Loading [MathJax]/jax/output/SVG/jax.js
Research article

An accelerated forward-backward algorithm with a new linesearch for convex minimization problems and its applications

  • We study and investigate a convex minimization problem of the sum of two convex functions in the setting of a Hilbert space. By assuming the Lipschitz continuity of the gradient of the function, many optimization methods have been invented, where the stepsizes of those algorithms depend on the Lipschitz constant. However, finding such a Lipschitz constant is not an easy task in general practice. In this work, by using a new modification of the linesearches of Cruz and Nghia [7] and Kankam et al. [14] and an inertial technique, we introduce an accelerated algorithm without any Lipschitz continuity assumption on the gradient. Subsequently, a weak convergence result of the proposed method is established. As applications, we apply and analyze our method for solving an image restoration problem and a regression problem. Numerical experiments show that our method has a higher efficiency than the well-known methods in the literature.

    Citation: Adisak Hanjing, Pachara Jailoka, Suthep Suantai. An accelerated forward-backward algorithm with a new linesearch for convex minimization problems and its applications[J]. AIMS Mathematics, 2021, 6(6): 6180-6200. doi: 10.3934/math.2021363

    Related Papers:

    [1] Luigi Montoro, Berardino Sciunzi . Qualitative properties of solutions to the Dirichlet problem for a Laplace equation involving the Hardy potential with possibly boundary singularity. Mathematics in Engineering, 2023, 5(1): 1-16. doi: 10.3934/mine.2023017
    [2] Juan-Carlos Felipe-Navarro, Tomás Sanz-Perela . Semilinear integro-differential equations, Ⅱ: one-dimensional and saddle-shaped solutions to the Allen-Cahn equation. Mathematics in Engineering, 2021, 3(5): 1-36. doi: 10.3934/mine.2021037
    [3] Francesca G. Alessio, Piero Montecchiari . Gradient Lagrangian systems and semilinear PDE. Mathematics in Engineering, 2021, 3(6): 1-28. doi: 10.3934/mine.2021044
    [4] Filippo Gazzola, Gianmarco Sperone . Remarks on radial symmetry and monotonicity for solutions of semilinear higher order elliptic equations. Mathematics in Engineering, 2022, 4(5): 1-24. doi: 10.3934/mine.2022040
    [5] Elena Beretta, M. Cristina Cerutti, Luca Ratti . Lipschitz stable determination of small conductivity inclusions in a semilinear equation from boundary data. Mathematics in Engineering, 2021, 3(1): 1-10. doi: 10.3934/mine.2021003
    [6] Huyuan Chen, Laurent Véron . Weak solutions of semilinear elliptic equations with Leray-Hardy potentials and measure data. Mathematics in Engineering, 2019, 1(3): 391-418. doi: 10.3934/mine.2019.3.391
    [7] Marco Cirant, Kevin R. Payne . Comparison principles for viscosity solutions of elliptic branches of fully nonlinear equations independent of the gradient. Mathematics in Engineering, 2021, 3(4): 1-45. doi: 10.3934/mine.2021030
    [8] Yuzhe Zhu . Propagation of smallness for solutions of elliptic equations in the plane. Mathematics in Engineering, 2025, 7(1): 1-12. doi: 10.3934/mine.2025001
    [9] Antonio Greco, Francesco Pisanu . Improvements on overdetermined problems associated to the p-Laplacian. Mathematics in Engineering, 2022, 4(3): 1-14. doi: 10.3934/mine.2022017
    [10] Italo Capuzzo Dolcetta . The weak maximum principle for degenerate elliptic equations: unbounded domains and systems. Mathematics in Engineering, 2020, 2(4): 772-786. doi: 10.3934/mine.2020036
  • We study and investigate a convex minimization problem of the sum of two convex functions in the setting of a Hilbert space. By assuming the Lipschitz continuity of the gradient of the function, many optimization methods have been invented, where the stepsizes of those algorithms depend on the Lipschitz constant. However, finding such a Lipschitz constant is not an easy task in general practice. In this work, by using a new modification of the linesearches of Cruz and Nghia [7] and Kankam et al. [14] and an inertial technique, we introduce an accelerated algorithm without any Lipschitz continuity assumption on the gradient. Subsequently, a weak convergence result of the proposed method is established. As applications, we apply and analyze our method for solving an image restoration problem and a regression problem. Numerical experiments show that our method has a higher efficiency than the well-known methods in the literature.



    The notion of intuitionistic fuzzy normed subring and intuitionistic fuzzy normed ideal was characterized by Abed Alhaleem and Ahmad in [10], after that the necessity has arisen to introduce the concepts of intuitionistic fuzzy normed prime ideals and intuitionistic fuzzy normed maximal ideals. Following the work of Emniyent and Şahin in [17] which outlined the concepts of fuzzy normed prime ideal and maximal ideal we implement the conception of intuitionistic fuzzy to prime and maximal normed ideals. After the establishment of fuzzy set by Zadeh [28] which showed that the membership of an element in a fuzzy set is at intervals [0, 1], many researchers investigated on the properties of fuzzy set because it handles uncertainty and vagueness, and due to its applications in many fields of studies. A lot of work has been done on various aspects and for the last 50 years, the relation betwee maximal and prime ideals has become the core of many researchers work. Swamy and Swamy in 1988 [27] presented the conceptions of fuzzy ideal and fuzzy prime ideal with truth values in a complete lattice fulfilling the infinite distributive law. Later, many researchers studied the generalization of fuzzy ideals and fuzzy prime (maximal) ideals of rings: Dixit et al [16], Malik and Mordeson in [22] and Mukherjee and Sen in [24]. The notion of intuitionistic fuzzy set was initiated by Atanassov [6], as a characterization of fuzzy set which assigned the degree of membership and the degree of non-membership for set elements, he also delineated some operations and connections over basic intuitionistic fuzzy sets. In [5], Atanassov introduced essential definitions and properties of the interval-valued intuitionistic fuzzy sets and the explanation of mostly extended modal operator through interval-valued intuitionistic fuzzy sets were presented in [4], and some of its main properties were studied. Banerjee and Basnet [13] investigated intuitionistic fuzzy rings and intuitionistic fuzzy ideals using intuitionistic fuzzy sets. In 2005 [20], an identification of intuitionistic fuzzy ideals, intuitionistic fuzzy prime ideals and intuitionistic fuzzy completely prime ideals was given. In [14], Bakhadach et al. implemented the terms of intuitionistic fuzzy ideals and intuitionistic fuzzy prime (maximal) ideals, investigated these notions to show new results using the intuitionistic fuzzy points and membership and nonmembership functions. The paper comprises the following: we begin with the preliminary section, we submit necessary notations and elementary outcomes. In Section 3, we characterize some properties of intuitionistic fuzzy normed ideals and identify the image and the inverse image of intuitionistic fuzzy normed ideals. In Section 4, we describe the notions of intuitionistic fuzzy normed prime ideals and intuitionistic fuzzy normed maximal ideals and we characterize the relation between the intuitionistic characteristic function and prime (maximal) ideals. In Section 5, the conclusions are outlined.

    We first include some definitions needed for the subsequent sections:

    Definition 2.1. [25] A linear space L is called a normed space if for any element r there is a real number r satisfying:

    r0 for every rL, when r=0 then r=0;

    α.r=|α|.r;

    r+vr+v for all r,vL.

    Definition 2.2. [18] A ring R is said to be a normed ring (NR) if it possesses a norm , that is, a non-negative real-valued function :NRR such that for any r,vR,

    1)r=0r=0,

    2)r+vr+v,

    3)r=r, (and hence 1A=1=1 if identity exists), and

    4)rvrv.

    Definition 2.3. [1] Let :[0,1]×[0,1][0,1] be a binary operation. Then is a t-norm if conciliates the conditions of commutativity, associativity, monotonicity and neutral element 1.

    We shortly use t-norm and write rv instead of (r,v).

    Two examples of continuous t-norm are: rv=rv and rv=min{r,v} [26].

    Proposition 2.4. [21] A t-norm T has the property, for every r,v[0,1]

    T(r,v)min(r,v)

    Definition 2.5. [19] Let :[0,1]×[0,1][0,1] be a binary operation. Then is a s-norm if conciliates the conditions of commutativity, associativity, monotonicity and neutral element 0.

    We shortly use s-norm and write rv instead of (r,v).

    Two examples of continuous s-norm are: rv=min(r+v,1) and rv=max{r,v} [26].

    Proposition 2.6. [21] A s-norm S has the property, for every r,v[0,1]

    max(r,v)S(r,v)

    Definition 2.7. [28] A membership function μA(r):X[0,1] specifies the fuzzy set A over X, where μA(r) defines the membership of an element rX in a fuzzy set A.

    Definition 2.8. [6] An intuitionistic fuzzy set A in set X is in the form IFSA={(r,μA(r),γA(r):rX}, such that the degree of membership is μA(r):X[0,1] and the degree of non-membership is γA(r):X[0,1], where 0μA(r)+γA(r))1 for all rX. We shortly use A=(μA,γA).

    Definition 2.9. [7] Let A be an intuitionistic fuzzy set in a ring R, we indicate the (α,β)-cut set by Aα,β={rR:μAα and γAβ} such that α+β1 and α,β[0,1].

    Definition 2.10. [23] The support of an intuitionistic fuzzy set A, is denoted by A and defined as A={r:μA(r)>0 and γA(r)<1}.

    Definition 2.11. [2] The complement, union and intersection of two IFSA=(μA,γA) and B=(μB,γB), in a ring R, are defined as follows:

    1)Ac={r,γA(r),μA(r):rR},

    2)AB={r,max(μA(r),μB(r)),min(γA(r),γB(r)):rR},

    3)AB={r,min(μA(r),μB(r)),max(γA(r),γB(r)):rR}.

    Definition 2.12. [12] Let NR be a normed ring. Then an IFS A={(r,μA(r),γA(r)):rNR} of NR is an intuitionistic fuzzy normed subring (IFNSR) of NR if:

    i. μA(rv)μA(r)μA(v),

    ii. μA(rv)μA(r)μA(v),

    iii. γA(rv)γA(r)γA(v),

    iv. γA(rv)γA(r)γA(v).

    Definition 2.13. [9] Let NR be a normed ring. Then an IFS A={(r,μA(r),γA(r)):rNR} of NR is an intuitionistic fuzzy normed ideal (IFNI) of NR if:

    i. μA(rv)μA(r)μA(v),

    ii. μA(rv)μA(r)μA(v),

    iii. γA(rv)γA(r)γA(v)),

    iv. γA(rv))γA(r)γA(v)}.

    Definition 2.14. [3] If A and B are two fuzzy subsets of the normed ring NR. Then the product AB(r) is defined by:

    AB(r)={r=vz(μA(v)μB(z)),ifr=vz0,otherwise

    Definition 2.15. [22] A fuzzy ideal A (non-constant) of a ring R is considered to be a fuzzy prime ideal if BCA for a fuzzy ideals B, C of R indicates that either BA or CA.

    In this section, we characterize several properties of intuitionistic fuzzy normed ideals and elementary results are obtained.

    Definition 3.1. [8] Let A and B be two intuitionistic fuzzy subsets of the normed ring NR. The operations are defined as:

    μAB(r)={r=vz(μA(v)μB(z)),ifr=vz0,otherwise

    and

    γAB(r)={r=vz(γA(v)γB(z)),ifr=vz1,otherwise

    Therefore, the intrinsic product of A and B is considered to be the intuitionistic fuzzy normed set AB=(μAB,γAB)=(μAμB,γAγB).

    Theorem 3.2. [10] Let A and B be two intuitionistic fuzzy ideals of a normed ring NR. Then AB is an intuitionistic fuzzy normed ideal of NR.

    Example 3.1. Let NR=Z the ring of integers under ordinary addition and multiplication of integers.

    Define the intuitionistic fuzzy normed subsets as A=(μA,γA) and B=(μB,γB), by

    μA(r)={0.7,ifr5Z0.2,otherwiseandγA(r)={0.1,ifr5Z0.4,otherwise
    μB(r)={0.8,ifr5Z0.3,otherwiseandγB(r)={0.2,ifr5Z0.7,otherwise

    As μAB(r)=min{μA(r),μB(r)} and γAB(r)=max{γA(r),γB(r)}. Then,

    μAB(r)={0.7,ifr5Z0.2,otherwiseandγAB(r)={0.2,ifr5Z0.7,otherwise

    It can be verified that A, B and AB are intuitionistic fuzzy normed ideals of NR.

    Lemma 3.3. Let A and B be an intuitionistic fuzzy normed right ideal and an intuitionistic fuzzy normed left ideal of a normed ring NR, respectively, then ABAB i.e, AB(r)AB(r)AB(r), where

    AB(r)={(r,μAB(r),γAB(r)):rNR}={(r,min{μA(r),μB(r)},max{γA(r),γB(r)}):rNR}.

    Proof. Let AB be an intuitionistic fuzzy normed ideal of NR. Assume that A is an intuitionistic fuzzy normed right ideal and B is an intuitionistic fuzzy normed left ideal. Let μAB(r)=r=vz(μA(v)μB(z)) and let γAB(r)=r=vz(γA(v)γB(z)).

    Since, A is an intuitionistic fuzzy normed right ideal and B is an intuitionistic fuzzy normed left ideal, we have

    μA(v)μA(vz)=μA(r)andμB(z)μB(vz)=μB(r)

    and

    γA(r)=γA(vz)γA(v)andγB(r)=γB(vz)γB(z).

    Thus,

    μAB(r)=r=vz(μA(v)μB(z))=min(μA(v),μB(z))min(μA(r),μB(r))μAB(r) (3.1)

    and

    γAB(r)=r=vz(γA(v)γB(z))=max(γA(v),γB(z))max(γA(r),γB(r))γAB(r). (3.2)

    By (3.1) and (3.2) the proof is concluded.

    Remark 3.4. The union of two intuitionistic fuzzy normed ideals of a ring NR needs not be always intuitionistic fuzzy normed ideal.

    Example 3.2. Let NR=Z the ring of integers under ordinary addition and multiplication of integers.

    Let the intuitionistic fuzzy normed subsets A=(μA,γA) and B=(μB,γB), define by

    μA(r)={0.85,ifr3Z0.3,otherwiseandγA(r)={0.2,ifr3Z0.4,otherwise
    μB(r)={0.75,ifr2Z0.35,otherwiseandγB(r)={0.3,ifr2Z0.5,otherwise

    It can be checked that A and B are intuitionistic fuzzy normed ideals of NR.

    As μAB(r)=max{μA(r),μB(r)} and γAB(r)=min{γA(r),γB(r)}. Then,

    μAB(r)={0.85,ifr3Z0.75,ifr2Z3Z0.35,ifr2Zorr3ZandγAB(r)={0.2,ifr3Z0.3,ifr2Z3Z0.4,ifr2Zorr3Z

    Let r=15 and v=4, then μAB(15)=0.85, μAB(4)=0.75 and γAB(15)=0.2, γAB(4)=0.3.

    Hence, μAB(154)=μAB(11)=0.35μAB(15)μAB(4)=min{0.85,0.75} and γAB(154)=γAB(11)=0.4γAB(15)γAB(4)=max{0.2,0.3}. Thus, the union of two intuitionistic fuzzy normed ideals of NR need not be an intuitionistic fuzzy normed ideal.

    Proposition 3.5. Let A=(μA,γA) be an intuitionistic fuzzy normed ideal of a ring NR, then we have for all rNR:

    i. μA(0)μA(r) and γA(0)γA(r),

    ii. μA(r)=μA(r) and γA(r)=γA(r),

    iii. If μA(rv)=μA(0) then μA(r)=μA(v),

    iv. If γA(rv)=γA(0) then γA(r)=γA(v).

    Proof. i. As A is an intuitionistic fuzzy normed ideal, then

    μA(0)=μA(rr)μA(r)μA(r)=μA(r)

    and

    γA(0)=γA(rr)γA(r)γA(r)=γA(r)

    ii. μA(r)=μA(0r)μA(0)μA(r)=μA(r) and μA(r)=μA(0(r))μA(0)μA(r)=μA(r).

    Therefore, μA(r)=μA(r)

    also,

    γA(r)=γA(0r)γA(0)γA(r)=γA(r) and γA(r)=γA(0(r))γA(0)γA(r)=γA(r).

    Therefore, γA(r)=γA(r).

    iii. Since μA(rv)=μA(0), then

    μA(v)=μA(r(rv))μA(r)μA(rv)=μA(r)μA(0)μA(r)

    similarly

    μA(r)=μA((rv)(v))μA(rv)μA(v)=μA(0)μA(v)μA(v)

    Consequently, μA(r)=μA(v).

    iv. same as in iii.

    Proposition 3.6. Let A be an intuitionistic fuzzy normed ideal of a normed ring NR, then A=(μA,μcA) is an intuitionistic fuzzy normed ideal of NR.

    Proof. Let r,vNR

    μcA(rv)=1μA(rv)1min{μA(r),μA(v)}=max{1μA(r),1μA(v)}=max{μcA(r),μcA(v)}

    Then μcA(rv)μcA(r)μcA(v).

    μcA(rv)=1μA(rv)1max{μA(r),μA(v)}=min{1μA(r),1μA(v)}=min{μcA(r),μcA(v)}

    Then μcA(rv)μcA(r)μcA(v).

    Accordingly, A=(μA,μcA) is an intuitionistic fuzzy normed ideal of NR.

    Proposition 3.7. If A is an intuitionistic fuzzy normed ideal of a normed ring NR, then A=(γcA,γA) is an intuitionistic fuzzy normed ideal of NR.

    Proof. Let r,vNR

    γcA(rv)=1γA(rv)1max{γA(r),γA(v)}=min{1γA(r),1γA(v)}=min{γcA(r),γcA(v)}

    Then γcA(rv)γcA(r)γcA(v).

    γAc(rv)=1γA(rv)1min{γA(r),γA(v)}=max{1μA(r),1γA(v)}=max{γcA(r),γcA(v)}

    Then γcA(rv)γcA(r)γcA(v).

    Therefore, A=(γcA,γA) is an intuitionistic fuzzy normed ideal of NR.

    Proposition 3.8. An IFSA=(μA,γA) is an intuitionistic fuzzy normed ideal of NR if the fuzzy subsets μA and γcA are intuitionistic fuzzy normed ideals of NR.

    Proof. Let r,vNR

    1γA(rv)=γcA(rv)min{γcA(r),γcA(v)}=min{(1γA(r)),(1γA(v))}=1max{γA(r),γA(v)}

    Then, γA(rv)γA(r)γA(v).

    1γA(rv)=γcA(rv)max{γcA(r),γcA(v)}=max{(1γA(r)),(1γA(v))}=1min{γA(r),γA(v)}

    Then, γA(rv)γA(r)γA(v).

    Consequently, A=(μA,γA) is an intuitionistic fuzzy normed ideal of NR.

    Definition 3.9. Let A be a set (non-empty) of the normed ring NR, the intuitionistic characteristic function of A is defined as λA=(μλA,γλA), where

    μλA(r)={1,ifrA0,ifrAandγλA(r)={0,ifrA1,ifrA

    Lemma 3.10. Let A and B be intuitionistic fuzzy sets of a normed ring NR, then:

    (i) λAλB=λAB (ii) λAλB=λAB (iii) If AB, then λAλB

    Theorem 3.11. For a non-empty subset A of NR, A is a subring of NR if and only if λA=(μλA,γλA) is an intuitionistic fuzzy normed subring of NR.

    Proof. Suppose A to be a subring of NR and let r,vNR. If r,vA, then by the intuitionistic characteristic function properties μλA(r)=1=μλA(v) and γλA(r)=0=γλA(v). As A is a subring, then rv and rvA. Thus, μλA(rv)=1=11=μλA(r)μλA(v) and μλA(rv)=1=11=μλA(r)μλA(v), also γλA(rv)=0=00=γλA(r)γλA(v) and γλA(rv)=0=00=γλA(r)γλA(v). This implies,

    μλA(rv)μλA(r)μλA(v)andμλA(rv)μλA(r)μλA(v),γλA(rv)γλA(r)γλA(v)andγλA(rv)γλA(r)γλA(v).

    Similarly we can prove the above expressions if r,vA.

    Hence, λA=(μλA,γλA) is an intuitionistic fuzzy normed subring of NR.

    Conversely, we hypothesise that the intuitionistic characteristic function λA=(μλA,γλA) is an intuitionistic fuzzy normed subring of NR. Let r,vA, then μλA(r)=1=μλA(v) and γλA(r)=0=γλA(v). So,

    μλA(rv)μλA(r)μλA(v)111,alsoμλA(rv)1,μλA(rv)μλA(r)μλA(v)111,alsoμλA(rv)1,γλA(rv)γλA(r)γλA(v)000,alsoγλA(rv)0,γλA(rv)γλA(r)γλA(v)000,alsoγλA(rv)0,

    then μλA(rv)=1, μλA(rv)=1 and γλA(rv)=0, γλA(rv)=0, which implies that rv and rvA. Therefore, A is a subring of NR.

    Theorem 3.12. Let I be a non-empty subset of a normed ring NR, then I is an ideal of NR if and only if λI=(μλI,γλI) is an intuitionistic fuzzy normed ideal of NR.

    Proof. Let I be an ideal of NR and let r,vNR.

    Case I. If r,vI then rvI and μλI(r)=1, μλI(v)=1 and γλI(r)=0, γλI(v)=0. Thus, μλI(rv)=1 and γλI(rv)=0. Accordingly, μλI(rv)=1=μλI(r)μλI(v) and γλI(rv)=0=γλI(r)γλI(v).

    Case II. If rI or vI so rvI, then μλI(r)=0 or μλI(v)=0 and γλI(r)=1 or γλI(v)=1. So, μλI(rv)=1μλI(r)μλI(v) and γλI(rv)=0γλI(r)γλI(v). Hence, λI=(μλI,γλI) is an intuitionistic fuzzy normed ideal of NR.

    On the hand, we suppose λI=(μλI,γλI) is an intuitionistic fuzzy normed ideal of NR. The proof is similar to the second part of the proof of Theorem 3.11.

    Proposition 3.13. If A is an intuitionistic fuzzy normed ideal of NR, then A is an ideal of NR where A is defined as,

    A={rNR:μA(r)=μA(0)andγA(r)=γA(0)}

    Proof. See [10] (p. 6)

    Lemma 3.14. Let A and B be two intuitionistic fuzzy normed left (right) ideal of NR. Therefore, AB(AB).

    Proof. Let rAB, then μA(r)=μA(0), μB(r)=μB(0) and γA(r)=γA(0), γB(r)=γB(0).

    μAB(r)=min{μA(r),μB(r)}=min{μA(0),μB(0)}=μAB(0)

    and

    γAB(r)=max{γA(r),γB(r)}=max{γA(0),γB(0)}=γAB(0)

    So, r(AB). Thus, AB(AB).

    Theorem 3.15. Let f:NRNR be an epimorphism mapping of normed rings. If A is an intuitionistic fuzzy normed ideal of the normed ring NR, then f(A) is also an intuitionistic fuzzy normed ideal of NR.

    Proof. Suppose A={(r,μA(r),γA(r)):rNR},

    f(A)={(v,f(r)=vμA(r),f(r)=vγA(r):rNR,vNR}.

    Let v1,v2NR, then there exists r1,r2NR such that f(r1)=v1 and f(r2)=v2.

    i.

    μf(A)(v1v2)=f(r1r2)=v1v2μA(r1r2)f(r1)=v1,f(r2)=v2(μA(r1)μA(r2))(f(r1)=v1μA(r1))(f(r2)=v2μA(r2))μf(A)(v1)μf(A)(v2)

    ii.

    μf(A)(v1v2)=f(r1r2)=v1v2μA(r1r2)f(r2)=v2μA(r2)μf(A)(v2)

    iii.

    γf(A)(v1v2)=f(r1r2)=v1v2γA(r1r2)f(r1)=v1,f(r2)=v2(γA(r1)γA(r2))(f(r1)=v1γA(r1))(f(r2)=v2γA(r2))γf(A)(v1)γf(A)(v2)

    iv.

    γf(A)(v1v2)=f(r1r2)=v1v2γA(r1r2)f(r2)=v2γA(r2)γf(A)(v2)

    Hence, f(A) is an intuitionistic fuzzy normed left ideal. Similarly, it can be justified that f(A) is an intuitionistic fuzzy normed right ideal. Then, f(A) is a intuitionistic fuzzy normed ideal of NR.

    Proposition 3.16. Define f:NRNR to be an epimorphism mapping. If B is an intuitionistic fuzzy normed ideal of the normed ring NR, then f1(B) is also an intuitionistic fuzzy normed ideal of NR.

    Proof. Suppose B={(v,μB(v),γB(v)):vNR}, f1(B)={(r,μf1(B)(r),γf1(B)(r):rNR}, where μf1(B)(r)=μB(f(r)) and γf1(B)(r)=γB(f(r)) for every rNR. Let r1,r2NR, then

    i.

    μf1(B)(r1r2)=μB(f(r1r2))=μB(f(r1)f(r2))μB(f(r1))μB(f(r2))μf1(B)(r1)μf1(B)(r2)

    ii.

    μf1(B)(r1r2)=μB(f(r1r2))=μB(f(r1)f(r2))μB(f(r2))μf1(B)(r2)

    iii.

    γf1(B)(r1r2)=γB(f(r1r2))=γB(f(r1)f(r2))γB(f(r1))γB(f(r2))γf1(B)(r1)γf1(B)(r2)

    iv.

    γf1(B)(r1r2)=γB(f(r1r2))=γB(f(r1)f(r2))γB(f(r2))γf1(B)(r2)

    Therefore, f1(B) is an intuitionistic fuzzy normed left ideal of NR. Similarly, it can be justified that f1(B) is an intuitionistic fuzzy normed right ideal. So, f1(B) is a intuitionistic fuzzy normed ideal of NR.

    In what follows, we produce the terms of intuitionistic fuzzy normed prime ideals and intuitionistic fuzzy normed maximal ideals and we investigate some associated properties.

    Definition 4.1. An intuitionistic fuzzy normed ideal A=(μA,γA) of a normed ring NR is said to be an intuitionistic fuzzy normed prime ideal of NR if for an intuitionistic fuzzy normed ideals B=(μB,γB) and C=(μC,γC) of NR where BCA indicates that either BA or CA, which imply that μBμA and γAγB or μCμA and γAγC.

    Proposition 4.2. An intuitionistic fuzzy normed ideal A=(μA,γA) is an intuitionistic fuzzy normed prime ideal if for any two intuitionistic fuzzy normed ideals B=(μB,γB) and C=(μC,γC) of NR satisfies:

    i. μAμBC i.e. μA(r)r=vz(μA(v)μB(z));

    ii. γAγBC i.e.γA(r)r=vz(γA(v)γB(z)).

    Theorem 4.3. Let A be an intuitionistic fuzzy normed prime ideal of NR. Then Im μA = Im γA∣=2; in other words A is two-valued.

    Proof. As A is not constant, Im μA∣≥2. assume that Im μA∣≥3. Aα,β={rR:μAα and γAβ} where α+β1. Let rNR and let B and C be two intuitionistic fuzzy subsets in NR, such that: μA(0)=s and k=glb{μA(r):rNR}, so there exists t,α Im(μA) such that kt<α<s with μB(r)=12(t+α), μC(r)={s,ifrAα,βk,ifrAα,β and γA(0)=c and h=lub{γA(r):rNR}, then there exists d,β Im(γA) such that c<β<dh with γB(r)=12(d+β) and γC(r)={c,ifrAα,βh,ifrAα,β for all rNR. Clearly B is an intuitionistic fuzzy normed ideal of NR. Now we claim that C is an intuitionistic fuzzy normed ideal of NR.

    Let r,vNR, if r,vAα,β then rvAα,β and μC(rv)=s=μC(r)μC(v), γC(rv)=c=γC(r)γC(v). If rAα,β and vAα,β then rvAα,β so, μC(rv)=k=μC(r)μC(v), γC(rv)=h=γC(r)γC(v). If r,vAα,β then rvAα,β so, μC(rv)k=μC(r)μC(v), γC(rv)h=γC(r)γC(v). Hence, μC(rv)μC(r)μC(v) and γC(rv)γC(r)γC(v) for all r,vNR.

    Now if rAα,β then rvAα,β, thus μC(rv)=s=μC(r)μC(v) and γC(rv)=c=γC(r)γC(v). If rAα,β, then μC(rv)k=μC(r)μC(v) and γC(rv)h=γC(r)γC(v). Therefore C is an intuitionistic fuzzy normed ideal of NR.

    To prove that BCA. Let rNR, we discuss the following cases:

    (i) If r=0, consequently

    μBC(0)=r=uv(μB(u)μC(v))12(t+α)<s=μA(0);
    γBC(r)=r=uv(γB(u)γC(v))12(d+β)>c=γA(0).

    (ii) If r0, rAα,β. Then μA(r)α and γA(r)β. Thus,

    μBC(r)=r=uv(μB(u)μC(v))12(t+α)<αμA(r);
    γBC(r)=r=uv(γB(u)γC(v))12(d+β)>βγA(r).

    Since μB(u)μC(v)μB(u) and γB(u)γC(v)γB(u).

    (iii) If r0, rAα,β. Then in that case u,vNR such that r=uv, uAα,β and vAα,β. Then,

    μBC(r)=r=uv(μB(u)μC(v))=kμA(r);
    γBC(r)=r=uv(γB(u)γC(v))=hγA(r).

    Therefore, in any case μBC(r)μA(r) and γBC(r)γA(r) for all rNR. Hence, BCA.

    Let a,bNR such that μA(a)=t, μA(b)=α and γA(a)=d, γA(b)=β. Thus, μB(a)=12(t+α)>t=μA(r) and γB(a)=12(d+β)<d=γA(r) which implies that BA. Also, μA(b)=α and γA(b)=β imply that bAα,β so, μC(b)=s>α and γC(b)=c<β, so CA. Therefore, neither BA nor CA. This indicates that A could not be an intuitionistic fuzzy normed prime ideal of NR, so its a contradiction. Thus, Im μA = ImγA∣=2.

    Proposition 4.4. If A is an intuitionistic fuzzy normed prime ideal of NR, so the following are satisfied:

    i. μA(0NR)=1 and γA(0NR)=0;

    ii. Im(μA)={1,α} and Im(γA)={0,β}, where α,β[0,1];

    iii. A is a prime ideal of NR.

    Theorem 4.5. Let A be a fuzzy subset of NR where A is two-valued, μA(0)=1 and γA(0)=0, and the set A={rNR:μA(r)=μA(0) and γA(r)=γA(0)} is a prime ideal of NR. Hence, A is an intuitionistic fuzzy normed prime ideal of NR.

    Proof. We have Im(μA)={1,α} and Im(γA)={0,β}. Let r,vNR. If r,vA, then rvA so, μA(rv)=1=μA(r)μA(v) and γA(rv)=0=γA(r)γA(v). If r,vA, then μA(rv)=αμA(r)μA(v) and γA(rv)=βγA(r)γA(v).

    Therefore, for all r,vNR,

    μA(rv)μA(r)μA(v)γA(rv)γA(r)γA(v)

    Similarly,

    μA(rv)μA(r)μA(v)γA(rv)γA(r)γA(v)

    Thus A is an intuitionistic fuzzy ideal of NR.

    Assume B and C be fuzzy ideals of NR where BCA. Assume that BA and CA. Then, we have r,vNR in such a way that μB(r)>μA(r) and γB(r)<γA(r), μC(v)>μA(v) and γC(r)<γA(r), so for all aA, μA(a)=1=μA(0) and γA(a)=0=γA(0), rA and vA. Since, A is a prime ideal of NR, we have nNR in such a way that rnvA. Let a=rnv then μA(a)=μA(r)=μA(v)=α and γA(a)=γA(r)=γA(v)=β, now

    μBC(a)=a=st(μB(s)μC(t))μB(r)μC(nv)μB(r)μC(v)>α=μA(a)[Since,μB(r)μA(r)=αandμC(nv)μC(v)μA(v)=α].

    and

    γBC(a)=a=st(γB(s)γC(t))γB(r)γC(nv)γB(r)γC(v)<β=γA(a)[Since,γB(r)γA(r)=βandγC(nv)γC(v)γA(v)=β].

    Which means that BCA. Which contradicts with the hypothesis that BCA. Therefore, either BA or CA. Then A is an intuitionistic fuzzy normed prime ideal.

    Theorem 4.6. Let P be a subset (non-empty) of NR. P is a prime ideal if and only if the intuitionistic characteristic function λP=(μλP,γλP) is an intuitionistic fuzzy normed prime ideal.

    Proof. presume that P is a prime ideal of NR. So by Theorem 3.12, λP is an intuitionistic fuzzy normed ideal of NR. Let A=(μA,γA) and B=(μB,γB) be any intuitionistic fuzzy normed ideals of NR with ABλP while AλP and BλP. Then there exist r,vNR such that

    μA(r)0,γA(r)1andμB(v)0,γB(v)1

    but

    μλP(r)=0,γλP(r)=1andμλP(v)=0,γλP(v)=1

    Therefore, rP and vP. Since P is a prime ideal, there exist nNR such that rnvP.

    Let a=rnv, then μλP(a)=0 and γλP(a)=1. Thus, μAb(a)=0 and γAB(a)=1. but

    μAB(a)=a=st(μA(s)μB(t))μA(r)μB(nv)μA(r)μB(v)min{μA(r),μB(v)}0[Since,μA(r)0andμB(v)0].

    and

    γAB(a)=a=st(γA(s)γB(t))γA(r)γB(nv)γA(r)γB(v)max{γA(r),γB(v)}1[Since,γA(r)1andγB(v)1].

    This is a contradiction with μλP(a)=0 and γλP(a)=1. Thus for any intuitionistic fuzzy normed ideals A and B of NR we have ABλP imply that AλP or BλP. So, λP=(μλP,γλP) is an intuitionistic fuzzy normed prime ideal of NR.

    Conversely, suppose λP is an intuitionistic fuzzy normed prime ideal. Let A and B be two intuitionistic fuzzy normed prime ideal of NR such that ABP. Let rNR, suppose μλAλB(r)0 and γλAλB(r)1, then μλAλB(r)=r=cd(μλA(c)μλB(d))0 and γλAλB(r)=r=cd(γλA(c)γλB(d))1. Then we have c,dNR such that r=cd, μλA(c)0, μλB(d)0 and γλA(c)1, γλB(d)1. Then, μλA(c)=1, μλB(d)=1 and γλA(c)=0, γλB(d)=0. Which implies cA and dB, therefore r=cdABP. Then, μλP(r)=1 and γλP(r)=0. Thus, for all rNR, μλAλB(r)μλP(r) and γλAλB(r)γλP(r). So, λAλBλP. Since λP is an intuitionistic fuzzy normed prime ideal. Then either λAλP or λBλP. Therefore, either AP or BP. Hence P is a prime ideal in NR.

    Definition 4.7. [15] Given a ring R and a proper ideal M of R, M is a maximal ideal of R if any of the following equivalent conditions hold:

    i. There exists no other proper ideal J of R so that MJ.

    ii. For any ideal J with MJ, either J=M or J=R.

    Definition 4.8. An intuitionistic fuzzy normed ideal A of a normed ring NR is said to be an intuitionistic fuzzy normed maximal ideal if for any intuitionistic fuzzy normed ideal B of NR, AB, implies that either B=A or B=λNR. Intuitionistic fuzzy normed maximal left (right) ideal are correspondingly specified.

    Proposition 4.9. Let A be an intuitionistic fuzzy normed maximal left (right) ideal of NR. Then, ImμA = ImγA∣=2

    Theorem 4.10. Let A be an intuitionistic fuzzy normed maximal left (right) ideal of a normed ring NR. Then A={rNR:μA(r)=μA(0) and γA(r)=γA(0)} is a maximal left (right) ideal of NR.

    Proof. As A is not constant, ANR. Then using Proposition 4.9, A is two-valued. Let Im(μA)={1,α} and Im(γA)={0,β}, where 0α<1 and 0<β1. Assume M to be a left ideal of NR in away that AM. Take B be an intuitionistic fuzzy subset of NR where if rM then μB(r)=1 and γB(r)=0 and if rM then μB(r)=c and γB(r)=d, where α<c<1 and 0<d<β. Then B is an intuitionistic fuzzy normed left ideal. Obviously AB. As A is an intuitionistic fuzzy normed maximal left ideal of NR then A=B or B=λNR. If A=B then A=M given that B=M. If B=λNR subsequently M=NR. Therefore, A is a maximal left ideal of NR.

    Theorem 4.11. If A is an intuitionistic fuzzy normed maximal left (right) ideal of NR, then μA(0)=1 and γA(0)=0.

    Proof. Suppose μA(0)1 and γA(0)0 and B to be an intuitionistic fuzzy subset of NR defined as B={rNR:μB(r)=h and γB(r)=k}, where μA(0)<h<1 and 0<k<γA(0). Then, B is an intuitionistic fuzzy normed ideal of NR. We can simply check that AB, BλNR and B={rNR:μB(r)=μB(0) and γB(r)=γB(0)}=NR. Hence, AB but AB and BλNR which contradicts with the assumption that A is an intuitionistic fuzzy normed maximal ideal of NR. Therefore, μA(0)=1 and γA(0)=0.

    Theorem 4.12. Let A be a intuitionistic fuzzy normed left (right) ideal of NR. If A is a maximal left (right) ideal of NR with μA(0)=1 and γA(0)=0, then A is an intuitionistic fuzzy normed maximal left (right) ideal of NR.

    Proof. By Proposition 4.9 A is two-valued. Let Im(μA)={1,α} and Im(γA)={0,β}, where 0α<1 and 0<β1. Define B to be an intuitionistic fuzzy normed left ideal of NR where AB. Hence, μB(0)=1 and γB(0)=0. Let rA. Then 1=μA(0)=μA(r)μB(r) and 0=γA(0)=γA(r)γB(r). Thus μB(r)=1=μB(0) and γB(r)=0=γB(0), hence rB then AB. Given that A a maximal left ideal of NR, then A=B or B=NR. If B=NR subsequently B=λNR. Therefore, A is an intuitionistic fuzzy normed maximal left ideal of NR.

    Remark 4.13. Let ANR and let 0α1 and 0β1. Let λAα,β be an intuitionistic fuzzy subset of NR where μλAα(r)=1 if rA, μλAα(r)=α if rA and γλAβ(r)=0 if rA, γλAβ(r)=β if rA. If α=0 and β=1, the λAα,β is the intuitionistic characteristic function of A, which identified by λA=(μλA,γλA). If NR is a ring and A is an intuitionistic fuzzy normed left (right) ideal of NR, then:

    - μλAα(0)=1, γλAβ(0)=0;

    - (λAα,β)=A, [(λAα,β)={rNR:μλAα(r)=μλAα(0), γλAβ(r)=γλAβ(0)}=A];

    - Im(μA)={1,α} and Im(γA)={0,β};

    - λAα,β is an intuitionistic fuzzy normed left (right) ideal of NR.

    In this article, we defined the intrinsic product of two intuitionistic fuzzy normed ideals and proved that this product is a subset of their intersection. Also, we characterized some properties of intuitionistic fuzzy normed ideals. We initiated the concepts of intuitionistic fuzzy normed prime ideal and intuitionistic fuzzy normed maximal ideal and we established several results related to these ideals. Further, we specified the conditions under which a given intuitionistic fuzzy normed ideal is considered to be an intuitionistic fuzzy normed prime (maximal) ideal. We generalised the relation between the intuitionistic characteristic function and prime (maximal) ideals.

    The author declares no conflict of interest in this paper



    [1] C. Byrne, Iterative oblique projection onto convex sets and the split feasibility problem, Inverse Probl., 18 (2002), 441-453. doi: 10.1088/0266-5611/18/2/310
    [2] H. H. Bauschke, P. L. Combettes, Convex analysis and monotone operator theory in Hilbert spaces, New York: Springer, 2011.
    [3] L. Bussaban, S. Suantai, A. Kaewkhao, A parallel inertial S-iteration forward-backward algorithm for regression and classification problems, Carpathian J. Math., 36 (2020), 35-44. doi: 10.37193/CJM.2020.01.04
    [4] D. P. Bertsekas, J. N. Tsitsiklis, Parallel and distributed computation numerical methods, Belmont: Athena Scientific, 1997.
    [5] A. Beck, M. Teboulle, A fast iterative shrinkage-thresholding algorithm for linear inverse problems, SIAM J. Imaging Sci., 2 (2009), 183-202. doi: 10.1137/080716542
    [6] P. L. Combettes, J. C. Pesquet, A Douglas-Rachford splitting approach to nonsmooth convex variational signal recovery, IEEE J. STSP, 1 (2007), 564-574.
    [7] J. Y. B. Cruz, T. T. A. Nghia, On the convergence of the forward-backward splitting method with linesearchs, Optim. Method. Softw., 31 (2016), 1209-1238. doi: 10.1080/10556788.2016.1214959
    [8] P. L. Combettes, V. R. Wajs, Signal recovery by proximal forward-backward splitting, Multiscale Model. Simul., 4 (2005), 1168-1200. doi: 10.1137/050626090
    [9] J. C. Dunn, Convexity, monotonicity, and gradient processes in Hilbert space, J. Math. Anal. Appl., 53 (1976), 145-158. doi: 10.1016/0022-247X(76)90152-9
    [10] I. Daubechies, M. Defrise, C. D. Mol, An iterative thresholding algorithm for linear inverse problems with a sparsity constraint, Commun. Pure Appl. Math., 57 (2004), 1413-1457. doi: 10.1002/cpa.20042
    [11] M. Figueiredo, R. Nowak, An EM algorithm for wavelet-based image restoration, IEEE T. Image Process, 12 (2003), 906-916. doi: 10.1109/TIP.2003.814255
    [12] A. Hanjing, S. Suantai, A fast image restoration algorithm based on a fixed point and optimization, Mathematics, 8, (2020), 378.
    [13] E. Hale, W. Yin, Y. Zhang, A fixed-point continuation method for l1-regularized minimization with applications to compressed sensing, Rice University: Department of Computational and Applied Mathematics, 2007.
    [14] K. Kankam, N. Pholasa, P. Cholamjiak, On convergence and complexity of the modified forward-backward method involving new linesearches for convex minimization, Math. Meth. Appl. Sci., 42 (2019), 1352-1362. doi: 10.1002/mma.5420
    [15] P. L. Lions, B. Mercier, Splitting algorithms for the sum of two nonlinear operators, SIAM J. Numer. Anal., 16 (1979), 964-979. doi: 10.1137/0716071
    [16] L. J. Lin, W. Takahashi, A general iterative method for hierarchical variational inequality problems in Hilbert spaces and applications, Positivity, 16 (2012), 429-453. doi: 10.1007/s11117-012-0161-0
    [17] B. Martinet, Régularisation d'inéquations variationnelles par approximations successives, Rev. Française Informat. Rech. Opér., 4 (1970), 154-158.
    [18] J. J. Moreau, Fonctions convexes duales et points proximaux dans un espace hilbertien, C. R. Acad. Sci. Paris Sér. A Math., 255 (1962), 2897-2899.
    [19] A. Moudafi, E. Al-Shemas, Simultaneous iterative methods for split equality problem, Trans. Math. Program. Appl., 1 (2013), 1-11.
    [20] Y. Nesterov, A method for solving the convex programming problem with convergence rate O(1/k2), Dokl. Akad. Nauk SSSR., 269 (1983), 543-547.
    [21] Y. Nesterov, Gradient Methods for Minimizing Composite Objective Function, CORE Report, 2007. Available from: http://www.ecore.be/DPs/dp 1191313936.pdf.
    [22] B. T. Polyak, Some methods of speeding up the convergence of iteration methods, USSR Comput. Math. Math. Phys., 4 (1964), 1-17.
    [23] R. T. Rockafellar, On the maximal monotonicity of subdifferential mappings, Pac. J. Math., 33 (1970), 209-216. doi: 10.2140/pjm.1970.33.209
    [24] R. T. Rockafellar, Monotone operators and the proximal point algorithm, SIAM J. Control Optim., 17 (1976), 877-898.
    [25] K. Scheinberg, D. Goldfarb, X. Bai. Fast first order methods for composite convex optimization with backtracking, Found. Comput. Math., 14 (2014), 389-417. doi: 10.1007/s10208-014-9189-9
    [26] S. Suantai, K. Kankam, P. Cholamjiak, A novel forward-backward algorithm for solving convex minimization problem in Hilbert spaces, Mathematics, 8 (2020), 42. doi: 10.3390/math8010042
    [27] S. Suantai, P. Jailoka, A. Hanjing, An accelerated viscosity forward-backward splitting algorithm with the linesearch process for convex minimization problems, J. Inequal. Appl., 2021 (2021), 42. doi: 10.1186/s13660-021-02571-5
    [28] R. Tibshirani, Regression shrinkage and selection via the lasso, J. R. Stat. Soc. B Methodol., 58 (1996), 267-288.
    [29] P. Tseng, A modified forward-backward splitting method for maximal monotone mappings, SIAM J. Control Optim., 38 (2000), 431-446. doi: 10.1137/S0363012998338806
    [30] K. Thung, P. Raveendran, A survey of image quality measures, In: Proceedings of the International Conference for Technical Postgraduates (TECHPOS), Kuala Lumpur, Malaysia, 14-15 December 2009, 1-4.
    [31] K. Tan, H. K. Xu, Approximating fixed points of nonexpansive mappings by the ishikawa iteration process, J. Math. Anal. Appl., 178 (1993), 301-308. doi: 10.1006/jmaa.1993.1309
    [32] K. Toh, S. Yun. An accelerated proximal gradient algorithm for nuclear norm regularized linear least squares problems, Pac. J. Optim., 6 (2010), 615-640.
    [33] Z. Wang, A. C. Bovik, H. R. Sheikh, E. P. Simoncelli, Image quality assessment: from error visibility to structural similarity, IEEE T. Image Process., 13 (2004), 600-612. doi: 10.1109/TIP.2003.819861
  • Reader Comments
  • © 2021 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(2502) PDF downloads(189) Cited by(2)

Figures and Tables

Figures(6)  /  Tables(4)

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog