## 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

### MSC:

 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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