Stability and sensitivity analysis in convex vector optimization. (English) Zbl 0654.49011

The following family of parametrized vector optimization problems is considered: \[ p-\min imize\quad f(x,u)=(f_ 1(x,u),\quad f_ p(x,u))\quad subject\quad to\quad x\in X(u)\subset {\mathbb{R}}\quad n, \] where \(f_ i: {\mathbb{R}}\) \(n\times {\mathbb{R}}\) \(m\to {\mathbb{R}}\) \((i=1,...,p)\), X is a set-valued map from \({\mathbb{R}}^ m \)to \({\mathbb{R}}^ n,\) and p is a nonempty pointed closed convex ordering cone in \({\mathbb{R}}^ p.\) The perturbation map for this family is a set-valued map W from \({\mathbb{R}}^ m \)to \({\mathbb{R}}^ p \)defined as follows: \[ Y(u):=\{y\in {\mathbb{R}}\quad p| y=f(x,u)\quad for\quad some\quad x\in X(u)\}, \]
\[ W(u):=Min_ PY(u)=\{\hat y\in Y(u)| (Y(u)-\hat y)\cap (-P)=\{0\}\}. \] It is assumed that the map X is convex (i.e., the graph of X is convex), while the function f is P-convex. Sufficient conditions for the upper and lower semicontinuity of the perturbation map W are obtained. Because of the convexity assumptions, these conditions are fairly simple if compared to those in the general case. Moreover, a complete characterization of the contingent derivative of the perturbation map is obtained under some additional assumptions.
Reviewer: M.Studniarski


49K40 Sensitivity, stability, well-posedness
90C31 Sensitivity, stability, parametric optimization
49J52 Nonsmooth analysis
54C60 Set-valued maps in general topology
90C25 Convex programming
49J45 Methods involving semicontinuity and convergence; relaxation
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