Structural stability of vector optimization problems

Paulo Mbunga

Preprint series: Institut für Mathematik, Humboldt-Universität zu Berlin (ISSN 0863-0976),

MSC 2000

90C29 Multi-objective and goal programming

90C31 Sensitivity, stability, parametric optimization

Abstract
We study global stability properties for vector optimization problems of the type: \begin{align} \tag*{$\mathcal{VOP}(f,H,G)$:} \min\left\{f(x)=\left(f_1(x),\dots,f_l(x)\right) \mid x\in M[H,G]\right\}, \end{align} where \begin{align*} M[H,G]:=\left\{x\in \R^n\mid h_i(x)=0,\quad g_j(x)\leq 0, \quad i\in I,j\in J\right\} \end{align*} with \begin{alignat*}{3} I &:= \{1,\dots,m\}, &\quad J &:= \{1,\dots,s\},&\quad L&:=\{1,\dots,l\}. \end{alignat*} We extend Guddat/Jongen's \cite{structstab} concept of structural stability of scalar nonlinear optimization problems to vector optimization problems. Under the assumption that $M[H,G]$ is compact we prove the necessary condition for the structural stability of a vector optimization problem, i.e. the scalar problem \begin{align} \tag*{$\mathcal{P}^{\max}(f,H,G)$:} \min\left\{\max_{l\in L}f_l(x)\mid x\in M[H,G] \right\} \end{align} has to be structurally stable.