Lipschitz-continuity of Pareto-optimal extreme points
The problem of a multicriteria vector optimization on a set \[ X(A, b)=\{ x\in{\mathbb R^{n}} \mid Ax\leq b \}\tag{1} \] where \(A\) is an \((m\times n)\)-matrix, \(A=( H, G)\), \(Hx=h\), \(Gx\leq g \) is considered. The solution of such a problem is a set of all non-predominated (effective) vectors. The set is a union of all non-predominated faces which are preassigned by the non-predominated extreme points. The research of stability properties of Pareto's sets of the optimal extreme points in relation to small perturbations of input data was carried out in the article . Are theorem on Lipshitz-semicontinuity from below in a point of a many-valued map \[ P:( A', b')\mapsto P( X(A',b'))\cap E( X(A', b')) \tag{2} \] which to every matrix \(A'\) and vector \(b'\) assigns a Pareto set of optimal extreme points \( E(X(A',b'))\) on the set \((X(A', b'))\) is proved.
scientific article; zbMATH DE number 1266904