Homotopy interior-point method for a general multiobjective programming problem. (English) Zbl 1244.90215

Summary: We present a combined homotopy interior-point method for a general multiobjective programming problem. For solving the KKT points of the multiobjective programming problem, the homotopy equation is constructed. We prove the existence and convergence of a smooth homotopy path from almost any initial interior point to a solution of the KKT system under some basic assumptions.


90C29 Multi-objective and goal programming
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[1] R. B. Kellogg, T. Y. Li, and J. Yorke, “A constructive proof of the Brouwer fixed-point theorem and computational results,” SIAM Journal on Numerical Analysis, vol. 13, no. 4, pp. 473-483, 1976. · Zbl 0355.65037
[2] S. Smale, “A convergent process of price adjustment and global newton methods,” Journal of Mathematical Economics, vol. 3, no. 2, pp. 107-120, 1976. · Zbl 0354.90018
[3] S. N. Chow, J. Mallet-Paret, and J. A. Yorke, “Finding zeroes of maps: homotopy methods that are constructive with probability one,” Mathematics of Computation, vol. 32, no. 143, pp. 887-899, 1978. · Zbl 0398.65029
[4] M. S. Gowda, “On the extended linear complementarity problem,” Mathematical Programming, vol. 72, no. 1, pp. 33-50, 1996. · Zbl 0853.90109
[5] F. Facchinei and J. S. Pang, Finite-Dimensional Variational Inequalities and Complementarity Problems, Springer, New York, NY, USA, 2003. · Zbl 1062.90002
[6] G. P. McCormick, “The projective SUMT method for convex programming,” Mathematics of Operations Research, vol. 14, no. 2, pp. 203-223, 1989. · Zbl 0675.90067
[7] R. D. C. Monteiro and I. Adler, “An extension of Karmarkar type algorithm to a class of convex separable programming problems with global linear rate of convergence,” Mathematics of Operations Research, vol. 15, no. 3, pp. 408-422, 1990. · Zbl 0708.90068
[8] Z. Lin, B. Yu, and G. Feng, “A combined homotopy interior point method for convex nonlinear programming,” Applied Mathematics and Computation, vol. 84, no. 2-3, pp. 193-211, 1997. · Zbl 0898.90100
[9] Z. H. Lin, D. L. Zhu, and Z. P. Sheng, “Finding a minimal efficient solution of a convex multiobjective program,” Journal of Optimization Theory and Applications, vol. 118, no. 3, pp. 587-600, 2003. · Zbl 1061.90102
[10] W. Song and G. M. Yao, “Homotopy method for a general multiobjective programming problem,” Journal of Optimization Theory and Applications, vol. 138, no. 1, pp. 139-153, 2008. · Zbl 1211.90219
[11] Z. S. Zhang, Introduction to Differential Topology, Beijing University Press, Beijing, China, 2002.
[12] G. L. Naber, Topological Methods in Euclidean Spaces, Cambridge University Press, Cambridge, UK, 1980. · Zbl 0437.55001
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