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

Tanino, Tetsuzo

SIAM J. Control Optimization 26, No. 3, 521-536 (1988).

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

Keywords:

parametrized vector optimization; perturbation map; upper and lower semicontinuity; contingent derivative