A constraint shifting homotopy method for computing fixed points on nonconvex sets

Volume 9, Issue 6, pp 3850--3857 http://dx.doi.org/10.22436/jnsa.009.06.32

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Authors

Zhichuan Zhu - Faculty of Statistics, Jilin University of Finance and Economics, Changchun, Jilin 130117, China. - School of Mathematics and Statistics, Northeast Normal University, Changchun, Jilin 130024, China. Li Yang - School of Science, Dalian University of Technology, panjin 124221, China.

Abstract

In this paper, a constraint shifting homotopy method for solving fixed point problems on nonconvex sets is proposed and the existence and global convergence of the smooth homotopy pathways is proved under some mild conditions. Compared with the previous results, the newly proposed homotopy method requires that the initial point needs to be only in the shifted feasible set not necessarily in the original feasible set, which relaxes the condition that the initial point must be an interior feasible point. Some numerical examples are also given to show the feasibility and effectiveness of our method.

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Keywords

• Fixed point
• self-mapping
• homotopy method
• nonconvex sets.

MSC

•  47H10
•  55M20

References

• [1] E. L. Allgower, K. Georg, Numerical Continuation Methods: An Introduction, Springer-Verlag, Berlin (1990)


• [2] R. Chen, Z. Zhu , Viscosity approximation fixed points for nonexpansive m-accretive operators, Fixed Point Theroy Appl., 2006 (2006), 10 pages.


• [3] S. N. Chow, J. Mallet-Paret, J. A. Yorke, Finding zeros of maps: homotopy methods that are constructive with probability one, Math. Comp., 32 (1978), 887-899.


• [4] R. B. Kellogg, T. Y. Li, J. A. Yorke, A constructive proof of the Brouwer fixed-point theorem and computational results, SIAM J. Num. Anal., 13 (1976), 473-483.


• [5] Z. Lin, B. Yu, D. Zhu, A continuation method for solving fixed points of self-mappings in general nonconvex sets, Nonlinear Anal., 52 (2003), 905-915.


• [6] G. L. Naber, Topological Method in Euclidean Space, Cambridge University Press, Cambridge-New York (1980)


• [7] M. Su, Z. Liu, Modified homotopy methods to solve fixed points of self-mapping in a broader class of nonconvex sets, Appl. Numer. Math., 58 (2008), 236-248.


• [8] M. Su, X, Qian , Existence of an interior path leading to the solution point of a class of fixed point problems, J. Inequal. Appl., 2015 (2015), 8 pages.


• [9] Y. Yao, R. Chen, Y.-C. Liou, A unified implicit algorithm for solving the triple-hierarchical constrained optimization problem, Math. Comput. Modelling, 55 (2012), 1506-1515.


• [10] Y. Yao, Y.-C. Liou, J.-C. Yao, Split common fixed point problem for two quasi-pseudocontractive operators and its algorithm construction , Fixed Point Theory Appl., 2015 (2015), 19 pages.


• [11] Y. Yao, M. Postolache, Y.-C. Liou, Z. Yao, Construction algorithms for a class of monotone variational inequalities, Optim. Lett., (in press),


• [12] B. Yu, Z. Lin, Homotopy Methods for a class of nonconvex Brouwer fixed-point problems, Appl. Math. Comput., 74 (1996), 65-77.


• [13] Z. Zhu, R. Chen, Strong Convergence on Iterative Methods of Cesáro Means for Nonexpansive Mapping in Banach Space, Abstr. Appl. Anal., 2014 (2014), 6 pages.


• [14] Z. Zhu, B. Yu, Y. Shang, A modified homotopy method for solving nonconvex fixed points problems, Fixed Point Theory, 14 (2013), 531-544.


• [15] Z. Zhu, B. Yu, L. Yang, Globally convergent homotopy method for designing piecewise linear deterministic contractual function, J. Ind. Manag. Optim., 10 (2014), 717-741.