A Kind of Stochastic Eigenvalue Complementarity Problems

With the development of computer science, computational electromagnetics have also been widely used. Electromagnetic phenomena are closely related to eigenvalue problems. On the other hand, in order to solve the uncertainty of input data, the stochastic eigenvalue complementarity problem, which is a general formulation for the eigenvalue complementarity problem, has aroused interest in research. So, in this paper, we propose a new kind of stochastic eigenvalue complementarity problem. We reformulate the given stochastic eigenvalue complementarity problem as a system of nonsmooth equations with nonnegative constraints. Then, a projected smoothing Newton method is presented to solve it. The global and local convergence properties of the given method for solving the proposed stochastic eigenvalue complementarity problem are also given. Finally, the related numerical results show that the proposed method is efficient.


Introduction
Computational electromagnetics is a science, which spans many subjects.It is an organic combination of mathematical theory, electromagnetic theory, and computer science.Electromagnetism is the classical dynamics theory.Electromagnetic phenomena are also closely related to eigenvalue problems and essentially can be deduced by Maxwell eigenvalue equation.The typical eigenvalue problem of the electromagnetic field is the resonant problem of the cavity and the guide wave problem of the waveguide.No matter the resonance problem of closed cavity, or the propagation problem of uniform guided waves, the wave equation is homogeneous without considering any source or field excitation process.For simplification, the conductors that constitute the cavity are idealized.So the field satisfies the homogeneous boundary condition on the boundary.There are many methods for solving computational electromagnetics, such as finite difference method and finite element method; see literatures [1][2][3][4][5][6][7] for details.In recent years, various methods in computational electromagnetics have been continuously improved.However, the input data are usually affected by many uncertain factors, such as the environment, which often results in the difference between calculation and measurement.So, we consider establishing a new model to solve some related electromagnetics problems.This is also the motivation of this paper.
Among the optimization model, as we all know, the eigenvalue complementarity problem arises from several important applications in engineering and physics, such as the study of the resonance frequency and the stability of dynamic systems.In the last few years, the eigenvalue complementarity problem has drawn increasing attention, in many literature systems, such as [8][9][10][11][12][13] and the references therein.Among them, in [8], the authors study an eigenvalue complementarity problem and find its origins in the solution of a contract problem in mechanics.In [9], the eigenvalue complementarity problems with symmetric real matrices are considered.The authors transform this problem into a differentiable optimization program involving the Rayleigh quotient on a simplex and find its stationary point by the spectral projected gradient algorithm.In [10][11][12][13], many methods are proposed to solve the eigenvalue complementarity problems, such as Levenberg-Marquardt method and the derivative-free projection method.In [14], the stability of dynamic system is studied.The model of eigenvalue complementarity problem is established and the finite element method is used to solve the model.
On the other hand, the stochastic linear complementarity problems, stochastic nonlinear complementarity problems, and some related stochastic optimization problems attract many researchers' attention.And a lot of related theories and algorithms are given, such as [15][16][17][18][19][20] and their references.The expected value (EV) to reformulate the problems is usually used.Many methods are effectual for solving the stochastic complementarity problems.The global and local convergence of these methods are also given.Besides, the expected residual minimization (ERM) also can reformulate the stochastic complementarity problems, such as those in [21][22][23].And the stochastic complementarity problems are eventually transformed into a certain minimal problem with constraints.
In order to reduce the difference between calculation and measurement in computational electromagnetics, we consider getting the approximate distribution, by statistics, of a large number of experimental data and establishing a new stochastic model.So, in this paper, we propose a new kind of stochastic eigenvalue complementarity problem and the smoothing Newton method is also proposed to solve the given problem.
The rest of this paper is organized as follows.In Section 2, we reformulate problem (1) as a nonsmooth system firstly and give the related preliminary results.In Section 3, we present the projected smoothing Newton method and give the convergence results.In Section 4, some preliminary numerical results are proposed.The last section is the conclusion.

Preliminary Results
In this section, we give the reformulation of problem (1) and some related preliminaries.
As ( 3) is a linear complementarity problem, we can use the complementarity functions to transform it.The complementarity function has many different forms, such as the functions given in [24][25][26][27][28][29].Among them, the Fischer-Burmeister (FB) function is used widely.By the FB function transformation, (3) is equivalent to Φ() = 0, where In this paper, we consider using the smoothing Fischer-Burmeister function, which is differentiable at any point.This smoothing method, which is an important method to solve nonsmooth problems, has been widely used in recent years, such as in [30][31][32][33].The idea of a smoothing method is to approximate nonsmooth functions by a sequence of smooth functions.There are many forms of smooth functions.Qi and Chen proposed a smooth approximation method for nonsmooth functions in [34].By introducing the smoothing approximation function of Φ  :   →   ,  > 0, smoothing methods are proposed by Chen et al.So far, a lot of progress has been made in the study of smoothing methods.In this paper, similar to [28], we use the smoothing Fischer-Burmeister function as where  > 0. Therefore, ( 3) is equivalent to Φ  () = 0, where Then, ( 3) and ( 4) are equivalent to the following problem: where and  = (  1 , . . .,    )  ∈   .Let  = (  , , ,   )  ∈  (+1)+2 and define a merit function of (9) as If (1) has a solution, then solving ( 9) is equivalent to finding a global solution of the following optimization problem: min  () where Next, we give some related definitions and propositions.
Definition 1. Assume that  :   →   is a locally Lipschitzian function.The -subdifferential of  at  is where   is the differentiable points set and   (  ) is the Jacobian of  at a point  ∈   .
Definition 2 (see [18]). is said to be semismooth at  if lim exist for any ℎ ∈   .

Projected Smoothing Newton Method and Convergence Analysis
In this section, we present the projected smoothing Newton method and establish the global and local convergence results of the given method.
In the following, we give the convergence results of Method 1.
Theorem 5. Let {  } be an infinite sequence generated by Method 1, and  * is an accumulation point of {  }.If  * is a BD-regular solution of (  ),  * is a solution of () = 0.

Numerical Results
In this section, we give some related numerical experiments.Because problem ( 1) is proposed for the first time, we give the following examples combined with the examples in literatures [1][2][3][4][5][6][7]38].In [7], the authors consider the complex shell inside the complicated enclosures of the electronic component and the high frequency radiation.The enclosure is assumed to be large compared to the wavelength.In this case, wavelengths have strong dependence on the properties of waves, including eigenvalues, eigenfunctions, scattering, and impedance matrices under small perturbations.The field fluctuations within the enclosure are described in a statistical sense using random matrix theory.In [38], in the microwave frequency band, the resonant circuit is usually realized by a resonant cavity.The cavity is filled with air or other mediums, and the electromagnetic oscillation in the cavity can be generated in the cavity by the way of probe and small hole.The waveguide cavity is formed by sealing the waveguides along the ends of the propagation direction with a conductor.Example 10 is converted from a two-dimensional rectangular resonant cavity problem in random system, which is proposed in [3,38].The program is written in Matlab 7.0.The parameters in Method 1 are taken as The stopping rules for Method 1 are ‖∇(  )‖ ≤ 1 − 3 or  max = 2 × 10 4 .
The initial points are randomly selected.In these tables of the numerical results,  * denotes the point of the minimization of (),  denotes the eigenvalue, and DIM denotes the dimension of the problem.
Example 7. We consider problem (1), where The numerical results of Example 7 are given in Table 1 and the values of () at each iteration are shown in Figure 1.
When  takes different values, the numerical results of Example 9 are shown in Table 3 and the values of () at each iteration are shown in Figure 3.
In the following, we consider the application of problem (1) in resonant cavity.
The calculation of the resonator can be described by the field equation and the boundary condition.With passivity  in the cavity, the field vector satisfies the homogeneous Helmholtz equation.The vector potential  and the vector potential   are used, and an arbitrary field vector  is taken as a leading vector.We have where Ψ and Ψ * satisfy On the inner wall  of the cavity, the boundary conditions can often be converted to Ψ *     = 0.
Besides, as we all know, generally speaking, microwave engineering can be divided into two categories: eigenvalue problems and noneigenvalue problems.The problems of the propagation of electromagnetic wave in the waveguide and the distribution of the midfield in the cavity are all eigenvalue problems.In [38], the eigenvalue problem here is as follows: The reason for such a conclusion is that if the problem is abstracted, the Laplace operator ∇ 2 acts on the function Ψ and the product of matrix  and vector  can be regarded as a linear operator  acting on an object .
In [3], the author considers the eigenvalue problem as follows: where  and  are matrices,  is the unknown quantity, and  is the eigenvalue.In a random system, different loop lengths can produce different resonant frequencies, resulting in multiple modes.So we denote the resonant frequencies as , where  is random.
In Example 10 of this paper, we transform the above eigenvalue problem into the eigenvalue complementarity problem (1) and consider the form in the stochastic system.Next, we give Example 10.The numerical results of Example 10 are shown in Table 4 and the values of () at each iteration are shown in Figure 4.

Conclusion
The numerical experiments show that the projected smoothing Newton method is effective in solving the eigenvalue complementarity problems, even for high dimensional problems.After several experiments, we find that the value of  in smoothing function has a great influence on the numerical results.When  = 0.01, the numerical results are optimal.Using the smoothing Newton method, the objective function is polished.So the objective function is guided everywhere.Besides, the projected smoothing Newton method is quadratic.In the application of the projected smoothing Newton method, we consider the two-dimension resonator problem.Therefore, using the new technology of optimization method to further improve the accuracy and efficiency of the given method, the applications in more complicated resonator problems are of great significance and practical value.

Figure 3 :
Figure 3: Change curve of numerical value of () at every iteration, when DIM takes different values.

Figure 4 :
Figure 4: Change curve of numerical value of () at every iteration.

Table 1 :
Numerical results of Example 7.

Table 2 :
Numerical results of Example 8.

Table 3 :
Numerical results of Example 9.

Table 4 :
Numerical results of Example 10.