for any admissible control of the evader.

    Lemma 3.1. If y(θ)Ω then there exist a strategy of the pursuer P such that x(θ)=y(θ) in the game (2.1)-(2.3).

    Proof. Let the pursuer's strategy be defined by

    U(t)=y0x0θ+(θt)v(t),  0tθ. (3.3)

    To show the admissibility of the strategy (3.3), we use the fact that y(θ)Ω, which means that

    2y0x0,y(θ)θρ22σ2θ2+||y0||2||x0||2.

    From this inequality we have

    2y0x0,θ0(θt)v(t)dt=2y0x0,y(θ)y0=2y0x0,y(θ)2y0x0,y0=2y0x0,y(θ)2||y0||2+2x0,y0θ(ρ22σ2θ2)+||y0||2||x0||22||y0||2+2x0,y0θ(ρ22σ2θ2)||y0||2||x0||2+2x0,y0=θ(ρ22σ2θ2)(||y0||2+||x0||22x0,y0)θ(ρ22σ2θ2)||y0x0||2.

    Therefore,

    2y0x0,θ0(θt)v(t)dtθ(ρ22σ2θ2)||y0x0||2. (3.4)

    Using (3.3) and the inequality (3.4) we have

    θ0||U(t)||2dt=θ0||y0x0θ+(θt)v(t)||2dt=θ0(||y0x0θ||2+2y0x0θ,(θt)v(t)+||(θt)v(t)||2)dt=θ0||y0x0||2θ2dt+2θ0y0x0θ,(θt)v(t)dt+θ0(θt)2||v(t)||2dt||y0x0||2θ+2θy0x0,θ0(θt)v(t)dt+2σ2θ2||y0x0||2θ+1θ(θ(ρ22σ2θ2)||y0x0||2)+2σ2θ2ρ2.

    This shows that the strategy (3.3) of the pursuer is admissible.

    We now show that the equality x(θ)=y(θ) is achievable, if the pursuer uses the strategy (3.3). Indeed,

    x(θ)=x0+θ0(y0x0θ+(θt)v(t))dt=x0+θ0(y0x0θ)dt+θ0(θt)v(t)dt=x0+y0x0+θ0(θt)v(t)dt=y0+θ0(θt)v(t)dt=y(θ).

    The proof of lemma (3.1) is complete.

    In this section, we present the main result of the paper. Firstly, we present some important results that are useful in the presentation of the main result.

    Lemma 4.1. (see [15], Lemma 9) If there exists a nonzero vector γ0 such that y0xi0,γ00, for all iI, and H(y0,r)iIH(xi0,Ri) then H(y0,r)iI0Xi, where

    I0={iI:S(y0,r)H(xi0,Ri)};
    Xi={{z2:2y0xi0,zR2ir2+||y0||2||xi0||2},xi0y0,{z2:2zy0,γ0Ri},xi0=y0.

    Lemma 4.2. (see [9], Assertion 5) Let infiiRi=R0>0, If there exists a nonzero vector γ0 such that y0xi0,γ00 for all iI and for any ε>0 the set iIH(xi0,Riε) does not contain the ball H(y0,r), then there exist a point ˉyS(y0,r) such that ||ˉyxi0||Ri for all iI.

    We define a positive number ϕ by

    ϕ=inf{l0:HE(y0,σ(θ33)1/2)iIHpi(xi0,ρiθ+l)}. (4.1)

    Theorem 4.1. If there exists a nonzero vector γ0 such that y0xi0,γ00, for all iI, then the number ϕ defined by (4.1) is the value of the game (2.1)-(2.3).

    Proof. To prove this theorem, we first introduce dummy pursuers whose state variables are zi, iI and motion described by the equations

    ˙zi=wi(t), zi(0)=xi0,

    where the control function wi(t) is such that

    (θ0||wi(s)||2ds)12ˉρi=ρi+ϕθ.

    The attainability domain of the dummy pursuer zi from the initial state xi0 is the ball

    HDi(xi0,ˉρiθ)=HDi(xi0,ρiθ+ϕ).

    Let strategies of the dummy pursuers zi,iI be defined as follows:

    For xi0y0, we set

    wi(t)={y0xi0θ+(θt)v(t),0tϑ0,ϑ<tθ, (4.2)

    where ϑ is the time such that

    ϑ0||y0xi0θ+(θt)v(t)||2dt=ˉρ2i.

    For xi0=y0, we set

    wi(t)=(θt)v(t). (4.3)

    Now, we define the strategy of the real pursuers by

    Ui(t)=ρi¯ρiwi(t), 0tθ. (4.4)

    In accordance with the payoff of the game, the number ϕ is the value of the game if the following inequalities hold

    supv()infiI||y(θ)xi(θ)||ϕinfu1(),,um(),infiI||y(θ)xi(θ)||. (4.5)

    In view of this, we prove the inequalities in (4.5). Firstly, we show that left hand side inequality of (4.5). By definition of ϕ, we have

    HE(y0,σ(θ33)1/2)i=1HPi(xi0,ρiθ+ϕ).

    By lemma (4.1), we have

    HE(y0,σ(θ33)1/2)i=1Xi,

    where

    ˉI={iI:S(y0,σ(θ33)1/2)HPi(xi0,ρiθ+ϕ)};
    Xi={{z2:2y0xi0,z(ρiθ+ϕ)2σ2θ33+||y0||2||xi0||2},xi0y0,{z2:2zy0,γρiθ+ϕ},xi0=y0.

    Consequently, the point y(θ)HE(y0,σ(θ33)1/2) belong to some half space Xj, jˉI.

    If xj0y0 and by the lemma (3.1) for strategy (4.2) of pursuers zj then we have the equality zj(θ)=y(θ) holding and

    θ0||wj(t)||2dtˉρ2j.

    For the other case, if xj0=y0;ˉρj=ρj+ϕθσ(θ33)1/2 and the dummy pursuer uses the strategy (4.3) then it is easy to show that zj(θ)=y(θ). This means that for each case, the equality zj(θ)=y(θ) is achieved.

    Now suppose that the real pursuers uses the strategies (4.4), we show that ||y(θ)xj(θ)||ϕ. Indeed,

    ||y(θ)xj(θ)||=||zj(θ)xj(θ)||=||xj0+θ0wj(t)dtxj0θ0uj(t)dt||=||θ0wj(t)dtθ0uj(t)dt||=||θ0wj(t)dtθ0ρj¯ρjwj(t)dt||θ0||(1ρj¯ρj)wj(t)||dt(¯ρjρj¯ρj)θ0||wj(t)||dt(¯ρjρ¯ρj)[(θ012dt)12(θ0||wj(t)||2dt)12](¯ρjρj¯ρj)θ¯ρj(¯ρjρj)θ=(ρj+ϕθρj)θ=ϕ.

    This proves the left hand inequality in (4.5). Which means that the value ϕ is guaranteed for the pursuers.

    Secondly, we prove the right hand inequality in (4.5). That is, we prove that the value ϕ is guaranteed for the evader. If ϕ=0, then the inequality follows for any admissible control of the evader. Suppose that ϕ0 and if ϵ be a non zero positive number such that ϵ<ϕ then by definition of ϕ the ball HE(y0,σ(θ3/3)1/2) is not contained in the set

    iIHpi(xi0,ρiθ+ϕϵ).

    Therefore, the existence of a point ˉyS(y0,σ(θ3/3)1/2), such that ||ˉyxi0||ρiθ+ϕ,iI, is guaranteed by the lemma (4.1). On another note, it is easy to show that

    ||xi(θ)xi0||ρiθ. (4.6)

    Consequently, we have

    ||ˉyxi(θ)||||ˉyxi0||||xi(θ)xi0||ρiθ+ϕρiθ=ϕ.

    This means that if the evader can be at point ˉy at time θ, then the right hand side of (4.5) is proved. Indeed, evader's control exists that takes it to the point ˉy for the time θ, since the point is contained in the attainability domain of the evader. In particular, let the evader's control be defined by

    v(t)=σ(θ33)1/2(θt)e,0tθ, e=ˉyy0||ˉyy0||.

    Then we have

    y(θ)=y0+θ0(θs)v(s)ds=y0+θ0(θs)2σ(θ33)1/2eds=y0+σ(θ33)1/2e=ˉy.

    Therefore, the value ϕ is guaranteed for the evader. This proves the right hand inequality of (4.5). The proof of the theorem is complete.

    Consider the game problem (2.1)-(2.3) and let ρi=1; σ=13; θ=9. We also assume the following initial positions of each of the ith pursuer and evader:

    xi0=(0,0,,19,), y0=0=(0,0,),

    where the number 19 is the ith coordinate of the initial position of the ith pursuer. Observe that ρiθ=3 and σ(θ33)1/2=9.

    The goal is to show the value of the game (2.1)-(2.3) is given by ϕ=7. To show this, it is sufficient to show that

    1. the following inclusion holds for any ϵ>0: HE(0,9)iIHPi(x0i,10+ϵ);

    2. the ball HE(0,9) is not contained in the set iIHPi(x0i,10).

    To show (1), we let y=(y1,y2,)HE(0,9). Therefore, we must have iI(yi)281. Then the vector y has a nonnegative coordinate or all the coordinates of the vector are negative.

    We consider the first case in which the vector y has a nonnegative coordinate yk. Then we have

    ||yxk0||=((k1i=1(yi)2)+(19yk)2+(i=k+1(yi)2))1/2=(i=1(yi)2+19219yk)1/2(100219yk)1/210<10+ϵ.

    This means that yHPk(x0k,10+ϵ). For the second case, since iI(yi)2<, then limkyk=0. Therefore, for the index k, we have

    ||yxk0||=(i=1(yi)2+19219yk)1/2(100219yk)1/2<10+ϵ.

    This also means that yHPk(x0k,10+ϵ).

    To show (2), it is obvious that for any index i, that

    ||yxi0||=(100219yi)1/2>10.

    This means that any vector ySE(0,9) with negatives coordinates does not belong to iIHPi(x0i,10). In view of the this and according to theorem 4.1, the game (2.1)-(2.3) has the value

    ϕ=inf{l0:HE(y0,σ(θ33)1/2)iIHpi(xi0,ρiθ+l)}=inf{l0:HE(0,9)iIHpi(xi0,3+l)}=7.

    The pursuit-evasion differential game problem of countably many pursuers and one evader has been studied in the Hilbert space 2. Control functions of the pursuers and the evader are subject to integral constraints. The value of the game is found and optimal strategies of the players are constructed. The problem considered in this work is uncommon in the literature but represent many real problems. It is a representation of pursuit problems involving objects with different dynamics and possibly information about the acceleration of one of the objects is not available. For further research, the evasion problem concerning the problem considered in this paper can be investigated.

    The second author was financially supported by Rajamangala University of Technology Phra Nakhon (RMUTP) Research Scholarship. We also acknowledged the valuable comments, suggestions and observations of the reviewers of the draft article.

    The authors declare that they have no competing interests in this paper.



    [1] M. Ahookhosh, K. Amini, S. Bahrami, A class of nonmonotone Armijo-type line search method for unconstrained optimization, Optimization, 61 (2012), 387–404. https://doi.org/10.1080/02331934.2011.641126 doi: 10.1080/02331934.2011.641126
    [2] E. G. Birgin, J. M. MartˊInez, M. Raydan, Nonmonotone spectral projected gradient methods on convex sets, SIAM J. Optimiz., 10 (2000), 1196–1211. https://doi.org/10.1137/S1052623497330963 doi: 10.1137/S1052623497330963
    [3] J. Barzilai, J. M. Borwein, Two-point step size gradient methods, IMA J. Numer. Anal., 8 (1988), 141–148. https://doi.org/10.1093/imanum/8.1.141 doi: 10.1093/imanum/8.1.141
    [4] S. Bonettini, Inexact block coordinate descent methods with application to non-negative matrix factorization, IMA J. Numer. Anal., 31 (2011), 1431–1452. https://doi.org/10.1093/imanum/drq024 doi: 10.1093/imanum/drq024
    [5] A. Cichocki, R. Zdunek, S. Amari, Hierarchical ALS algorithms for nonnegative matrix and 3D tensor factorization, In: Independent Component Analysis and Signal Separation, Heidelberg: Springer, 2007,169–176. https://doi.org/10.1007/978-3-540-74494-8_22
    [6] A. Cristofari, M. D. Santis, S. Lucidi, F. Rinaldi, A two-stage active-set algorithm for bound-constrained optimization, J. Optim. Theory Appl., 172 (2017), 369–401. https://doi.org/10.1007/s10957-016-1024-9 doi: 10.1007/s10957-016-1024-9
    [7] Y. H. Dai, On the nonmonotone line search, J. Optim. Theory Appl., 112 (2002), 315–330. https://doi.org/10.1023/A:1013653923062 doi: 10.1023/A:1013653923062
    [8] Y. H. Dai, L. Z. Liao, R-Linear convergence of the Barzilai-Borwein gradient method, IMA J. Numer. Anal., 22 (2002), 1–10. https://doi.org/10.1093/imanum/22.1.1 doi: 10.1093/imanum/22.1.1
    [9] P. Deng, T. R. Li, H. J. Wang, D. X. Wang, S. J. Horng, R. Liu, Graph regularized sparse non-negative matrix factorization for clustering, IEEE Transactions on Computational Social Systems, 10 (2023), 910–921. https://doi.org/10.1109/TCSS.2022.3154030 doi: 10.1109/TCSS.2022.3154030
    [10] P. Deng, F. Zhang, T. R. Li, H. J. Wang, S. J. Horng, Biased unconstrained non-negative matrix factorization for clustering, Knowl.-Based Syst., 239 (2022), 108040. https://doi.org/10.1016/j.knosys.2021.108040 doi: 10.1016/j.knosys.2021.108040
    [11] N. Gillis, The why and how of nonnegative matrix factorization, 2014, arXiv: 1401.5226. https://doi.org/10.48550/arXiv.1401.5226
    [12] R. Glowinski, Numerical methods for nonlinear variational problems, Heidelberg: Springer, 1984. https://doi.org/10.1007/978-3-662-12613-4
    [13] P. H. Gong, C. S. Zhang, Efficient nonnegative matrix factorization via projected Newton method, Pattern Recogn., 45 (2012), 3557–3565. https://doi.org/10.1016/j.patcog.2012.02.037 doi: 10.1016/j.patcog.2012.02.037
    [14] N. Z. Gu, J. T. Mo Incorporating nonmonotone strategies into the trust region method for unconstrained optimization, Comput. Math. Appl., 55 (2008), 2158–2172. https://doi.org/10.1016/j.camwa.2007.08.038
    [15] N. Y. Guan, D. C. Tao, Z. G. Luo, B. Yuan NeNMF: An optimal gradient method for nonnegative matrix factorization, IEEE T. Signal Proces., 60 (2012), 2882–2898. https://doi.org/10.1109/TSP.2012.2190406
    [16] L. X. Han, M. Neumann, U. Prasad, Alternating projected Barzilai-Borwein methods for nonnegative matrix factorization, Electronic Transactions on Numerical Analysis, 36 (2009), 54–82. https://doi.org/10.1007/978-0-8176-4751-3_16 doi: 10.1007/978-0-8176-4751-3_16
    [17] G. Hu, B. Du, X. F. Wang, G. Wei, An enhanced black widow optimization algorithm for feature selection, Knowl.-Based Syst., 235 (2022), 107638. https://doi.org/10.1016/j.knosys.2021.107638 doi: 10.1016/j.knosys.2021.107638
    [18] G. Hu, J. Y. Zhong, G. Wei, SaCHBA_PDN: Modified honey badger algorithm with multi-strategy for UAV path planning, Expert Syst. Appl., 223 (2023), 119941. https://doi.org/10.1016/j.eswa.2023.119941 doi: 10.1016/j.eswa.2023.119941
    [19] G. Hu, J. Y. Zhong, G. Wei, C. T. Chang, DTCSMO: An efficient hybrid starling murmuration optimizer for engineering applications, Comput. Method. Appl. M., 405 (2023), 115878. https://doi.org/10.1016/j.cma.2023.115878 doi: 10.1016/j.cma.2023.115878
    [20] G. Hu, J. Wang, M. Li, A. G. Hussien, M. Abbas, EJS: Multi-strategy enhanced jellyfish search algorithm for engineering applications, Mathematics, 11 (2023), 851. https://doi.org/10.3390/math11040851 doi: 10.3390/math11040851
    [21] G. Hu, R. Yang, X. Q. Qin, G. Wei, MCSA: Multi-strategy boosted chameleon-inspired optimization algorithm for engineering applications, Comput. Method. Appl. M., 403 (2022), 115676. https://doi.org/10.1016/j.cma.2022.115676 doi: 10.1016/j.cma.2022.115676
    [22] G. Hu, X. N. Zhu, G. Wei, C. Chang, An marine predators algorithm for shape optimization of developable Ball surfaces, Eng. Appl. Artif. Intel., 105 (2021), 104417. https://doi.org/10.1016/j.engappai.2021.104417 doi: 10.1016/j.engappai.2021.104417
    [23] Y. K. Huang, H. W. Liu, S. S. Zhou, Quadratic regularization projected alternating Barzilai-Borwein method for nonnegative matrix factorization, Data Min. Knowl. Disc., 29 (2015), 1665–1684. https://doi.org/10.1007/s10618-014-0390-x doi: 10.1007/s10618-014-0390-x
    [24] Y. K. Huang, H. W. Liu, S. Zhou, An efficint monotone projected Barzilai-Borwein method for nonnegative matrix factorization, Appl. Math. Lett., 45 (2015), 12–17. https://doi.org/10.1016/j.aml.2015.01.003 doi: 10.1016/j.aml.2015.01.003
    [25] D. D. Lee, H. S. Seung, Learning the parts of objects by non-negative matrix factorization, Nature, 401 (1999), 788–791. https://doi.org/10.1038/44565 doi: 10.1038/44565
    [26] D. D. Lee, H. S. Seung, Algorithms for non-negative matrix factorization, Advances in Neural Processing Information Systems, 13 (2001), 556–562.
    [27] X. L. Li, H. W. Liu, X. Y. Zheng, Non-monotone projection gradient method for non-negative matrix factorization, Comput. Optim. Appl., 51 (2012), 1163–1171. https://doi.org/10.1007/s10589-010-9387-6 doi: 10.1007/s10589-010-9387-6
    [28] H. W. Liu, X. L. Li, Modified subspace Barzilai-Borwein gradient method for non-negative matrix factorization, Comput. Optim. Appl., 55 (2013), 173–196. https://doi.org/10.1007/s10589-012-9507-6 doi: 10.1007/s10589-012-9507-6
    [29] C. J. Lin, Projected gradient methods for non-negative matrix factorization, Neural Comput., 19 (2007), 2756–2779. https://doi.org/10.1162/neco.2007.19.10.2756 doi: 10.1162/neco.2007.19.10.2756
    [30] H. Nosratipour, A. H. Borzabadi, O. S. Fard, On the nonmonotonicity degree of nonmonotone line searches, Calcolo, 54 (2017), 1217–1242. https://doi.org/10.1007/s10092-017-0226-3 doi: 10.1007/s10092-017-0226-3
    [31] D. Kim, S. Sra, I. S. Dhillon, Fast Newton-type methods for the least squares nonnegative matrix approximation problem, SIAM International Conference on Data Mining, 1 (2007), 343–354. https://doi.org/10.1137/1.9781611972771.31 doi: 10.1137/1.9781611972771.31
    [32] P. Paatero, U. Tapper, Positive matrix factorization: A non-negative factor model with optimal utilization of error estimates of data values, Environmetrics, 5 (1994), 111–126. https://doi.org/10.1002/env.3170050203 doi: 10.1002/env.3170050203
    [33] M. Raydan, On the Barzilai-Borwein choice of steplength for the gradient method, IMA J. Numer. Anal., 13 (1993), 321–326. https://doi.org/10.1093/imanum/13.3.321 doi: 10.1093/imanum/13.3.321
    [34] M. Raydan, The Barzilai and Borwein gradient method for the large-scale unconstrained minimization problem, SIAM J. Optimiz., 7 (1997), 26–33. https://doi.org/10.1137/S1052623494266365 doi: 10.1137/S1052623494266365
    [35] D. X. Wang, T. R. Li, P. Deng, J. Liu, W. Huang, F. Zhang, A generalized deep learning algorithm based on NMF for multi-view clustering, IEEE T. Big Data, 9 (2023), 328–340. https://doi.org/10.1109/TBDATA.2022.3163584 doi: 10.1109/TBDATA.2022.3163584
    [36] D. X. Wang, T. R. Li, P. Deng, F. Zhang, W. Huang, P. F. Zhang, et al., A generalized deep learning clustering algorithm based on non-negative matrix factorization, ACM T. Knowl. Discov. D., 17 (2023), 1–20. https://doi.org/10.1145/3584862 doi: 10.1145/3584862
    [37] D. X. Wang, T. R. Li, W. Huang, Z. P. Luo, P. Deng, P. F. Zhang, et al., A multi-view clustering algorithm based on deep semi-NMF, Inform. Fusion, 99 (2023), 101884. https://doi.org/10.1016/j.inffus.2023.101884 doi: 10.1016/j.inffus.2023.101884
    [38] Z. J. Wang, Z. S. Chen, L. Xiao, Q. Su, K. Govindan, M. J. Skibniewski, Blockchain adoption in sustainable supply chains for Industry 5.0: A multistakeholder perspective, J. Innov. Knowl., 8 (2023), 100425. https://doi.org/10.1016/j.jik.2023.100425 doi: 10.1016/j.jik.2023.100425
    [39] Z. J. Wang, Z. S. Chen, S. Qin, K. S. Chin, P. Witold, M. J. Skibniewski, Enhancing the sustainability and robustness of critical material supply in electrical vehicle market: An AI-powered supplier selection approach, Ann. Oper. Res., 2023 (2023), 102690. https://doi.org/10.1007/s10479-023-05698-4 doi: 10.1007/s10479-023-05698-4
    [40] Z. J. Wang, Y. Y. Sun, Z. S. Chen, G. Z. Feng, Q. Su, Optimal versioning strategy of enterprise software considering the customer cost-acceptance level, Kybernetes, 52 (2023), 997–1026. https://doi.org/10.1108/K-04-2021-0339 doi: 10.1108/K-04-2021-0339
    [41] Z. J. Wang, Y. Y. Sun, Q. Su, M. Deveci, K. Govindan, M. J. Skibniewski, et al., Smart contract application in resisting extreme weather risks for the prefabricated construction supply chain: prototype exploration and assessment, Group Decis. Negot., (2024). https://doi.org/10.1007/s10726-024-09877-x
    [42] Y. H. Xiao, Q. J. Hu, Subspace Barzilai-Borwein gradient method for large-scale bound constrained optimization, Appl. Math. Optim., 58 (2008), 275–290. https://doi.org/10.1007/s00245-008-9038-9 doi: 10.1007/s00245-008-9038-9
    [43] Y. H. Xiao, Q. J. Hu, Z. X. Wei, Modified active set projected spectral gradient method for bound constrained optimization, Appl. Math. Model., 35 (2011), 3117–3127. https://doi.org/10.1016/j.apm.2010.09.011 doi: 10.1016/j.apm.2010.09.011
    [44] Y. Y. Xu, W. T. Yin, A block coordinate descent method for regularized multi-convex optimization with applications to nonnegative tensor factorization and completion, SIAM J. Imaging Sci., 6 (2013), 1758–1789. https://doi.org/10.1137/120887795 doi: 10.1137/120887795
    [45] H. C. Zhang, W. W. Hager, A nonmonotone line search technique and its application to unconstrained optimization, SIAM J. Optimiz., 14 (2004), 1043–1056. https://doi.org/10.1137/S1052623403428208 doi: 10.1137/S1052623403428208
    [46] R. Zdunek, A. Cichocki, Fast nonnegative matrix factorization algorithms using projected gradient approaches for large-scale problems, Comput. Intel. Neurosc., 2008 (2008), 939567. https://doi.org/10.1155/2008/939567 doi: 10.1155/2008/939567
  • This article has been cited by:

    1. Jewaidu Rilwan, Pasquale Fotia, Massimiliano Ferrara, On game value of a differential game problem with Grönwall-type constraints on players control functions, 2023, 1593-8883, 10.1007/s10203-023-00389-y
    2. Abbas Ja'afaru Badakaya, Aminu Sulaiman Halliru, Jamilu Adamu, A game problem of pursuit-evasion with nth order differential equations describing players' dynamics, 2022, 9, 2330-7706, 193, 10.1080/23307706.2021.1941336
    3. Jamilu Adamu, Aminu Sulaiman Halliru, Bala Ma’aji Abdulhamid, Pursuit differential game problem with integral and geometric constraints., 2022, 2714-4704, 83, 10.46481/jnsps.2022.379
    4. Abbas Ja'afaru Badakaya, Aminu Sulaiman Halliru, Jamilu Adamu, Game value for a pursuit-evasion differential game problem in a Hilbert space, 2022, 9, 2164-6066, 1, 10.3934/jdg.2021019
    5. Somayeh Sharifi, Abbas Ja’afaru Badakaya, Mehdi Salimi, On game value for a pursuit-evasion differential game with state and integral constraints, 2022, 39, 0916-7005, 653, 10.1007/s13160-022-00501-6
    6. Abbas Ja’afaru Badakaya, Sani Musa Tsoho, Mehdi Salimi, On Some -Catch Pursuit Differential Games with Different Players’ Dynamic Equations, 2024, 0971-3514, 10.1007/s12591-023-00673-8
    7. Bashir Mai Umar, Jewaidu Rilwan, Maggie Aphane, Kanikar Muangchoo, Pursuit and Evasion Linear Differential Game Problems with Generalized Integral Constraints, 2024, 16, 2073-8994, 513, 10.3390/sym16050513
    8. D. N. Madhavan,, I. A. Alias,, G. Ibragimov,, R. M. Hasim,, Some Results on Pursuit Games for an Infinite System of Ternary Differential Equations, 2024, 18, 1823-8343, 567, 10.47836/mjms.18.3.07
  • Reader Comments
  • © 2024 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(882) PDF downloads(46) Cited by(0)

Figures and Tables

Figures(4)  /  Tables(2)

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog