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Counterexample and optimality conditions in differentiable multiobjective programming. (English) Zbl 1017.90099

By giving a simple counterexample the authors show that the sufficient optimality conditions for Pareto optimal solutions of differentiable multiobjective programming problems (MOP) stated in Majundar (1997) are not correct in general. Modifications of the above mentioned results are developed based on the assumption that the functions involved at a (weak) Pareto optimal solution are convex, pseudoconvex and quasiconvex, respectively. In particular, in the modification of the sufficient optimality theorem for a Pareto optimal solution of (MOP), the assumption of pseudoconvexity of the objective functions is replaced by strict pseudoconvexity.

MSC:

90C29 Multi-objective and goal programming
90C26 Nonconvex programming, global optimization
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References:

[1] Majumdar, A. A. K., Optimality Conditions in Differentiable Multiobjective Programming, Journal of Optimization Theory and Applications, Vol. 92, No. 2, pp. 419–427, 1997. · Zbl 0886.90122
[2] Singh, C., Optimality Conditions in Multiobjective Differentiable Programming, Journal of Optimization Theory and Applications, Vol. 53, No. 1, pp. 115–123, 1987. · Zbl 0593.90071
[3] Mangasarian, O. L., Nonlinear Programming, McGraw-Hill Book Company, New York, NY, 1969.
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