INTRODUCTION
Linear variation inequality problem is a fundamental field of research in operation research and plays a significant role in differential equation, mechanics, cybernetics, game theory, economy and finance, transport and many other fields. Linear variation inequality problem can be described: Suppose K is a nonempty closed convex subset of R^{n}, we want to find a vector x*εK such that the inequality
Holds for all yεK, where M = (M_{ij})_{n*n} is a n*n real matrix, p = (p1, p2,…, pn)TεR^{n}. So far, there are several methods developed for solving the problem, such as iterative methods, projection methods, Newton methods and so on (Dafermos, 1983; Taji et al., 1993; Marcotte and Wu, 1995; Jian Jinbao and Yanlian, 1997). It usually claims parallel solving linear variation inequality problem in engineering and technology, but the iterative algorithm that based on traditional digital computer is hard to satisfy the demands of parallel solving. Furthermore, the convergence of iterative methods has much to do with the selection of initial values. So it has important actual meaningful to parallel solving the problem (1) by studying and using evolution strategies.
The evolution strategies algorithm (Schwefel and Back, 1995a,b; Zhi and Tao, 2000; Qingxia, 1997) with global convergence is another research hotspot in design optimization in recent years. Evolution strategies is an intelligence optimization algorithm that imitative natural selection and heredity mechanism, which was put forward by (Schwefel, 1965) in order to research the fluid mechanics of wind tunnel. Evolution strategies is a self adaptation and stochastic algorithm, which originates from the Evolution Theory Darwinian. The algorithm comes from the simple natural evolution model. A population is made up of a test point set of objective function search space. Every individual's fitness can be decided by the objective function value of the corresponding test points. Every individual's reproduction rate depends on the relative fitness of the individual in current population. They select parents (test points) in the population according to the reproduction rate and selection operator. The selected parents can generate the offspring by recombination and mutation. The recombination stochastic combine the genetic information of parents while the mutation stochastic change the genetic information of parents. The two obvious features are hidden parallelism and population global search. Furthermore, evolution strategies are extremely robust, which has unique predominant performance for solving some complex nonlinear systems. The algorithm has been commonly used in disposing varied optimizations at present, such as the training and design of neural network, system identification, robotic control, machine learning and many other fields.
This study presents a solution to solve linear variation inequality by evolution strategies based on the global convergence and parallel search of evolution strategies algorithm. The algorithm can actualize parallel search in the feasible region and convergent to the global optimal solution, which efficiently overcome the defect that the traditional methods can't parallel solute the linear variation inequality in engineering technique. The numerical computation results indicate that the algorithm has high convergence rate and high accuracy. It is a very effective way to solve linear variation inequality, which is comfortable for facilitating practical application.
THE THEORY OF LINEAR VARIATION INEQUALITIES
In this study, we select the 2norm as the vector norm. For explicitness, we
define:
K = {xεR^{n}  ¨Ox = (x_{1},x_{2},L,x_{n})^{T},
x_{i}ε [a_{i},b_{i}], I =1,2,…,n} 
Where a_{i}≤=b_{i}q = (q_{1},q_{2},…,q_{n})^{T}ε.
We define the function:
Fk (q) = (F_{k}^{(1)} (q_{1}),
F_{k} ^{(2)} (q_{2}), L, F_{k}^{ (I)}
(q_{i}), L, F_{k.}^{(n)} (q_{n}))^{T} 
Where
Obviously, F_{K}^{(I)} (q_{i}) is a monotone continuous function whose global Lipschitz constant is 1.
Lemma 1: (Eaves, 1971; Mosco, 1985) The necessary and sufficient condition for x* is a solution to the problem (1) is if and only if
Lemma 2: (Zhanquan et al., 1999). The Eq. (2)
has the solution in K.
Proof: Let H(x) = F_{K}(xMxp), then H is a continuous operator from R^{n }into itself, so it is a compact operator. H is a mapping from K to K too, so H (x) has a fixed point in K at least, namely the equation (2) has the solution in K.
STANDARD EVOLUTION STRATEGIES^{ }
We describe the process of standard evolution strategies simply (Qingxia, 1997):
Step 1: Determine the expressive form of the problem. Every individual of the expression is made up of goal variation X and standard deviation σ, each part also has n components. Namely,
(X, σ) = ((x _{1}, x _{2}, ...,
x_{i}, ...., x_{n}), (σ_{1}, σ_{2},
...., σ_{i}, .... σ_{n})) 
The relations between X and σ are
where (x_{i}, σ_{i}) denotes the i_{th} component
of the above generation's individual, denotes the i_{th} component of
the offspring's individual, N(0,1) denotes the stochastic numbers that obeys
standard normal distribution, N_{i}(0,1) denotes recreating the stochastic
numbers that obeys standard normal distribution according to the i_{th}
components, r^{i} denotes global coefficient which always choose 1 and
r denotes local coefficient which always choose 1. The above formula indicates
that the new individuals are generated from the old individuals.
Step 2: Initialize population with random and compute these fitness. The initial population of the evolution strategies is made up of μ individuals. Every individual (X, σ) includes n x_{i }and σ_{i} components. The method of generating initial population generates randomly. In order to compare with the traditional methods, we can start from some starting point (X (0), σ (0)), which generates μ initial individuals by multiple mutation. The initial point can randomly select from the feasible region and the standard deviation of the initial individual is σ (0) = 3.0.
Step 3: Compute the fitness of the initial individuals. If it satisfies the conditions, then terminate. If not, carry on.
Step 4: Generate the new population with the following operations in
accordance with the evolution strategies.
• 
Recombination. Generate the new individuals by exchanging the goal variations
and the standard deviations of the two parent individuals. The goal variation
adopts discrete recombination in general and the standard deviation always
adopts mean recombination. 
• 
Mutation. We adds stochastic variations to the recombinant individuals
and generates the new individuals in term of the formula (3). 
• 
Compute the new individuals' fitness. 
• 
Selection. We select the good individuals, to make up of the next generation
population in term of the strategy of selection (μ, λ). 
Step 5: Repeat execute the step 4 until it satisfies the termination conditions and then select the best individual as the result of the evolution strategies.
IMPROVED EVOLUTION STRATEGIES
We know the one dimension Cauchy density function concentrates on the origin's neighborhood, which can be defined (Gehlhaar and Fogel, 1997):
where t denotes a proportional parameter. The corresponding distribution function
can be defined:
The density function f_{t}(x) is similar to the Gaussian density function.
The differences between the Cauchy density function and the Gaussian density
function are as following: The Cauchy distribution is a little smaller than
the Gaussian distribution in vertical direction, but the Cauchy distribution
is next to horizontal axis in horizontal direction and it is slower for the
Cauchy distribution. So, the Cauchy distribution can be seen infinite. The relations
between the Cauchy density function and the Gaussian density function can be
seen in Fig. 1.
The Cauchy distribution easily generates a random number which is far away
the origin, according that the Cauchy distribution has the high two wings probability
characteristics. The random numbers generated by the Cauchy distribution have
more breadth distribution than the Gaussian distribution.

Fig. 1: 
Compared
the Cauchy distribution (t =1) with the Gaussian distribution (π
(0,1)) 
It means that the
evolution maybe jump out the local minimal area with the Cauchy distribution
quickly using Cauchy mutation generate the offspring. However, the Cauchy distribution's small middle part is its weakness which
cuts down the performance in accurate local search. But the Gaussian mutation
operator can make the population meticulous search in local space. In order
to exert their advantage, it presents an eclectic scheme in the thirteenth documentary,
which combines the Cauchy mutation operator with the Gaussian mutation operator
and defines a new mutation operator which is called mean mutation operator.
So, the new mutation form is
where ½(ηi + Ni(0,1)) is mean mutation operator and the parameter çi is a proportional parameter which obeys the Cauchy distribution when t equals 1. It is used to renew each component. The other parameters equal the parameters of the formula (3).
THE ALGORITHM PROCESS OF SOLVING LINEAR VARIATION INEQUALITY BY EVOLUTION
STRATEGIES
Solving the linear variation inequality can be converted function optimization in the field K. Obviously, the solution x* to the variation inequality is the best solution to the function optimization.
Step 1: Determine the expression form of the problem. Every individual of the expression is made up of goal variation X and standard deviation σ, which each part also has n components. Namely,
where x_{1}, x_{2}, …, x_{n}
are the components of the vector x*, n denotes the dimension of the vector
x*.
Step 2: Initialize population with random. The initial population of the evolution strategies is made up of μ individuals. Every individual includes x_{i }and σ_{i} components. The method of generating initial population is random. In order to compare with the traditional methods, we can start from some starting point (X(0),σ (0)) which generates μ initial individuals by multiple mutation. The initial point can randomly select from the feasible region and the standard deviation of the initial individual is σ (0) = 30.

Fig. 2: 
a)The numerical distribution map of the fitness of initial
population's individual; b) the numerical distribution map of the fitness
of the population's individuals when termination and c) the variety of the
best fitness of each generation in evolution 
Step 3: Compute the fitness x* is the solution to the formula (1) when
x* = F_{K} (x*Mx*p) according to the formula (2). Let
y' = x  F_{K} ( x Mx p) 
we know y' = y'_{1}, y'_{2}, …, y'_{n}, y'εK,
so we select the fitness function
The solution x* to the formula (1) is the value which makes f (x*) =
0. Suppose e = f (x), then the accurate solution x* to the formula (1) is the
value of x which makes e = 0. The smaller of the value of e is, the better of
the similar to the solution to the formula (1) is. We select the fitness function
e and select ε near to zero for the termination conditions. The evolution
terminate when the value of the fitness is smaller than ε.
Step 4: If it satisfies the conditions, then terminate. If not, carry on.
Step 5: Generate the new population with the following operations in
accordance with evolution strategies.
• 
Recombination. Generate the new individuals by exchanging the goal variation
and the standard deviation of the two parent individuals. The goal variation
always adopts discrete recombination and the standard deviation always adopts
mean recombination. 
• 
Mutation. Add stochastic variations to the recombinant individuals and
generate the new individuals in term of the formula (4). 
• 
Compute the new individuals' fitness as the step 3. 
• 
Select. Select the good individuals and make them composite the next generation
population in term of the strategy of selection (μ, λ) (Fig.
2). 
Step 6: Repeat execute the step 5 until it satisfies the termination conditions and select the best individual as the result of the evolution strategies.
SIMULATION RESULT
In this study, we solve the linear variation inequalities by (Table
1) evolution strategies for Example.
Let n = 2, μ = 15, λ =100, ε = 10^{–10}. 
K = [0, 2]x[0, 2]x[0, 2]x[0, 2]x[0, 2]x[0, 2]x[0,
2]x[0, 2] 
Table 1: 
The simulated results of the solution of the liner variation
inequalities (5) 

We want to find a vector x*εK such that the inequality holds for all yεK.
(y x*)^{T} (Mx* +p)≥0 (5) 
CONCLUSIONS
This study presents a solution to linear variation inequality by evolution strategies based on the advantage of evolution strategies such as group search and global convergence. We select the mean mutation operator in order to get quick convergence. The Algorithm avoids the trivial deviate operation in solving linear variation inequalities with traditional methods, whose parallelism satisfies the question of parallel solving the linear variation inequality in engineering technique. Global convergence to the best solution, high accuracy, high convergence rate and parallelism make the algorithm easily use in engineering.