Abstract
In this paper, the generic uniqueness of Pareto weakly efficient solutions, especially Pareto-efficient solutions, of vector optimization problems is studied by using the nonlinear and linear scalarization methods, and some further results on the generic uniqueness are proved. These results present that, for most of the vector optimization problems in the sense of the Baire category, the Pareto weakly efficient solution, especially the Pareto-efficient solution, is unique. Furthermore, based on these results, the generic Tykhonov well-posedness of vector optimization problems is given.
Highlights
The uniqueness of the solution is critical to the stability and calculation of the solution, but it is difficult to guarantee the uniqueness of solutions, even for the optimal solution of scalar optimization problems
Beer [2] extended the generic uniqueness to constrained optimization problems in the Cech complete space. e generic uniqueness of optimal solutions was proved for some classes of infinite dimensional linear programming problems, and some generic uniqueness results were proposed in linear optimization problems
If the vector a is a Pareto-efficient point of the function f, the result still holds
Summary
The uniqueness of the solution is critical to the stability and calculation of the solution, but it is difficult to guarantee the uniqueness of solutions, even for the optimal solution of scalar optimization problems. Some results of existence of Pareto-efficient solutions for vector optimization problems are restated as follows (see, e.g., [27, 28]).
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