Mathematical Problems in Engineering

Volume 2017 (2017), Article ID 4067202, 10 pages

https://doi.org/10.1155/2017/4067202

## A Parameter Perturbation Homotopy Continuation Method for Solving Fixed Point Problems with Both Inequality and Equality Constraints

^{1}School of Mathematics, Luoyang Normal University, Luoyang 471934, China^{2}Key Laboratory of Symbolic Computation and Knowledge Engineering of Ministry of Education, Jilin University, Changchun 130012, China^{3}Section of Mathematics, Aviation University of Air force, Changchun 130022, China^{4}School of Mathematics, Nankai University, Tianjin 300071, China

Correspondence should be addressed to Menglong Su

Received 9 December 2016; Accepted 23 January 2017; Published 13 February 2017

Academic Editor: Maria L. Gandarias

Copyright © 2017 Menglong Su et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### 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