Homotopy method for a general multiobjective programming problem. (English) Zbl 1211.90219

Summary: In this paper, we present a combined homotopy interior-point method for a general multiobjective programming problem. The algorithm generated by this method associated to Karush-Kuhn-Tucker points of the multiobjective programming problem is proved to be globally convergent under some basic assumptions.


90C29 Multi-objective and goal programming
90C51 Interior-point methods
Full Text: DOI


[1] Kuhn, H.W., Tucker, A.W.: Nonlinear programming. In: Neyman, J. (ed.) Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability. University of California Press, Berkeley (1951) · Zbl 0044.05903
[2] Abadie, M.: Generalized Kuhn–Tucker conditions for mathematical programming. SIAM J. Control. 7, 232–241 (1969) · Zbl 0182.53101
[3] Lin, C.Y., Dong, J.L.: Methods and Theories in Multiobjective Optimization. Jinlin Education Press, Changchun (1992) (in Chinese)
[4] Maeda, T.: Second-order conditions for efficiency in nonsmmoth multiobjective optimization problems. J. Optim. Theory Appl. 122, 521–538 (2004) · Zbl 1082.90106
[5] Kellogg, R.B., Li, T.Y., Yorke, J.A.: A constructive proof of the Brouwer fixed-point theorem and computational results. SIAM J. Numer. Analysis 18, 473–483 (1976) · Zbl 0355.65037
[6] Chow, S.N., Mallet-Paret, J., Yorke, J.A.: Finding zeros of maps: Homotopy methods that are constructive with probability one. Math. Comput. 32, 887–899 (1978) · Zbl 0398.65029
[7] Allgower, E.L., Georg, K.: Numerical Continuation Methods: An Introduction. Springer, Berlin (1990) · Zbl 0717.65030
[8] Garcia, C.B., Zangwill, W.I.: Pathways to Solutions, Fixed Points and Equilibria. Prentice-Hall, Englewood Cliffs (1981) · Zbl 0512.90070
[9] Megiddo, N.: Pathways to the optimal set in linear programming. In: Progress in Mathematical Programming, Interior Point and Related Methods, pp. 131–158. Springer, New York (1988)
[10] Kojima, M., Mizuno, S., Yoshise, A.: A primal-dual interior point algorithm for linear programming. In: Megiddo, N. (ed.) Progress in Mathematical Programming, Interior Point and Related Methods, pp. 29–47. Springer, New York (1988) · Zbl 0708.90049
[11] McCormick, G.P.: The projective SUMT method for convex programming. Math. Oper. Res. 14, 203–223 (1989) · Zbl 0675.90067
[12] Monteiro, R.D.C., Adler, I.: An extension of karmarkar type algorithm to a class of convex separable programming problems with global linear rate of convergence. Math. Oper. Res. 15, 408–422 (1990) · Zbl 0708.90068
[13] Lin, Z.H., Yu, B., Feng, G.C.: A combined homotopy interior method for convex nonlinear programming. Appl. Math. Comput. 84, 93–211 (1997) · Zbl 0898.90100
[14] Lin, Z.H., Li, Y., Yu, B.: A combined homotopy interior point method for general nonlinear programming problems. Appl. Math. Comput. 80, 209–224 (1996) · Zbl 0883.65054
[15] Lin, Z.H., Zhu, D.L., Sheng, Z.P.: Finding a minimal efficient solution of a convex multiobjective program. J. Optim. Theory Appl. 118, 587–600 (2003) · Zbl 1061.90102
[16] Di, S.: Classical optimality conditions under weaker assumptions. SIAM J. Optim. 6, 178–197 (1996) · Zbl 0841.46041
[17] Naber, G.L.: Topological Methods in Euclidean Spaces. Cambridge University Press, London (1980) · Zbl 0437.55001
[18] Zhang, Z.S.: Introduction to Differential Topology. Beijing University Press, Beijing (2002) (in Chinese)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.