On well-posedness and Hausdorff convergence of solution sets of vector optimization problems
The paper considers well-posedness of vector optimization problems (VOP) of minimizing a function \(F(x)=(f_1(x), \ldots, f_l(x))\) subject to \(x \in X\), where \(f\) is continuous and \(X\) closed. The notion of exact and \(\epsilon\)-weakly efficient solutions (in an additive sense) is considered. Similarly, for the (weighted sum) scalarization of the vector optimization problem, optimal and \(\epsilon\)-approximate solutions are considered. Well-posedness of the (VOP) in terms of the Hausdorff distance is defined. The main results of the paper are: 1. The Hausdorff distance between \(\epsilon\)-approximate solutions of the weighted sum scalarization to the exact solution set converges to 0 if and only if the weighted sum scalarization is well-posed. 2. For a convex VOP such that the solution set of the weighted sum scalarization is nonempty for all weights and such that the weighted sum problem is well-posed over the weight set \(\Lambda\) then the VOP is well-posed. The presented results are stronger than those obtained by \textit{S. Deng} [RAIRO, Oper. Res. 37, No. 3, 195--208 (2003; Zbl 1070.90095)]. Examples are used to show that they are the strongest possible results.
scientific article; zbMATH DE number 6062516