Fixed-Point Iteration Method for Solving the Convex Quadratic Programming with Mixed Constraints

The present paper is devoted to a novel smoothing function method for convex quadratic programming problem with mixed constrains, which has important application in mechanics and engineering science. The problem is reformulated as a system of non-smooth equations, and then a smoothing function for the system of non-smooth equations is proposed. The condition of convergences of this iteration algorithm is given. Theory analysis and primary numerical results illustrate that this method is feasible and effective.


Introduction
Consider the following convex quadratic programming problem where 11 12 is an n n × symmetrical positive semi-definite partitioned matrix, A i j = are the partitioned matrix of Q and A respectively.As is well known, this problem models a large spectrum of applications in computer science.Operation research and engineering as the nonlinear programming is solved eventually based on the convex quadratic programming problem.Therefore, the study on its effective solvers becomes a prolonged research subject.
Common effective algorithm is proposed related to certain schemes of convex quadratic programming problem, see [1][2][3][4][5].Moreover, these methods are not polynomial algorithm in [6].And the theory and algorithm of quadratic programming are presented completely.According to the optimization condition (KKT) and the duality theory of quadratic programming, the fixed-point iteration method is obtained for equality constrained (see [7]) and inequality constrained (see [8]) convex quadratic programming problem respectively.In fact, equality and inequality can be converted with inputting artificial variables and slack variables.The main default is to increase the scale and numerical difficulty while introduces these variables.Motivated by smoothing methods of OPEN ACCESS AM [9][10][11] and fixed-point method of [7,8], the paper mainly concerns about the mixed constraint quadratic programming and the fixed-point iteration method is given.And also the rank of the coefficient matrix is not full.The method considering is fulfilled efficiently.Thus the proposing question in [8] is solved.

Scheme of Algorithm
According to the dual theory, the dual problem of primal problem (1) is 1 max 2 where is the optimal solution of (1),there exists a , , , , is the optimal solution of (1) and ( ) , , , x y x y is the optimal solution of (2) }.
It is easy to verify that F is a close set.Suppose that the feasible region * F is not empty.A necessary and sufficient condition for a optimal solution is that: 1) Primal feasibility The scheme are rewritten below ( ) ( ) T 0, 0, 0 The component form is Where 0 r > .The question ( 5) is non-smooth optimization problem, and the Newton-type methods cannot be applied directly.In the paper, we converted the problem into a smooth optimization.
At the beginning, we introduce a function mapping R into R + , and have the following properties.Property 1 Let : R R θ + → be a real function, and such that: is strictly convex and differentiable; 2) We assumed that ( ) is a convex and differentiable function, which indicate one can apply the classical Newton-type algorithm directly.And the assumption ( ) > means that 0 is not a sub-gradient of θ at 0.
The hypothesis ( 3) and ( 4) indicate the approximate function ( ) has the same numerical properties as the max-function.
We consider the following function, and ( ) describe as above ( ) Proof According to property 1 and (6), the convexity and differentiability are straightforward.Nothing that ( ) ( ) The property (2) implies ( ) be a function given as (6), then That is, the function ( ) p t θ uniformly converges to the max-function, when the adjustable parameter p approaches 0 .

Proof
It is easily known that satisfies the properties (1) to ( 4), together with theorem 1 and 2. Thus, the max-function in (3) replaced by , and the fixed-point iteration which basing on smoothing function is obtained.

Algorithm
Step 1 Set a initial point 0 Step 2 Apply fixed-point algorithms to solve ( ) ( ) Step 3 Terminal condition: ( ) ( ) Step 4 Let The Jacobian matrix in ( 7) is shown ,

Convergence Analyses
Now we discuss the convergence of the algorithm.To begin with, two simple lemmas are introduced that is the basis of our theoretical analysis.Since their proof can be found in reference [12], we here omit the proof due to space.
Lemma 1 [12] A necessary and sufficient condition for the fixed-point iteration method to be convergent is that the spectral radius of Jacobian matrix B satisfies ( ) Lemma 2 [12] Suppose M is a matrix, whose eigenvalues can be ordered from the smallest to largest, i.e.  be the eigenvalues of the Jacobian matrix M , if one of the following conditions is true, then we can select the proper parameter k r makes the algorithm convergent: (1) The real part of j λ is positive, that is Re are eigenvalues of the Jacobian matrix in (8), and matrix ' M s eigenvalues denote as ( ) for all positive number of r ; while 0 λ ≠ , we know .For all discussed above, associate with Lemma 1, we know that the matrix M has the eigenvalues of ( ) According to the theorem above, the algorithm is always convergent if the parameter r is chosen properly.Thus the recent relevant results in reference [8] are extended.

Numerical Experiment and Conclusion
In this section, we implement algorithm for solving system of inequalities in MATLAB7.0 in order to demonstrate the behavior of the algorithm.Due to page limit, we omit to test the effectiveness of our method in generating highly quadratic programming problems.
Example 1 Consider , , 0 The exact optimal solution is , , 0 The exact optimal solution is T , and the optimal value of objective function is * 55 4 z = .
The results are shown in Table 1.
From the table, we can see that the algorithm is very simple and only applies the traditional fixed-point method.The implementation of this algorithm is rather easy.The exact optimal solution is ( ) T * 0 0 0 0 0 0 x = , and the optimal value of objective function is * 0 z = .Appling the method proposed, the iterations consuming time are shown in Table 2.

Theorem 1
Let be a function given as(6), then for any arbitrary where the spectral radius ( ) the eigenvalue of matrix T B B .
V is diagonal matrix with nonnegative diagonal elements, then the maximum norm of the matrix VM 's eigenvalue such that For t fixed, we can compute

Table 2 .
Experiments and calculated data.