Abstract
In this paper, we propose a parameter perturbation homotopy continuation method for solving fixed point problems on more general nonconvex sets with both inequality and equality constraints. By adopting appropriate techniques, we make the initial points not certainly in the set consisting of the equality constraints. This point can improve the computational efficiency greatly when the equality constraints are complex. In addition, we also weaken the assumptions of the previous results in the literature so that the method proposed in this paper can be applied to solve fixed point problems in more general nonconvex sets. Under suitable conditions, we obtain the global convergence of this homotopy continuation method. Moreover, we provide several numerical examples to illustrate the results of this paper.
1. Introduction
Fixed point theorems have been widely applied to many areas such as mechanics, physics, transportation, control, economics, differential equations, and optimization. Particularly the algorithm construction of computing a fixed point has attracted much attention and lots of results have appeared. The results of related research mentioned above are found in the literature (see [1–10], etc. and the references therein). Although the homotopy method has become a popular tool in dealing with fixed point problems (see [11–15], etc. and the references therein), this method still requires certain convexity assumptions. These convexity assumptions prevent the homotopy method from being extended to provide a constructive proof of the general Brouwer fixed point theorem, which does not require the convexity of the subsets in . Recently, on a class of nonconvex subset that satisfies the normal cone condition, by introducing the ideas of Karmarkar’s interior point method into the homotopy method, Yu and Lin [16] proposed a combined homotopy interior point method to give a constructive proof of the general Brouwer fixed point theorem and hence solve fixed point problems numerically. In [17], by introducing mappings and , we further extended the results in [16] to more general nonconvex sets with both inequality and equality constraint functions. In 2013, by introducing functions , , and , , Zhu et al. [18] further extended the results in [17] to more general nonconvex sets. It should be pointed out that, in [18], the initial point must satisfy the equation . In fact, it is difficult to select such an initial point when the constraint function is complex. This point reduces the computational efficiency of the homotopy interior point method greatly. In this paper, we propose a parameter perturbation homotopy continuation method for solving fixed point problems with both inequality and equality constraints. By using this new homotopy continuation method, we can not only remove the assumption that the matrix is of full row rank in [18] but also can make the initial point only satisfy , without satisfying the equation . Moreover, we mainly extend the results in [19, 20] to more general nonconvex sets and provide a practical perturbation for the inequality constraints compared with the results in [20]. Under commonly used conditions in the literature, we obtain the global convergence of the parameter perturbation continuation method. Moreover, we provide several numerical examples to illustrate the results of this paper.
This paper is organized as follows. Section 2 contains the basic definitions and the required preliminary materials widely used in what follows. Section 3 is the main part, which exhibits a convergence proof of the parameter perturbation continuation method. Section 4 presents the use of the reduced predictor-corrector algorithms given by Allgower and Georg [11] to compute a number of experimental examples, which illustrate the results of this paper.
2. Preliminaries
In this section, we need the following notations: , , , , and , , , and .
In [18], the initial point is confined in the interior of ; thus must satisfy as well as . Once the equality constraint function is complex, it is difficult to select such an initial point . This point may reduce the computational efficiency of the homotopy interior point method greatly. In this paper, we attempt to construct a new homotopy equation, which not only makes us remove the assumption that is of full row rank in [18] but also makes the initial point only satisfy , without satisfying the equation . To this end, we construct a new homotopy equation as follows: where , , and and , . Besides, , , are constants. Set , , , , , and .
Next, we make the following assumptions:
is nonempty and is bounded.
, , and , ; besides, for any and for any and , if , then
, if then , , and . In addition, for any is a matrix of full column rank.
, we have Besides, , we have
Example 1. To find a fixed point of self-mapping in , in this example, set and ; the original feasible set (See Figure 1) is so small that we cannot find an initial point easily. However, set , , , , , and ; thenWhen , the perturbated feasible set (see Figure 2) becomes big enough to find an initial point more easily than before.
For given , rewrite as . The zero-point set of is
Remark 2. In [18], to guarantee that the homotopy smooth curve is diffeomorphic to a unit interval , we need to prove that the matrixis nonsingular, where is the homotopy in [18]. This requires that the matrix be of full row rank. In new homotopy (1), the matrixis nonsingular without requiring the assumption that the matrix is of full row rank, where
Remark 3. When , the homotopy in [18] becomesIt is easy to see that the initial point must satisfy as well as . This point makes the choice of initial points become difficult when the function is complex. However, from new homotopy (1), when , we haveFrom (12), we conclude that only needs to satisfy , , without satisfying the equation .
In the following, we recall some basic definitions and results from differential topology, which will be used in our main result of this paper.
The inverse image theorem (see [21]) tells us that if is a regular value of the map , then consists of some smooth curves. And the regularity of can be obtained by the following lemma.
Lemma 4 (parameterized Sard’s theorem). Let and be open sets, and let be a map, where . If is a regular value of , then, for almost all , 0 is a regular value of .
3. Convergence Analysis
In this section, we are devoted to giving the global convergence analysis of the parameter perturbation homotopy continuation method.
Lemma 5. Let be defined as in (1), let , , and , , be functions, let assumptions ()–() hold, and let , , and , , be functions. Then, for almost all , is a regular value of map , and there exists a curve of dimension 1 such that
Proof. When is considered as a variable, let the Jacobian matrix of be denoted by , for any :whereBecause , , and is a matrix of full column rank, then is a matrix of full row rank. Moreover, it is easy to show that is of full row rank and thus 0 is a regular value of . By the parameterized Sard’s theorem, for almost all , 0 is a regular value of the map . By the inverse image theorem, consists of some smooth curves. Because has a unique solution , then there exists a curve (denoted by ) of dimension 1 such that
Lemma 6. Let be defined as in (1), let , , and , , be functions, let assumptions ()–() hold, and let , , and , , be functions. Then, for almost all , is a bounded curve.
Proof. Assume that is an unbounded curve. Then there exists a sequence of points such that . Because and are bounded, hence there exists a subsequence of points (denoted also by ) such that , , and as . From homotopy equation (1), we haveLet From (19), we haveIf , rewrite (17) asBy assumption , the previous three parts in the left-hand side of (22) are bounded, but the other two parts tend to infinity as ; this is impossible. Now we can assume that . At the same time we have . From (21), it is easy to show that .
If , from (17), when , then , , and , . Thereforewhich contradicts assumption .
If , rewrite (17) asWhen , because and , , are bounded, then the right-hand side of (24) is bounded. However, if , , by assumption , the left-hand side of (24) tends to infinity. This results in a contradiction.
When , because the nonempty index set , the proof of is similar to that of .
Now we give the main results of this section.
Theorem 7. Let be defined as in (1), let , , and , , be functions, let assumptions hold, and let , , and , , be functions. Then, for any mapping : satisfying , has a fixed point in . In addition, for almost all , homotopy equation (1) determines a smooth curve starting from . The limit set of is nonempty; and -component of any point in is a fixed point of in .
Proof. By Lemma 5, for almost all , there exists a smooth curve starting from . Because is nonsingular, by the classification theorem of one-dimensional smooth manifold, is diffeomorphic to a unit interval.
Let . Then the following three cases are possible: (a).(b).(c).Because has a unique solution in , so case (b) is impossible.
In case (c), first, we prove that . If , then there exist and a sequence of points such that as . From the third equation in (1), we haveWhen , because and are bounded, the left-hand side of (25) tends to 0. At the same time, the right-hand side of (25) tends to , which is strictly less than 0. This results in a contradiction.
Then we prove that . If , then there exist and a sequence of points such that ; as . This contradicts Lemma 6, so case (c) is also impossible.
From the above discussion, we obtain that case (a) is the unique possible case. When , from (1), is a solution of the equationWhen , we have and , so . From the third equation in (26), we obtain . Then the first equation in (26) yieldsIt follows from assumption and that . Then we easily show that by using assumption .
When , we have , , , , and , so . Because , , from the third equation in (26), , . Then (26) yieldsIt follows from assumption and that By using assumption , one obtains and for , and thus . This completes the proof.
4. Numerical Results
For almost all , by Theorem 7, the homotopy generates a curve by differentiating the first equation of (13); we get the following theorem.
Theorem 8. The homotopy path is determined by the following initial value problem to the ordinary differential equationwhere is the arc length of the curve .
Remark 9. Let , , and . Thus, (30) can be rewritten asBy solving the equation in (31), we obtain a solution . Thus, (30) becomes the following initial value problem:By Theorems 7 and 8, for almost all , the solution curve of (30) exists. Because the initial value problem (32) is equivalent to (30), the solution of (32) also exists for every .
Now, we discuss how to solve the initial value problem (32). Generally, if we use the numerical methods developed to address the initial value problems of ordinary differential equations to solve (32) alone, then the step length must be sufficiently small to ensure that the sequence is close enough to the solution curve. Using such methods may greatly increase the computational cost. Because the goal is to try to find a point (when is approximately zero) instead of tracking the smooth curve very precisely, it is desirable to combine the numerical methods for solving initial value problems of ordinary differential equations with other methods to develop more efficient methods that are later called predictor-corrector methods [11].
Below, we describe the implementation of a standard predictor-corrector method in detail. Suppose that we have obtained a sequence of points , , starting with an initial guess . To obtain , we must compute the tangent vector . By the basic theory of homotopy methods, the vector satisfies the following conditions:(a).(b) keeps the sign of the determinant of invariant.We then consider the following system:which can enable us to obtain a unit tangent vector at . One of the main solution strategies for (33) is based on the QR decomposition. For more details, the reader is referred to [11]. By Theorem 7, the matrix is of full row rank. From (33), the unit tangent vector at a point on has two opposite directions: one (the positive direction) makes increase, and the other (the negative direction) makes decrease. Because the negative direction will lead us back to the initial point, we must go along the positive direction. In the following discussion, for convenience, is denoted by . The criterion that determines the positive direction is based on condition (b). In the first iteration, the sign is determined by the following lemma.
Lemma 10. If is smooth, then the positive direction at the initial guess satisfies
Proof. Because wherethe tangent vector at satisfieswhere and . From (37), it is easy to show that . Therefore, the determinant of is Because , , and , the sign of the determinant is .
Therefore, by using the Euler method, for some small step length (not too small), we are able to obtain a predictor point . Here, we do not substitute more complicated algorithms for the Euler method because the predictor point does not need to be close enough to the smooth curve if only it lies in the interior of the convergence domain of Newton’s method during the corrector phase.
Next, we may perform a corrector step. Set which is the Moore-Penrose inverse of . The corrector phase then tries to identify a corrector point on the path . The corrector step is usually carried out by Newton’s method that uses the Moore-Penrose inverse of , starting with and proceeding until is approximately zero. The following pseudocode describes the basic steps of a generic predictor-corrector method.
Algorithm 11 (Euler-Newton method).
Step 0. Provide an initial guess , an initial step length , and three small positive numbers , , and . Set .
Step 1. Compute the direction of the predictor step: (a)Compute a unit tangent vector .(b)Determine the direction of the predictor step as follows: If the sign of the determinant is , then . If the sign of the determinant is , then .Step 2. Compute a corrector point . If , then let , and go to Step 3. If , then let , and go to Step 3. If , then let , , and go to Step 2.Step 3. If , then stop. Otherwise, , and go to Step 1.
By using homotopy (1) and the predictor-corrector algorithm, we give several numerical examples to illustrate the work in this paper. In each example, we set , , and . The behaviors of homotopy paths are shown in Figures 3, 4, and 5. Computational results are given in Table 1, where denotes the initial guess, denotes the number of iterations, denotes the value of when the algorithm stops, and denotes the fixed point.
Example 12 (see Example 1). In this example, we select two initial points and in . The pathways generated by homotopy equation (1) are shown in Figure 3. When , we can find a fixed point of in following the homotopy pathways and , respectively.
Example 13. To find a fixed point of self-mapping in in this example, we introduce the functions and . Assumptions ()–() are easily verified. We set and select two initial points and in . The initial points can be more easily selected in than in , because contains the equality constraint . The pathways generated by homotopy equation (1) are shown in Figure 4. When , we can find a fixed point of in following the homotopy pathways and , respectively.
Example 14. To find a fixed point of self-mapping in , In this example, we introduce the functions and . Assumptions ()–() are easily verified. We set and select two initial points and in . The initial points can be more easily selected in than in , because contains the equality constraint . The pathways generated by homotopy equation (1) are shown in Figure 5. When , we can find a fixed point of in following the homotopy pathways and , respectively.
5. Conclusions
In this paper, we apply appropriate perturbations to the constraint functions and hence develop a parameter perturbation homotopy continuation method for solving fixed point problems with both inequality and equality constraints. Our results make initial points of the algorithm be chosen more easily than the previous results in the literature and improve the efficiency of the algorithm greatly. Because fixed point problems have been widely applied to many areas such as mechanics, physics, transportation, control, economics, differential equations, and optimization, our results may be useful to propose a powerful solution tool for dealing with these practical problems. In the future, we want to propose new techniques to extend our results to more general nonconvex sets and extend the method proposed in this paper to solve the periodicity problems of some important differential equations.
Competing Interests
The authors declare that there are no competing interests regarding the publication of this paper.
Acknowledgments
This work was supported by National Natural Science Foundation of China (no. 11671188 and no. U1304103).