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AN η-APPROXIMATION METHOD FOR NONSMOOTH MULTIOBJECTIVE PROGRAMMING PROBLEMS

Part of: Existence theories

Published online by Cambridge University Press:  01 January 2008

TADEUSZ ANTCZAK
TADEUSZ ANTCZAK*
Affiliation:
Faculty of Mathematics and Computer Science, University of Łódź, Banacha 22, 90-238 Łódź, Poland (email: antczak@math.uni.lodz.pl)
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Abstract

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In this paper, a new approach to a characterization of solvability of a nonlinear nonsmooth multiobjective programming problem with inequality constraints is introduced. A family of η-approximated vector optimization problems is constructed by a modification of the objective and the constraint functions in the original nonsmooth multiobjective programming problem. The connection between (weak) efficient points in the original nonsmooth multiobjective programming problem and its equivalent η-approximated vector optimization problems is established under V-invexity. It turns out that, in most cases, solvability of a nonlinear nonsmooth multiobjective programming problem can be characterized by solvability of differentiable vector optimization problems.

Keywords

nonsmooth multiobjective programmingV-invex function with respect to ηη-approximated vector optimization problem(weak) Pareto optimal point

MSC classification

Secondary: 90C26: Nonconvex programming, global optimization 90C29: Multi-objective and goal programming 90C46: Optimality conditions, duality 49J52: Nonsmooth analysis
Type
Research Article
Information
The ANZIAM Journal , Volume 49 , Issue 3 , January 2008 , pp. 309 - 323
Copyright
Copyright © Australian Mathematical Society 2008

References

[1]Antczak, T., “A new approach to multiobjective programming with a modified objective function”, J. Global Optim. 27 (2003) 485–495.CrossRefGoogle Scholar
[2]Antczak, T., “An η-approximation method in nonlinear vector optimization”, Nonlinear Anal. 63 (2005) 225–236.CrossRefGoogle Scholar
[3]Antczak, T., “Saddle points criteria and duality in multiobjective programming via an η-approximation method”, J. Aust. Math. Soc. B 47 (2005) 155–172.Google Scholar
[4]Antczak, T., “An η-approximation approach in nonlinear vector optimization with univex functions”, Asia-Pac. J. Oper. Res. 23 (2006) 525–542.CrossRefGoogle Scholar
[5]Antczak, T., “Optimality and duality for nonsmooth multiobjective programming problems with V-r-invexity”, J. Global Optim. to appear.Google Scholar
[6]Clarke, F. H., Optimization and nonsmooth analysis, in Canadian Mathematical Society Series of Monographs and Advanced Texts (John Wiley & Sons, New York, 1983).Google Scholar
[7]Craven, B. D., “Invex functions and constrained local minima”, Bull. Austral. Math. Soc. 24 (1981) 357–366.CrossRefGoogle Scholar
[8]Craven, B. D., “Quasimin and quasisaddlepoint for vector optimization”, Numer. Funct. Anal. Optim. 11 (1990) 45–54.CrossRefGoogle Scholar
[9]Egudo, R. R. and Hanson, M. A., “On sufficiency of Kuhn–Tucker conditions in nonsmooth multiobjective programming”, FSU Technical Report No. M-888, 1993.Google Scholar
[10]Giorgi, G. and Guerraggio, A., “The notion of invexity in vector optimization: smooth and nonsmooth case”, in Generalized convexity, generalized monotonicity, Proc. Fifth Symp. on Generalized convexity of Luminy, France (eds. J. P. Crouzeix, J. E. Martinez-Legaz and M. Volle), (Kluwer Academic, Dordrecht, 1997).Google Scholar
[11]Hanson, M. A., “On sufficiency of the Kuhn–Tucker conditions”, J. Math. Anal. Appl. 80 (1981) 545–550.CrossRefGoogle Scholar
[12]Jeyakumar, V. and Mond, B., “On generalized convex mathematical programming”, J. Aust. Math. Soc. B 34 (1992) 43–53.CrossRefGoogle Scholar
[13]Kim, D. S. and Schaible, S., “Optimality and duality for invex nonsmooth multiobjective programming problems”, Optimization 53 (2004) 165–176.CrossRefGoogle Scholar
[14]Lee, G. M., “Nonsmooth invexity in multiobjective programming”, J. Inf. Optim. Sci. 15 (1994) 127–136.Google Scholar
[15]Luc, D. T., Theory of vector optimization, Volume 319 of Lecture Notes in Economics and Mathematical Systems (Springer, Berlin, 1989).CrossRefGoogle Scholar
[16]Reiland, T. W., “Nonsmooth invexity”, Bull. Austral. Math. Soc. 42 (1990) 437–446.CrossRefGoogle Scholar
[17]Ruiz-Canales, P. and Rufián-Lizana, A., “A characterization of weakly efficient points”, Math. Program. 68 (1995) 205–212.CrossRefGoogle Scholar
[18]Sawaragi, Y., Nakayama, H. and Taninio, T., Theory of multiobjective optimization (Academic Press, Orlando, FL, 1985).Google Scholar
[19]Singh, C., “Optimality conditions in multiobjective differentiable programming”, J. Optim. Theory Appl. 53 (1987) 115–123.CrossRefGoogle Scholar
[20]Steuer, R. E., Multiple criteria optimization: theory, computation, and application (Wiley, New York, 1986).Google Scholar
[21]Taninio, T., “Stability and sensitivity analysis in convex vector optimization”, SIAM J. Control Optim. 26 (1988) 521–536.CrossRefGoogle Scholar
[22]Weir, T., Mond, B. and Craven, B. D., “On duality for weakly minimized vector valued optimization problems”, Optimization 17 (1986) 711–721.CrossRefGoogle Scholar