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Research article

Alternating direction method for the fixed point problem of set-valued mappings with second-order cone double constraints

  • Received: 06 September 2022 Revised: 28 November 2022 Accepted: 08 December 2022 Published: 03 January 2023
  • MSC : 37C25, 65K99

  • This paper focuses on solving a class of equilibrium problems, namely, the fixed point problem of set-valued mappings with second-order cone double constraints. Under certain conditions, the variational inequality form of the fixed point problem of set-valued mappings with second-order cone double constraints is obtained by using the generalized saddle point theory three times. The alternating direction method is used to solve the fixed point problem of set-valued mappings with second-order cone double constraints, and the global convergence of the algorithm is proved. Finally, numerical results of solving five examples with an inexact alternating direction method are given, and the feasibility and effectiveness of the algorithm are demonstrated by comparing with other algorithms.

    Citation: Na Mi, Juhe Sun, Li Wang, Yu Liu. Alternating direction method for the fixed point problem of set-valued mappings with second-order cone double constraints[J]. AIMS Mathematics, 2023, 8(3): 6389-6406. doi: 10.3934/math.2023323

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  • This paper focuses on solving a class of equilibrium problems, namely, the fixed point problem of set-valued mappings with second-order cone double constraints. Under certain conditions, the variational inequality form of the fixed point problem of set-valued mappings with second-order cone double constraints is obtained by using the generalized saddle point theory three times. The alternating direction method is used to solve the fixed point problem of set-valued mappings with second-order cone double constraints, and the global convergence of the algorithm is proved. Finally, numerical results of solving five examples with an inexact alternating direction method are given, and the feasibility and effectiveness of the algorithm are demonstrated by comparing with other algorithms.



    A set-valued mapping is a set-to-set correspondence. Problems in many fields can be abstracted as set-valued mapping models, such as sub-differential mapping of convex functions, optimal solution mapping for optimization problems with parametric variables, optimal approximation mapping, parametric mapping in cybernetics and normal dual mapping in functional analysis. These are common examples that can be abstracted into a set-valued mapping model, which shows that it is very meaningful to carry out research on this; see [1,2,3,4,5,6,7,8].

    The fixed point problem of set-valued mappings with double constraints appears frequently in many fields, such as equilibrium constraint problems [9,10,11,12], n-person games [13,14] and bi-level programming problems [12,15,16,17]. Such problems are different from simple single-objective optimization problems. The constraint conditions are often more complex and diverse, and they are relatively difficult to deal with.

    At present, there are relatively few studies on the fixed point problem of set-valued mappings with double constraints. In [18], the fixed point problem of set-valued mappings with double constraints is discussed, and the properties of symmetric and skew symmetric functions are given. The differential feedback control gradient-type method for solving equilibrium problems with double constraints is innovatively proposed, and its global convergence is proved. However, Antipin [18] only gives the theoretical part. Wang and Zhang [19] used the augmented Lagrange function of the generalized saddle point problem to construct a numerical algorithm on the basis of Antipin's idea. The study [19] proved that the algorithm has global convergence, and some numerical examples are given. However, numerical examples are not well integrated with practical applications.

    Motivated by the above and by [20,21,22], this paper uses the projection idea to give a class of inexact alternating direction methods to solve the fixed point problem of set-valued mappings with second-order cone double constraints, and it obtains very good results.

    In this paper, we introduce some basic knowledge, properties, definitions and theorems in Section 2. In Section 3, we use the generalized saddle point theory three times to obtain the variational inequality form of the fixed point problem of set-valued mappings with second-order cone double constraints. We also solve the variational inequality problem with the exact alternate method and prove the convergence of the algorithm in Section 4. We use the inexact alternating direction method to solve the variational inequality problem and prove the convergence of the algorithm in Section 5. At last, in Section 6, we solve certain types of complex mathematical optimization problems and give examples to illustrate the feasibility and effectiveness of the inexact alternating direction method.

    We will give some basic definitions and properties in this section, which can be found in [18,19,20,21].

    Definition 2.1. [18] A function g from Rn×Rn to Rm is called symmetric onto Ω0×Ω0 if it obeys the equality

    g(v,w)=g(w,v),   wΩ0,  vΩ0.

    Symmetric functions have the following properties:

    Property 2.1. [18] The matrices of gradient-restrictions of vector symmetric functions g(v,w) with respect to variables v and w onto the diagonal of the square Ω0×Ω0 are identical. That is,

    Twg(v,v)=Tvg(v,v),   vΩ0.

    Note that the gradient-restrictions Twg(v,w)v=w and Tvg(v,w)v=w are the gradients of the function g(v,v), and "" is a differential operator.

    Property 2.2. [18] The operator 2wg(v,w)|v=w is potential, i.e., 2Twg(v,v)=Tg(v,v).

    Definition 2.2. [18] A function Φ(v,w) from Rn×Rn to R is called skew symmetric onto Ω0×Ω0 if it obeys the equality

    Φ(w,w)Φ(w,v)Φ(v,w)+Φ(v,v)0,   vΩ0, wΩ0.

    Definition 2.3. [19] If the differentiable map f:RnR is convex, then for any x,yRn,

    f(x),yxf(y)f(x)f(y),yx.

    Definition 2.4. [20] Assuming that ΩRn is a non-empty closed convex set, any projection of xRn on Ω is defined as

    ΠΩ(x)=argminyΩ{yx}.

    Definition 2.5. [21] If KnRn(n1) satisfies

    Kn={(x1;x2)|x1R,x2Rn1,x1x2},

    it is called an n-dimensional second-order cone, also called an ice cream cone.

    Definition 2.6. [21] If x=(x1;x2)R×Rn1, then its projection on the second-order cone is

    ΠK(x)={12(1+x1x2)(x2;x2), if|x1|<x2,x, ifx2x1,0, ifx2x1.

    Definition 2.7. [20] (The non-expansive property of projection) Ω is a closed convex set;

    ΠΩ(u)ΠΩ(v)uv,   u,vΩ.

    Definition 2.8. Π is an orthogonal projection if and only if there is a subspace W of V such that Π maps all elements in V to W, and Π is an identity transformation on W.

    The fixed point problem of set-valued mappings with second-order cone double constraints is as follows: Find vΩ satisfying

    vargmin{Φ(v,w)|g(v,w)K,wΩ}, (3.1)

    where Φ:Rn× RnR, g:Rn× RnRm are two maps, K=Km1×Km2××Kmp, m1,m2,,mp1, and m1+m2++mp=m, Kmi, i=1,2,,p, is an mi-dimensional second-order cone. ΩRn is a non-empty closed convex set. Assume that Φ(v,w) and each component function of vector-valued function g(v,w) are convex in wΩ for any vΩ. It is assumed that the solution mapping w(v)argmin{Φ(v,w)|g(v,w)K,wΩ} is well-defined for any vΩ, and the solution set of the original problem is Ω={vΩ|vw(v)}Ω. Ω is non-empty, which can be obtained from the continuity of Φ(v,w) and the convex property of Φ(v,w) in wΩ for any vΩ; see [25].

    Under the condition that the objective function Φ(v,w) is a skew symmetric function and the constraint condition g(v,w) is a symmetric function, the original problem (3.1) is transformed into the form of a variational inequality by using the generalized saddle point three times.

    Assuming vΩ is the solution of the original problem (3.1), then

    Φ(v,v)Φ(v,w), g(v,w)K,  wΩ. (3.2)

    Now, let Ψ(v,w)=Φ(v,w)Φ(v,v), and then (3.2) can be rewritten as

    Ψ(v,w)0, g(v,w)K,  wΩ. (3.3)

    If Φ(v,w) is a skew symmetric function, according to Definition 2.2, we have that

    Φ(w,w)Φ(v,w)Φ(w,v)+Φ(v,v)0,  wΩ,  vΩ.

    Setting v=v in the above inequality, we obtain

    Φ(w,w)Φ(v,w)Φ(w,v)+Φ(v,v)0,  wΩ,

    which is

    Φ(w,w)Φ(w,v)Φ(v,w)Φ(v,v),  wΩ.

    Combining the above inequality with (3.3), we can get

    Φ(w,w)Φ(w,v)Φ(v,w)Φ(v,v)0,  wΩ.

    Therefore,

    Φ(w,w)Φ(w,v)0, g(v,w)K,  wΩ.

    If g(v,w) is a symmetric function, according to Definition 2.1, the above inequality can be transformed into

    Φ(w,w)Φ(w,v)0, g(w,v)K,  wΩ,

    which is

    Φ(v,v)Φ(v,v)0, g(v,v)K,  vΩ.

    The above inequality implies that

    Ψ(v,v)0, g(v,v)K,  vΩ. (3.4)

    Comparing (3.3) and (3.4), we can see that (v,v) is the saddle point of the function Ψ(v,w) on the symmetric set {g(v,w)K|w,vΩ×Ω}, that is,

    {Ψ(v,v)Ψ(v,v)Ψ(v,w),g(v,v)K, g(v,w)K,  w,vΩ. (3.5)

    Obviously, inequality (3.5) is a generalized saddle point problem. On the one hand, according to Definition 2.3, if the function Φ is convex differentiable in wΩ for any v, then the original problem (3.1) can be described equivalently as follows: Find vΩ satisfying

    wΦ(v,v),wv0, g(v,w)K,  wΩ. (3.6)

    On the other hand, since the function Φ is skew symmetric,  w,vΩ, the monotonicity of wΦ(v,w)|v=w is as follows:

    wΦ(v,v),wvΦ(v,w)Φ(v,v)wΦ(w,w),wv,

    which implies that

    wΦ(w,w)wΦ(v,v),wv0,  w,vΩ. (3.7)

    Setting v=v in (3.7), we get

    wΦ(w,w)wΦ(v,v),wv0,  wΩ. (3.8)

    Combining (3.8) with (3.6), we can obtain

    wΦ(w,w),wv0, g(v,w)K,  wΩ.

    Since g(v,w) is a symmetric function, the above inequality can be rewritten as

    wΦ(v,v),vv0, g(v,v)K,  vΩ. (3.9)

    Comparing (3.6) with (3.9), we can see that (v,v) is the saddle point of the function wΦ(v,v),wv on the symmetric set {g(v,w)K|w,vΩ×Ω}. Hence,

    {wΦ(v,v),vvwΦ(v,v),vvwΦ(v,v),wv,g(v,v)K, g(v,w)K,  w,vΩ. (3.10)

    Obviously, inequality (3.10) is a generalized saddle point problem. The Lagrange function of (3.10) has the following form:

    L(v,w,λ)=wΦ(v,v),wv+λ,g(v,w), (3.11)

    where λK.

    If the feasible region {w|g(v,w)K,wΩ} satisfies the regular condition at w=v, for example, the slater condition holds; then, inequality (3.10) is a convex optimization problem on w. Let v be a feasible solution to the problem (3.10), and we have λΛ(v) such that the function L(v,w,λ) satisfies

    L(v,v,λ)L(v,v,λ)L(v,w,λ),  wΩ,  λK. (3.12)

    Obviously, inequality (3.12) is a generalized saddle point problem.

    For the left side of (3.12):

    wΦ(v,v),vv+λ,g(v,v)wΦ(v,v),vv+λ,g(v,v),  λK,

    which implies

    λλ,g(v,v)0,  λK.

    For the right side of (3.12):

    wΦ(v,v),vv+λ,g(v,v)wΦ(v,v),wv+λ,g(v,w),  wΩ,

    which implies

    wΦ(v,v),wv+λ,g(v,w)g(v,v)0,  wΩ,
    wΦ(v,v),wv+λ,Twg(w,v)(wv)0,  wΩ,
    wΦ(v,v)+wg(v,v)λ,wv0,  wΩ.

    Through the above discussion, we have

    {wΦ(v,v)+wg(v,v)λ,wv0, wΩ,λλ,g(v,v)0, λK. (3.13)

    As we all know, by the equivalent relationship between the projection operator and the variational inequality, inequality (3.13) can be transformed into

    {v=ΠΩ(vα(wΦ(v,v)+wg(v,v)λ)),λ=ΠK(λ+αg(v,v)), (3.14)

    where α>0 is a parameter, and ΠΩ() and ΠK() are projection operators onto the set Ω and second-order cone K, respectively.

    After the above discussion, it is easy to find that when the objective function Φ(v,w) is a skew symmetric function, and the constraint g(v,w) is a symmetric function, v is the solution of the fixed point problem of set-valued mappings with second-order cone double constraints if and only if v satisfies inequality (3.13) or (3.14).

    The classic methods for solving the variational inequality problem (3.13) are the alternating direction method, the projection shrinkage method and the external gradient method. The alternating direction method is more suitable for dealing with large-scale problems that solve the original problem by solving a series of sub-variational inequalities in each iteration.

    Solving the variational inequality problem (3.13) is equivalent to solving the projection equation (3.14). Let the residual function be

    e(u)=(e1(u)e2(u))=(vΠΩ(vα(wΦ(v,v)+wg(v,v)λ))λΠK(λ+αg(v,v))), (4.1)

    where u=(vλ), and α>0 is a parameter. Hence, solving the problem (3.13) is equivalent to finding the zero point of the residual function (4.1). [21] proposes an alternating direction method for structural variational inequalities with equality constraints. Referring to its ideas, we propose Algorithm 1 for the fixed point problem of set-valued mappings with the second-order cone double constraints.

    Algorithm 1. Alternating direction method
    Step 0. Given ϵ1>0,α>0 and λ0K, set k=0.
    Step 1. Find vk+1Ω such that
    (wvk+1)T(wΦ(vk+1,vk+1)+wg(vk+1,vk+1)ΠK(λk+αg(vk+1,vk+1))0,  wΩ.
    Step 2. Update the multiplier
    λk+1=ΠK(λk+αg(vk+1,vk+1)).
    Step 3. Convergence verification: If rk=e(uk+1)ϵ1, the iteration is terminated. Otherwise, proceed to the next step.
    Step 4. Set k=k+1, and return to Step 1.

    On the one hand, it can be obtained in Algorithm 1:

    {vk+1=ΠΩ(vk+1α(wΦ(vk+1,vk+1)+wg(vk+1,vk+1)ΠK(λk+αg(vk+1,vk+1))),λk+1=ΠK(λk+αg(vk+1,vk+1)). (4.2)

    On the other hand, according to the residual function e(uk+1),

    e(uk+1)=(vk+1ΠΩ(vk+1α(wΦ(vk+1,vk+1)+wg(vk+1,vk+1)λk+1))λk+1ΠK(λk+1+αg(vk+1,vk+1))). (4.3)

    Substituting the iteration points in Eq (4.2) into Eq (4.3), the following equation holds by using the non-expansive property of projection.

    e1(uk+1)=vk+1ΠΩ(vk+1α(wΦ(vk+1,vk+1)+wg(vk+1,vk+1)ΠK(λk+1+αg(vk+1,vk+1)))=ΠΩ(vk+1α(wΦ(vk+1,vk+1)+wg(vk+1,vk+1)ΠK(λk+αg(vk+1,vk+1)))ΠΩ(vk+1α(wΦ(vk+1,vk+1)+wg(vk+1,vk+1)ΠK(λk+1+αg(vk+1,vk+1)))αwg(vk+1,vk+1)(ΠK(λk+1+αg(vk+1,vk+1)ΠK(λk+αg(vk+1,vk+1))αwg(vk+1,vk+1)λkλk+1.
    e2(uk+1)=λk+1ΠK(λk+1+αg(vk+1,vk+1)=ΠK(λk+αg(vk+1,vk+1)ΠK(λk+1+αg(vk+1,vk+1)λkλk+1.

    With the help of e(u)2e1(u)2+e2(u)2, we have

    e(uk+1)2(1+α2wg(vk+1,vk+1)2)λkλk+12. (4.4)

    Therefore, in order to prove that the alternating direction method is convergent, it is necessary to prove limkλk+1λk2=0.

    Theorem 4.1. Let u=(v,w,λ) be the solution of problem (3.1), and let {uk}={vk,wk,λk} be the sequence generated by Algorithm 1. If limkλk+1λk2=0 holds, then the algorithm is convergent.

    Proof. The iterative system in Algorithm 1 can be equivalently expressed as the following variational inequalities:

    wΦ(vk+1,vk+1)+wg(vk+1,vk+1)λk+1,vvk+10, (4.5)
    λk+1λkαg(vk+1,vk+1),λλk+10. (4.6)

    Inequalities (4.5) and (4.6) can be expressed in an equivalent manner as

    wΦ(vk+1,vk+1),vvk+1+wg(vk+1,vk+1)λk+1,vvk+10, (4.7)
    λk+1λk,λλk+1αg(vk+1,vk+1),λλk+10. (4.8)

    Let w=vk+1 in the first inequality of (3.13), and adding it to (4.7), we get

    wΦ(vk+1,vk+1)wΦ(v,v),vvk+1+wg(vk+1,vk+1)λk+1wg(v,v)λ,vvk+10. (4.9)

    According to the monotonicity of wΦ(v,w)|v=w, we have

    wg(vk+1,vk+1)λk+1wg(v,v)λ,vvk+10. (4.10)

    In view of the convexity of g and Property 2.2, we obtain

    g(vk+1,vk+1)g(v,v),λλk+10. (4.11)

    Since λK, g(v,w)K. Therefore, λ,g(v,v)=0, λk+1,g(v,v)0, and

    λk+1λk,λλk+1αg(vk+1,vk+1)g(v,v),λλk+10. (4.12)

    Combining (4.11) and (4.12), we have λk+1λk,λλk+10. So, the following formula holds:

    λλk2=λλk+1+λk+1λk2λλk+12+λk+1λk2. (4.13)

    Summing (4.13) from k=0 to k=n yields

    nk=0λλk2nk=0λλk+12+nk=0λk+1λk2, (4.14)

    which is

    λ0λ2λλn+12+nk=0λk+1λk2. (4.15)

    Therefore, the series k=0λk+1λk2 is bounded. According to the convergence of the series, limkλk+1λk2=0 can be obtained.

    In summary, limke(uk+1)2=0, and the convergence of Algorithm 1 is proved.

    Algorithm 1 is a theoretical algorithm. From a practical point of view, we will give an inexact alternating direction method to solve our problem, which is what we will discuss in the next section.

    Although the alternating direction method can decompose the large-scale original problem into a series of sub-problems for solution, it still needs to solve the sub-variational inequalities during each iteration. Due to the difficulty of solving the variational inequalities, this section proposes the new alternating direction method of projection. It can be approximately solved according to the properties of orthogonal projection and second-order cone projection in each iteration. Compared with the precise solution of sub-problems in the previous section, inexact solution is more feasible and faster.

    Using Definition 2.8 of orthogonal projection, the residual function can be expressed as

    e(u)=(e1(u)e2(u))=(vΠΩ(vα(wΦ(v,v)+wg(v,v)λ))λΠK(λ+αg(v,v)))=(α(wΦ(v,v)+wg(v,v)λ)λΠK(λ+αg(v,v))).

    Then, solving the problem (3.13) is equivalent to finding the zero point of the above residual function. We propose an inexact alternating direction method, namely, Algorithm 2, which improves Algorithm 1 and is feasible.

    Algorithm 2. Inexact alternating direction method
    Step 0. Given ϵ1>0,α>0 and λ0K, set k=0.
    Step 1. Find vk+1Ω such that
    vk+1vk+α(wΦ(vk+1,vk+1)+wg(vk+1,vk+1)ΠK(λk+αg(vk+1,vk+1)))=0.
    Step 2. Update multiplier
    λk+1=ΠK(λk+αg(vk+1,vk+1)).
    Step 3. Convergence verification: If rk=e(uk+1)ϵ1, the iteration is terminated. Otherwise, proceed to the next step.
    Step 4. Set k=k+1, return to Step 1.

    This article selects ϵ1=106.

    Let Gk(v)=vvk+α(wΦ(v,v)+wg(v,v)ΠK(λk+αg(v,v))). Because the projection operator ΠK() is semi-smooth (see [23,24]), the equation Gk() in the inexact alternating direction method is also semi-smooth. If any element in Gk(vk+1) is nonsingular, and vk is sufficiently close to vk+1 when k is sufficiently large, then the first iterative formulation in Algorithm 2 can be solved by applying the semi-smooth Newton method of Qi and Sun [23].

    Newton method
    Step 0. Given ϵ0>0 and ξ0=vk, set j=0.
    Step 1. Determine whether Gk(ξj)ϵ0 is satisfied. If it is satisfied, the iteration is terminated,
    and let vk+1=ξj. Otherwise, proceed to the next step.
    Step 2. Calculate dj=[Gk(ξj)]1Gk(ξj).
    Step 3. Set ξj+1=ξj+dj, j=j+1, return to Step 1.

    This article selects ϵ0=106. On the one hand, it can be obtained with Algorithm 2:

    {vk+1vk+α(wΦ(vk+1,vk+1)+wg(vk+1,vk+1)ΠK(λk+αg(vk+1,vk+1)))=0,λk+1=ΠK(λk+αg(vk+1,vk+1)). (5.1)

    On the other hand, according to the residual function e(uk+1),

    e(uk+1)=(α(wΦ(vk+1,vk+1)+wg(vk+1,vk+1)λk+1))λk+1ΠK(λk+1+αg(vk+1,vk+1))). (5.2)

    Combining Eqs (5.1) and (5.2), the following equations are established:

    e1(uk+1)=α(wΦ(vk+1,vk+1)+wg(vk+1,vk+1)ΠK(λk+1+αg(vk+1,vk+1))0=α(wΦ(vk+1,vk+1)+wg(vk+1,vk+1)ΠK(λk+1+αg(vk+1,vk+1))vk+1+vkα(wΦ(vk+1,vk+1)+wg(vk+1,vk+1)ΠK(λk+1+αg(vk+1,vk+1))αwg(vk+1,vk+1)(ΠK(λk+1+αg(vk+1,vk+1)ΠK(λk+αg(vk+1,vk+1))+vk+1vkαwg(vk+1,vk+1)λkλk+1+vk+1vk.
    e2(uk+1)=λk+1ΠK(λk+1+αg(vk+1,vk+1)=ΠK(λk+αg(vk+1,vk+1)ΠK(λk+1+αg(vk+1,vk+1)λk+1λk.

    With the help of e(u)2e1(u)2+e2(u)2, we have

    e(uk+1)2(1+2α2wg(vk+1,vk+1)2)λkλk+12+2vk+1vk2.

    As in Section 4, in order to illustrate that Algorithm 2 is convergent, it is necessary to prove limkλk+1λk2=0 and limkvk+1vk2=0.

    Theorem 5.1. Let u=(v,w,λ) be the solution of problem (3.1), and let {uk}={vk,wk,λk} be the sequence generated by Algorithm 2. If limkλk+1λk2=0 and limkvk+1vk2=0 hold, then the algorithm is convergent.

    Proof. The iterative system in the inexact alternating direction method can be equivalently expressed as the following variational inequality:

    vk+1vk+α(wΦ(vk+1,vk+1)+wg(vk+1,vk+1)λk+1),vvk+1=0, (5.3)
    λk+1λkαg(vk+1,vk+1),λλk+10. (5.4)

    (5.3) and (5.4) can be expressed in an equivalent manner as

    vk+1vk,vvk+1+αwΦ(vk+1,vk+1),vvk+1+αwg(vk+1,vk+1)λk+1,vvk+1=0, (5.5)
    λk+1λk,λλk+1αg(vk+1,vk+1),λλk+10. (5.6)

    Let w=vk+1 in the first inequality of (3.13). Adding it to the (5.5), we get

    vk+1vk,vvk+1+αwΦ(vk+1,vk+1)wΦ(v,v),vvk+1+αwg(vk+1,vk+1)λk+1wg(v,v)λ,vvk+10.

    According to the monotonicity of wΦ(v,w)|v=w, we have

    vk+1vk,vvk+1+αwg(vk+1,vk+1)λk+1wg(v,v)λ,vvk+10. (5.7)

    In view of the convexity of g and Property 2.2, we obtain

    vk+1vk,vvk+1+α2g(vk+1,vk+1)g(v,v),λλk+10. (5.8)

    Since λK, g(v,w)K. Therefore, λ,g(v,v)=0, λk+1,g(v,v)0, and

    λk+1λk,λλk+1αg(vk+1,vk+1)g(v,v),λλk+10. (5.9)

    Combining (5.8) and (5.9), we have vk+1vk,vvk+1+12λk+1λk,λλk+10. So, the following inequation holds:

    12λλk2+vvk212λλk+12+12λk+1λk2+vvk+12+vk+1vk2. (5.10)

    Summing (5.10) from k=0 to k=n yields

    12λ0λ2+v0v212λλn+12+vvn+12+12nk=0λk+1λk2+nk=0vk+1vk2. (5.11)

    Therefore, the series k=0λk+1λk2 and k=0vk+1vk2 are bounded. According to the convergence of the series, limkλk+1λk2=0 and limkvk+1vk2=0 can be obtained.

    In summary, limke(uk+1)2=0, and the convergence of the inexact alternating direction method is proved.

    The mathematical programming problem with equilibrium constraints is a constraint optimization problem with variational inequalities or complementary systems in the constraints. The function f is the overall objective function of the minimization problem, F is the internal equilibrium function, and Z is the joint upper-level feasible region. For each given xX, X={xRn|(x,y)Z}, and the range C(x) of y is defined. Assuming that Xdom(C), the mathematical programming problem with equilibrium constraints can be expressed as

    minf(x,y)
    s.t (x,y)Z
    yΩ, (6.1)

    where, for each xX, Ω is the solution set of the variational inequality defined by VI(F(x,),C(x)). That is, yΩ is equivalent to yC(x) and satisfies F(x,y),vy0,  vC(x).

    If there is a real-valued function Φ(x,y) that satisfies F(x,y)=yΦ(x,y), in this special case, problem (6.1) is usually regarded as a bi-level optimization problem. Then, the bottom optimization problem of the bi-level optimization problem can be expressed as

    minΦ(x,y)
    yΩ. (6.2)

    Therefore, for any given vector x, there is yC(x), C={xΩ|g(x,y)K},

    yargmin{Φ(x,y)|g(x,y)K,xΩ}.

    It can represent the optimal solution mapping S(x) of the above problem (6.2), and the solution mapping S is a set-valued mapping from Rn to Rm. Next, we will solve Examples 1–5 with Algorithm 2.

    Example 1. Economic equilibrium model

    minf(v,w)
    s.t (v,w)Z
    Nw+Mv+m,wv0,  wΩ
    vC(w),

    where C(w)={wΩ|g(v,w)K}, g(v,w)=v+w. The bottom optimization problem can be processed as

    vargmin{N2w+Mv+m,w|g(v,w)K,wΩ}. (6.3)

    Here, Φ(v,w)=N2w+Mv+m,w is a skew symmetric function, and g(v,w)=v+w is a symmetric function.

    In this example, α=0.5 is used, and then the expression of Hj in step j in Newton method is as follows:

    Hj=In+α(N+M)+2α2ΠK(λk+2αξj).

    Table 1 shows the numerical results of Example 1, where Iter is the number of iterations of the solution process, and Time is the CPU calculation time when the program terminates.

    Table 1.  Numerical calculation results of problem (6.3).
    n K Iter Time rk ϵ0 ϵ1
    50 K50 21 5.468750e-01 1.797038e-07 106 106
    100 K50×K50 28 8.437500e-01 6.575534e-07 106 106
    200 K50×K50×K100 28 1.281250e+00 8.227579e-07 106 106

     | Show Table
    DownLoad: CSV

    The more general situation of Nash equilibrium n-person game: Let fi(xi,xi) be the economic cost function of the i-th player, iI. This function not only depends on the strategy xiXi of this player, where Xi=(xi)iI is a convex set, but it also depends on the strategies xiXi, Xi=(xj)iIi of other players. Then, the equilibrium point of the n-person game is the solution contained in the following set:

    xiargmin{fi(xi,xi)|xiXi}.

    Let function Φ(v,w)=ni=1fi(xi,xi), where v=(xi),w=(xi), g(v,w)=(gi(xi,xi)), i=1,2,,n. Here, (v,w)=(xi,xi)Ω×Ω, Ω=X1×X2××Xn, and then

    vargmin{Φ(v,w)|g(v,w)K,wΩ}.

    Example 2. Cournot duopoly model

    vargmin{Φ(v,w)|g(v,w)K,wΩ}, (6.4)

    where Φ(v,w)=wT(1001)w+vT(0110)wwTu, (u>0), is a skew symmetric function, and g(v,w)=((w11)2(v11)2(w21)2(v21)2) is a symmetric function.

    Let N=(2002), M=(0110). In this example, α=0.5 is used, and then the expression of Hj in step j in Newton method is as follows:

    Hj=In+α(N+M)2αdiag1in{ΠK(λkiα(ξji1)2)4α(ξji1)2ΠK(λki2α(ξji1)2)}.

    Table 2 shows the numerical results of Example 2.

    Table 2.  Numerical calculation results of problem (6.4).
    n K Iter Time rk ϵ0 ϵ1
    2 K2 26 6.093750e-01 9.330831e-07 106 106

     | Show Table
    DownLoad: CSV

    Consider the following bi-level programming problem:

    min(v,w)f(v,w)
    s.t h(v,w)0
    vΩ, (6.5)

    where Ω represents the solution set of the underlying problem, which is

    minvΦ(v,w)
    s.t g(v,w)K. (6.6)

    Then, the set of reasonable reflections of the underlying problem is

    Ω={v|vargmin{Φ(v,w)|g(v,w)K,wΩ}}.

    Example 3. Consider the following bi-level programming problem in R2:

    minwv
    s.t (v,w)0
    vargmin{v22|v+wK,wΩ}. (6.7)

    Obviously, problem (6.7) has the only optimal solution (0,0).

    Example 4. Consider the bottom optimization problem in the bi-level programming problem in Rn:

    vargmin{w22|g(v,w)K,wΩ}, (6.8)

    where Φ(v,w)=w22 is a skew symmetric function, and g(v,w)=(w1sin(v1)+v1sin(w1)w2sin(v2)+v2sin(w2)wnsin(vn)+vnsin(wn)) is a symmetric function.

    In this example, α=0.5 is used, and then the expression of Hj in step j in Newton method is as follows:

    Hj=(1+α)In+αdiag1in{[2cos(ξji)ξjisin(ξji)]ΠK(λki+2αξjisin(ξji))}+2α2diag1in{[sin(ξji)+ξjicos(ξji)]2ΠK(λki+2αξjisin(ξji))}.

    Table 3 shows the numerical results of Example 3.

    Table 3.  Numerical calculation results of problem (6.8).
    n K Iter Time rk ϵ0 ϵ1
    50 K50 11 5.312500e-01 5.085837e-07 106 106
    100 K50×K50 17 1.218750e+00 5.810239e-07 106 106
    200 K50×K50×K100 17 3.843750e+00 8.325520e-07 106 106

     | Show Table
    DownLoad: CSV

    To illustrate the effectiveness and feasibility of the algorithm, we compared Algorithm 2 with the algorithm in [19] under the condition of achieving the same accuracy, and the numerical results are shown below. Here, ALM denotes the augmented Lagrange method in [19], and IADM denotes Algorithm 2 in this paper.

    Table 4.  Comparison of numerical calculation results of Algorithm 2 and algorithm in [19].
    Ex1(n=50) Ex4(n=50) Ex1(n=100) Ex4(n=100)
    Method Time Iter Time Iter Time Iter Time Iter
    ALM 7.187500e-01 23 6.562500e-01 11 9.843750e-01 23 1.187500e+00 12
    IADM 5.468750e-01 21 5.312500e-01 11 8.437500e-01 28 1.218750e+00 17
    Ex1(n=200) Ex4(n=200) Ex2(n=2)
    Method Time Iter Time Iter Time Iter
    ALM 1.859375e+00 23 4.218750e+00 18 6.250000e-01 42
    IADM 1.281250e+00 28 3.843750e+00 17 6.093750e-01 26

     | Show Table
    DownLoad: CSV

    As can be seen from the above table, Algorithm 2 is better than the algorithm in [19].

    Let K be a nonempty closed and convex subset of a Hilbert space E and f:E×ER be a bi-function satisfying f(x,x)=0 for any xK. The equilibrium problem for the bi-function f on K is defined as follows:

    Find xK such that f(x,y)>0,  yK. (6.9)

    Problem (6.9) is equivalent to the following optimization problem with non-negative optimal value:

      minf(x,y)
    s.t  yK. (6.10)

    That is,

    xargmin{f(x,y)yK}. (6.11)

    Example 5. Consider that f:E×ER is defined by

    f(x,y)=Ax+By+c,yx,  yK, (6.12)

    where A=(3.1200023.6000003.5200023.3000003), B=(1.6100011.6000001.5100011.5000002), c=(12121), and K={xR55xi5}. It is easy to verify that f(x,y) is a skew symmetric function. Let a=(5,5,5,5,5)T, g(x)=xa, and h(x)=xa. We have

    xargmin{Ax+By+c,yxyK}, (6.13)

    where K={xR5g(x)0,h(x)0}.

    In this example, α=0.5 is used, and then the expression of Hj in step j in Newton method is as follows:

    Hj=I5+α(A+B)αI5(βγ),

    where β={0,ifpα(ξ+c)<0,α,ifpα(ξ+c)>0,0 or α,ifpα(ξ+c)=0, and γ={0,ifμ+α(ξc)<0,α,ifμ+α(ξc)>0,0 or α,ifμ+α(ξc)=0.Note that pR5+ and μR5+ are Lagrange multipliers of problem (6.13).

    Table 5 shows the numerical results of Example 5.

    Table 5.  Numerical calculation results of problem (6.13).
    n Iter Time rk ϵ0 ϵ1
    5 23 0.468750 6.202562e-07 106 106

     | Show Table
    DownLoad: CSV

    Table 6 shows the numerical results of [26].

    Table 6.  Numerical calculation results of [26].
    n λn Iter Time ϵ1
    5 1n+1 102 0.9632 106
    5 1n+1 18 0.1919 106
    5 1log(n+2) 13 0.1279 106
    5 1(n+1)log(n+3) 123 1.1854 106
    5 log(n+3)n+1 77 0.7494 106
    5 1loglog(n+20) 18 0.1797 106

     | Show Table
    DownLoad: CSV

    By comparing with [26], we can see that in the cases of 1n+1, 1(n+1)log(n+3) and log(n+3)n+1, the algorithm of this paper has obvious advantages.

    This paper mainly studies the fixed point problem of set-valued mappings with second-order cone double constraints by using the alternating direction method. With the help of generalized saddle point theory, the variational inequality problems of the fixed point problem of set-valued mappings with second-order cone double constraints are obtained. By using the projection idea, a new alternating direction method is proposed. The new alternating direction method requires only simple projection calculations for each iteration, avoiding the problem of solving sub-variational inequalities. Finally, five complex problems with second-order cone double constraints in practical applications are given. An inexact alternating direction method is used to solve them, which further illustrates the practicability and effectiveness of the algorithm. Furthermore, the results for fixed-point problems can also be applied to [27,28,29].

    Although the research in this paper has achieved some results, there are still many questions worthy of our continued research. For example, the convergence rate of the new alternating direction method still needs to be estimated. In practical applications, the number of variables is often more than two, and the problem of which alternating direction method is applicable to multiple separable variables is also the focus of future research.

    This work was supported by the general project of Liaoning Provincial Department of Education under project No. LJKMZ20220524 and the National Natural Science Foundation of China under project No. 10771026.



    [1] H. Kuk, T. Tanino, M. Tanaka, Sensitivity analysis in parametrized convex vector optimization, J. Math. Anal. Appl., 202 (1996), 511–522. https://doi.org/10.1006/jmaa.1996.0331 doi: 10.1006/jmaa.1996.0331
    [2] P. H. Sach, Sufficient conditions for generalized convex set-valued maps, Optimization, 37 (1996), 293–304. http://doi.org/10.1080/02331939608844223 doi: 10.1080/02331939608844223
    [3] W. Song, Weak subdifferential of set-valued mappings, Optimization, 52 (2003), 263–276. https://doi.org/10.1080/0233193031000120051 doi: 10.1080/0233193031000120051
    [4] H. Leiva, N. Merentes, K. Nikodem, J. L. Sanchez, Strongly convex set-valued maps, J. Glob. Optim., 57 (2013), 695–705. https://doi.org/10.1007/s10898-013-0051-4 doi: 10.1007/s10898-013-0051-4
    [5] I. Beg, A. R. Butt, Fixed point of set-valued graph contractive mappings, J. Inequal. Appl., 2013 (2013), 1–7. https://doi.org/10.1186/1029-242X-2013-252 doi: 10.1186/1029-242X-2013-252
    [6] S. J. Li, X. Q. Yang, G. Y. Chen, Nonconvex vector optimization of set-valued mappings, J. Math. Anal. Appl., 283 (2003), 337–350. https://doi.org/10.1016/S0022-247X(02)00410-9 doi: 10.1016/S0022-247X(02)00410-9
    [7] N. Sisarat, R. Wangkeeree, T. Tanaka, Sequential characterizations of approximate solutions in convex vector optimization problems with set-valued maps, J. Glob. Optim., 77 (2020), 273–287. https://doi.org/10.1007/s10898-019-00864-0 doi: 10.1007/s10898-019-00864-0
    [8] M. Abbas, B. Ali, C. Vetro, A Suzuki type fixed point theorem for a generalized multivalued mapping on partial Hausdorff metric spaces, Topology Appl., 160 (2013), 553–563. https://doi.org/10.1016/j.topol.2013.01.006 doi: 10.1016/j.topol.2013.01.006
    [9] F. M. Yao, Research on several theories and applications of mathematical programming with equilibrium constraints, Harbin University of Science and Technology, 2007.
    [10] A. H. Bajgiran, J. Jang, A study of subsidizing a biofuel supply chain to incentivize the production of advanced biofuel: an equilibrium problem with equilibrium constraints approach, Int. J. Energy Res., 45 (2021), 16932–16946. https://doi.org/10.1002/er.6914 doi: 10.1002/er.6914
    [11] E. Allevi, A. J. Conejo, G. Oggioni, R. Riccardi, C. Ruiz, Evaluating the strategic behavior of cement producers: an equilibrium problem with equilibrium constraints, Eur. J. Oper. Res., 264 (2018), 717–731. https://doi.org/10.1016/j.ejor.2017.06.043 doi: 10.1016/j.ejor.2017.06.043
    [12] A. S. Antipin, The convergence of proximal methods to fixed points of extremal mappings and estimates of their rate of convergence, Comput. Maths. Math. Phys., 35 (1995), 539–551.
    [13] J. F. Nash, Equilibrium points in n-person games, In: Proceedings of the National Academy of Sciences of the United States of America, 36 (1950), 48–49. https://doi.org/10.1073/pnas.36.1.48
    [14] D. J. Goehring, J. P. Kahan, The uniform n-person prisoner's dilemma game: construction and test of an index of cooperation, J. Conflict Resolut., 20 (1976), 111–128. https://doi.org/10.1177/002200277602000104 doi: 10.1177/002200277602000104
    [15] Q. X. Cheng, Y. H. Chen, Z. Y. Liu, A bi-level programming model for the optimal lane reservation problem, Expert Syst. Appl., 189 (2022), 116–147. https://doi.org/10.1016/j.eswa.2021.116147 doi: 10.1016/j.eswa.2021.116147
    [16] B. L. Lin, J. P. Wu, J. X. Wang, J. S. Duan, Y. N. Zhao, A bi-level programming model for the railway express cargo service network design problem, Symmetry, 10 (2018), 227. https://doi.org/10.3390/sym10060227 doi: 10.3390/sym10060227
    [17] M. Wei, B. Sun, W. Z. Jin, A bi-level programming model for uncertain regional bus scheduling problems, J. Transp. Syst. Eng. Inform. Tech., 13 (2013), 106–112. https://doi.org/10.1016/S1570-6672(13)60120-8 doi: 10.1016/S1570-6672(13)60120-8
    [18] A. Antipin, Differential equations for equilibrium problems with coupled constraints, Nonlinear Anal., 47 (2001), 1833–1844. https://doi.org/10.1016/S0362-546X(01)00314-5 doi: 10.1016/S0362-546X(01)00314-5
    [19] L. Wang, F. Shan, L. W. Zhang, An implementable augmented Lagrange method for solving fixed point problems with coupled constraints, Nonlinear Anal., 74 (2011), 1761–1768. https://doi.org/10.1016/j.na.2010.10.048 doi: 10.1016/j.na.2010.10.048
    [20] A. Bnouhachem, M. H. Xu, M. Khalfaoui, Z. H. Sheng, A new alternating direction method for solving variational inequalities, Comput. Math. Appl., 62 (2011), 626–634. https://doi.org/10.1016/j.camwa.2011.05.043 doi: 10.1016/j.camwa.2011.05.043
    [21] B. S. He, H. Yang, S. L. Wang, Alternating direction method with self-adaptive penalty parameters for monotone variational inequalities, J. Optimiz. Theory Appl., 106 (2000), 337–356. https://doi.org/10.1023/A:1004603514434 doi: 10.1023/A:1004603514434
    [22] M. Sun, A new projection-type alternating direction method for monotone variational inequality problems, J. Oper. Res. Soc. Japan, 52 (2009), 1–10. https://doi.org/10.15807/jorsj.52.1 doi: 10.15807/jorsj.52.1
    [23] L. Q. Qi, J. Sun, A nonsmooth version of Newton's method, Math. Program., 58 (1993), 353–367. https://doi.org/10.1007/BF01581275 doi: 10.1007/BF01581275
    [24] F. Facchinei, J. S. Pang, Finite-dimensional variational inequalities and complementarity problems, New York: Springer, 2003. https://doi.org/10.1007/b97544
    [25] J. P. Aubin, H. Frankowska, Set-valued analysis, Boston: Birkhäuser, 1990.
    [26] H. ur Rehman, N. Pakkaranang, A. Hussain, N. Wairojjana, A modified extra-gradient method for a family of strongly pseudomonotone equilibrium problems in real Hilbert spaces, J. Math. Comput. Sci., 22 (2021), 38–48. https://doi.org/10.22436/jmcs.022.01.04 doi: 10.22436/jmcs.022.01.04
    [27] K. Muangchoo, A new strongly convergent algorithm to solve pseudo-monotone equilibrium problems in a real Hilbert space, J. Math. Comput. Sci., 24 (2022), 308–322. https://doi.org/10.22436/jmcs.024.04.03 doi: 10.22436/jmcs.024.04.03
    [28] K. Muangchoo, Explicit Halpern-type iterative algorithm for solving equilibrium problems with applications, J. Math. Comput. Sci., 25 (2022), 115–132. https://doi.org/10.22436/jmcs.025.02.02 doi: 10.22436/jmcs.025.02.02
    [29] J. N. Ezeora, P. C. Jackreece, Iterative solution of split equilibrium and fixed point problems in real Hilbert spaces, J. Nonlinear Sci. Appl., 14 (2021), 359–371. https://doi.org/10.22436/jnsa.014.05.06 doi: 10.22436/jnsa.014.05.06
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