Abstract
By using slack variables and minimum function, we first reformulate the system of equalities and inequalities as a system of nonsmooth equations, and, using smoothing technique, we construct the smooth operator. A new noninterior continuation method is proposed to solve the system of smooth equations. It shows that any accumulation point of the iteration sequence generated by our algorithm is a solution of the system of equalities and inequalities. Some numerical experiments show the feasibility and efficiency of the algorithm.
1. Introduction
In this paper, we consider the following system of equalities and inequalities: where and . Define with , for any . Throughout this paper, we assume that is continuously differentiable.
Problems taking the form (1) have been studied extensively due to its various applications in data analysis, set separation problems, computer aided design problems, and image reconstructions.
Recently, a class of popular numerical methods, namely, the so-called noninterior continuation methods, has been studied extensively for complementarity, variational inequality, and mathematical programming problems; see, for example, [1–7]. However, as we observe, there are few noninterior continuation methods available for the system of equalities and inequalities given by (1).
In this paper, we first reformulate (1) as a system of nonsmooth equations by using slack variables and minimum function, and, using smoothing technique, we construct the smooth equations. Then, a noninterior continuation method for (1) by modifying and extending the method of Huang [1] is proposed. Under suitable assumptions, we show that the proposed algorithm is globally linearly convergent. We also report some preliminary numerical results, which demonstrate that the algorithm is effective for solving (1).
The organization of this paper is as follows. In Section 2, we reformulate (1) as a system of smooth equations. In Section 3, we propose a noninterior continuation method for solving (1). Global convergence is analyzed in Section 4. Some preliminary computational results are reported in Section 5.
We introduce some notations. All vectors are column vectors, the superscript denotes transpose, (resp., ) denotes the nonnegative (resp., positive) orthant in . denotes identity matrix. For , denotes the 2-norm of . For a continuously differentiable function , we denote the Jacobian of at by .
2. Equivalent Smoothing Reformulation of (1)
In this section, we give the equivalent smoothing reformulation of (1) and discuss some associated properties of the reformulation. Firstly, we introduce the NCP function and the smoothing function. A function is called an NCP function, if it possesses the following property: One well-known NCP function is the minimum function [9], which is defined as follows: Accordingly, the smoothing function associated with is [4]
For (1), we introduce a slack variable . Then, (1) is equivalent to the following system of equations: Based on the minimum function, we reformulate (5) into the following equivalent system of nonlinear equation: where , .
Since the function in (6) is nonsmooth, the noninterior continuation method cannot be directly applied to solve (6). In order to make (6) solvable by the noninterior continuation method, we will use the smoothing technique and construct the smooth approximation of as . Consider where , , , , and = . Thereby, it is obvious that, if and , then solves (1). It is not difficult to see that, for any , the function is continuously differentiable. Let denote the Jacobian of the function ; then, for any , where and . Here, we use to denote the -dimensional zero vector and to denote the zero matrix for any positive integers and . Thus, we can solve approximately the smooth system by using Newton’s method at each iteration and then obtain a solution of by reducing the parameter to zero so that a solution of (1) can be found.
3. Algorithm
In this section, we propose a noninterior continuation algorithm. Some basic properties are given. In particular, we show that the algorithm is well defined.
Algorithm 1 (a noninterior continuation algorithm).
Consider the following. Step 0. Choose . Take any and ; choose such that . Set . Step 1. If , then stop.. Step 2. If , then set and , and go to Step 4; otherwise, compute by
Step 3. Let be maximum of the values such that
Set . Step 4. Set the following:
Let be the minimum of the values such that
and set . Set and go to Step 1.
Remark 2. Algorithm 1 is a modified version of Huang’s algorithm in [1]. It is easy to see that, if , for any , then Algorithm 1 does not solve the Newton equation (9) and does not perform the line search (10) in the th iteration. Thus, Algorithm 1 only needs to solve at most a linear system of equations at each iteration. Algorithm 1 can be started easily. In fact, we can choose any as the starting point of our algorithm and then set
Define . We will use the following assumption.
Assumption 3. is invertible for any and .
The next result plays an important role in establishing the well-definedness and the local quadratic convergence of Algorithm 1.
Lemma 4. (i) is continuously differentiable at any .
(ii) For any and , we have
Theorem 5. Suppose that is a continuously differentiable function and Assumption 3 is satisfied. Then Algorithm 1 is well defined.
Proof. For any square matrix , we use to denote the determinant of . It is easy to see from (8) that , for any and . Furthermore, it is easy to see that is positive semidefinite. Thus, by Assumption 3, we obtain that is nonsingular, for any and . Hence, Step 2 is well defined.
Now we prove that Step 3 is well defined. For any , define
From and Lemma 4(i), we know that is continuously differentiable at . Thus, by (15), we have
Then by (9), (15) and (16),
Since , then . For sufficient small, we can get , this shows that Step 3 is well defined.
Next we show that Step 4 is well defined. If , it follows from Lemma 4(ii) and (7) that
Then, by , (10)–(12), and (18), we have
If , similarly we also obtain that (19) holds. Thus, from (19), we know that there exists a minimal such that (12) holds; that is, Step 4 is well defined.
Therefore, Algorithm 1 is well defined.
4. Convergence of Algorithm 1
In this section, we analyze the global convergence properties of Algorithm 1. We show that any accumulation point of the iteration sequence is a solution of the system .
Theorem 6. Suppose that is a continuously differentiable function and is an accumulation point of the iteration sequence generated by Algorithm 1. If Assumption 3 is satisfied, then , and hence is a solution of .
Proof. Since the sequence is monotonically decreasing and bounded from below by zero, then . If , we obtain the desired result. Suppose . Without loss of generality, we assume that . If there exists an infinite subset such that , , then, by Step 2 of Algorithm 1, we have , for any . It follows from Step 4 that
Let ; we have that
which is a contradiction. Therefore, without loss of generality, we may assume that holds, for any . From the assumption, is continuously differentiable and (7); it is not difficult to see that is continuously differentiable in both and for any . Since , by assumption, then we have
The steplength does not satisfy (10); that is,
By taking in the above inequality, we have
It follows from (9) that
By substituting (25) into (24), we obtain that , which contradicts . This proves .
Next, we prove that is a solution of . In view of the Algorithm 1, we have
Then, by taking the limit on both sides of (26) based on the continuity of , we have . Hence, .
5. Numerical Experiments
In this section, we implement Algorithm 1 for solving the system of equalities and inequalities in MATLAB in order to see the behavior of our noninterior continuation algorithm. All the program codes were written in MATLAB and run in MATLAB 7.5 environment. All numerical experiments were done at a PC with CPU of 1.6 GHz and RAM of 512 MB.
In numerical implementation, we adopt the similar strategy to [10]; the function defined by (7) is replaced by where is a given constant. It is easy to see that such a change does not destroy any theoretical results obtained in this paper. In order to obtain an interior solution of (1), we solve the following system of equalities and inequalities: where is a sufficiently small number and is a vector of all ones. The parameters used in Algorithm 1 were as follows: , , , , and ; the parameter and the starting point are chosen according to the ones listed in Tables 1, 2, 3, and 4. Set and . We used as the stopping criterion.
We consider the following four examples.
Example 1. Consider (1), where with and
Example 2. Consider (1), where with and
Example 3. Consider (1), where with and
Example 4. Consider (1), where with and
The first example only contains inequalities; the other examples contain equalities and inequalities. Instead of these three examples, we use Algorithm 1 to solve the following problems.
Example 1′. Consider (1), where with and
Example 2′. Consider (1), where with and
Example 3′. Consider (1), where with and
Example 4′. Consider (1), where with and
The numerical results are listed in Tables 1, 2, 3, 4, and 5, where Exam denotes the tested examples, ST denotes the starting point , denotes the value of the parameter given in (27), CPU denotes the CPU time for solving the underlying problem in second, IT denotes the total number of iterations, − represents iteration number in more than 1000, and SOL denotes the solution obtained by Algorithm 1.
From Tables 1, 2, 3, and 4, it is easy to see that all problems that we tested can be solved efficiently. In Table 5, we compare our proposed algorithm with the algorithm in [8]. The numerical results illustrate that our algorithm is more effective.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This work is supported by the National Natural Science Foundation of China (Grants nos. 11326186, 61101208, and 11241005), the Fundamental Research Funds for the Central Universities, and a Project of Shandong Province Higher Educational Science and Technology Program, China (no. J13LI05).