Abstract
In recent years, fixed-point theorems have attracted increasing attention and have been widely investigated by many authors. Moreover, determining a fixed point has become an interesting topic. In this paper, we provide a constructive proof of the general Brouwer fixed-point theorem and then obtain the existence of a smooth path which connects a given point to the fixed point. We also present a non-interior point homotopy algorithm for solving fixed-point problems on a class of nonconvex sets by numerically tricking this homotopy path.
1. Introduction
In recent years, fixed-point theorems have attracted increasing attention and have been widely investigated by many authors (e.g., [1–4] and the references therein) because these theorems play important roles in mechanics, physics, differential equations, and so on. The determination of a constructive proof of the fixed-point theorem and therefore finding a fixed point became an attractive topic. The homotopy method, as a globally convergent algorithm, is a powerful tool in handling fixed-point problems (e.g., [5–10] and the references therein). The general Brouwer fixed-point theorem states that if a bounded closed subset in is diffeomorphic to the closed unit ball, then any continuous self-mapping in it has a fixed point. However, the abovementioned results generally require certain convexity assumptions; thus, the traditional homotopy method cannot be used to handle the general Brouwer fixed-point theorem. Until 1996, Yu and Lin [11] combined the ideas of interior point methods and homotopy methods to propose a homotopy interior path-following method (see [12–17] for more details) that provides a constructive proof of the general Brouwer fixed-point theorem on a class of nonconvex subsets, without constructing a homeomorphism for transforming the bounded closed set to the closed unit ball. In [18, 19], the authors furthermore extended the results in [11] to more general nonconvex sets with inequality and equality constraint functions by replacing the gradient mappings with the newly introduced mappings.
The expansion of the scope of initial point selection to improve the computational efficiency of an algorithm is an important research area. In [20], we applied appropriate perturbations to the constraint functions and developed a new homotopy method to expand the scope of initial point selection, but involving the inequality constraint cases only. In [21], using similar perturbations to the inequality constraints in [20], we mainly extended the results in [18] to unbounded cases by providing a set of unbounded conditions. It should be pointed out that the results in [18, 19] excluded the initial point selection; in addition, the researchers excluded the equality constraint cases, although the results in [20, 21] expanded the scope of initial point selection. It is difficult to construct appropriate perturbations and to guarantee the regularity of the homotopy matrix for the existence of the equality constraints. To overcome the difficulties mentioned above, we apply new perturbations to the equality constraints and construct a new homotopy matrix to guarantee its regularity. Therefore, we develop a non-interior point homotopy path-following method for solving fixed-point problems with inequality and equality constraints. We can select initial points easily and thus considerably improve the computational efficiency of the algorithm by using the new approach.
This paper is organized as follows. Section 2 introduces two parameters and constructs appropriate perturbations to the constraint functions to develop a non-interior point homotopy path-following method for solving fixed-point problems with equality and inequality constraints. Section 3 presents several experimental examples to illustrate the results in this paper.
2. Main Results
In this section, we use the following notations: , , , , , , , , , , , , , and .
In [18], we extended the results in [11] to more general sets under the following assumptions:
is nonempty and is bounded.
For any , ifthen , , , and , .
The weak normal cone condition of : for any , we have
For any , is of full column rank and is nonsingular.
In this study, we introduce the following parameters to construct appropriate perturbations to the constraint functions:where is an arbitrarily given point. Then, set , , , , , , , and .
To solve fixed-point problems in more general nonconvex sets, we also introduce the continuous mappings and which satisfy the following conditions:
is nonempty and is bounded.
, , , ; besides, for any , if , then
For any , ifthen , , , , , . Besides, the matrix is nonsingular.
When , for any , we have From the geometric perspective in , we explain that the results in [18] are extended to more general nonconvex sets. Set ; note that is the linear combinations of , = , and , , and the set is not a bending cone surrounded by several radials or beelines; thus, many nonconvex sets cannot satisfy assumption in [18]. However, set . Note that , , and , , are the special cases of , , and , ; in many cases, may be a bending cone surrounded by several curves because , , and , , are arbitrary functions of , , . This point enables many nonconvex sets to not satisfy assumption but to satisfy assumption .
To solve fixed-point problems, we construct the following new homotopy equation:where , , , , and .
We rewrite as for a given . The zero-point set of isLemmas 1–4 will be used in the proof of our main results.
Lemma 1 (see [22]). Let be a map and 0 a regular value of . Then, is a manifold of dimension 1.
Lemma 2 (see [22]). A manifold of dimension 1 is diffeomorphic to a loop or an interval.
Lemma 3 (transversality theorem). Let , , and be smooth manifolds with dimensions , , and , respectively. Let be a submanifold of codimension (i.e., the dimension of ). Consider a smooth map . If is transversal to , then, for almost all , is transversal to . Recall that a smooth map is transversal to ifwhere is the Jacobi matrix of and and denote the tangent spaces of and at , respectively.
In this paper, ; thus, Lemma 3 corresponds to Lemma 4.
Lemma 4 (parameterized Sard’s theorem). Let and be open sets and be a map, where . If is a regular value of , then, for almost all , 0 is a regular value of .
Lemma 5. Let be defined as in (7); let , , and , , be functions; let assumptions (C1)–(C4) hold; and let , , and, , be functions. Then, for almost all , 0 is a regular value of map , and consists of some smooth curves. Among them, a smooth curve, denoted by , starts from .
Proof. We denote by when is considered as a variable. Let the Jacobian matrix of be denoted by . For any ,whereBecause is a matrix of full row rank, is of full row rank. Therefore, is also of full row rank, and 0 is a regular value of . By Lemma 4, for almost all , we obtain that 0 is a regular value of map . By Lemma 1, consists of several smooth curves. Because , then a curve of dimension 1, denoted by , starts from .
Lemma 6. Let be defined as in (7); let , , and , , be functions; let assumptions (C1)–(C4) 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 the homotopy equation (7), we obtainLet If , from (12), one obtainsThe sixth to ninth parts in the left-hand side of (16) tend to infinity as , but the other five parts are bounded, which is impossible. Therefore, the projection of the smooth curve onto the -plane is also bounded. Now, we can assume that . Simultaneously, .
If case (c) holds, 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 (14), we obtainWhen , the index set is When , the index set is If , from (12), we obtainBy assumptions , , and (20), we obtainwhere . Therefore, from (20) and (21), we obtainwhich contradicts assumption .
If , from (12), we obtainWhen , because and , , are bounded, the right-hand side of (23) is bounded. But, by assumption , if , , then the left-hand side of (23) tends to be infinite. This results in a contradiction.
If , then the proof is similar to that of (12) because the nonempty index set .
Now, we aim to prove our main result, that is, Theorem 7.
Theorem 7. Let be defined as in (7) and let assumptions (C1)–(C4) hold. Then, for almost all , there exists a curve of dimension 1 such thatWhen , tends to be point , the -component of which is a fixed point of in .
Proof. By Lemmas 5 and 6, we obtain that there exists a curve starting from . For any , it is easy to show that is nonsingular. By Lemma 2, we conclude that is diffeomorphic to a unit interval.
Let be a limit point of , and then the following cases may occur:
(a) .
(b) .
(c) .
When , the homotopy equation (7) becomesFrom (26) and (27), we obtain . Then, assumption , together with (25), yields that . By assumption , we obtain , . Besides, it follows from (27) that . Then, (25)–(27) have a unique solution in ; therefore, case (b) is impossible.
In case (c), we prove that . If , then there exist and a sequence of points such that as . From (14), we obtainWhen , because and are bounded, the left-hand side of (28) tends to be 0. Simultaneously, the right-hand side of (28) tends to be , 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; thus, case (c) is also impossible.
From the above discussion, we conclude that case (a) is the only possible case. Therefore, is a solution of (7) when , and is a fixed point of in .
By differentiating the first equation in (24), we obtain Theorem 8, which, together with Theorem 7, can reduce various predictor-corrector algorithms (see [8] and the references therein).
Theorem 8. The homotopy path is determined by the following initial value problem to the ordinary differential equation:where is the arc length of the curve .
In implementing the predictor-corrector algorithm, we must proceed along the positive direction of the unit tangent vector at a point on . The criterion that determines the positive direction is based on the condition that maintains the sign of the determinant of . In the first iteration, the sign is determined by the following lemma.
Lemma 9. If is smooth, then the positive direction at the initial guess satisfies .
Proof. Note that :whereThe tangent vector at satisfieswhere and . From (32), we obtain .
Therefore, the determinant of isBecause , , and , the sign of the determinant is .
3. Numerical Results
In this section, the numerical results provided below are obtained through the predictor-corrector algorithm. In each example, we set , , and . The behavior of homotopy paths is graphically illustrated, which can deliver a visual insight into the performance of our computer code. Computational results are summarized in Table 1, where denotes the initial guess, is the number of iterations, is the value of when the algorithm stops, and is the fixed point.
Example 1. One finds a fixed point of self-mapping inLet and , and then assumptions are satisfied. Based on the results of this study, we select initial points and which are not confined to the interior of the feasible sets. We obtain a fixed point of in when , following the discrete homotopy pathways and , as illustrated in Figure 1.
Example 2. One finds a fixed point of self-mapping in the set Let and , and then assumptions are satisfied. Based on the results of this study, we can select initial points and which are not confined to the interior of the feasible sets. We obtain a fixed point of in when , following the discrete homotopy pathways and , as illustrated in Figure 2.
Example 3. One finds a fixed point of self-mapping in the set Let functions and , and then assumptions are satisfied. Based on the results of this study, we can select initial points and which are not confined to the interior of the feasible sets. We obtain a fixed point of in when , following the discrete homotopy pathways and , as illustrated in Figure 3.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
This work was supported by the National Natural Science Foundation of China (nos. 11671188 and U1304103), Innovation Scientists Technicians Troop Construction Projects of Henan Province (no. C20150027), and Innovation Scientists Technicians Troop Construction Projects of Luoyang Normal University (no. 2014-CXTD-001).