Approximating Extreme Points of Infinite Dimensional Convex Sets

Approximating Extreme Points of Infinite Dimensional Convex Sets

The property that an optimal solution to the problem of minimizing a continuous concave function over a compact convex set in IRn is attained at an extreme point is generalized by the Bauer Minimum Principle to the infinite dimensional context. The problem of approximating and characterizing infinite dimensional extreme points thus becomes an important problem. Consider now an infinite dimensio...

Extreme Points of Polar Convex Sets

Extreme Points in Convex Sets of Symmetric Matrices

This paper deals with the following problem: What are the extreme points of a convex set K of n X n matrices, which is the intersection of the set S„ of symmetric matrices of nonnegative type, with another convex subset of symmetric matrices HI In the case where the facial structure of H is known, we expose a general method to determine the extreme points of K (Theorem 1). Then, we apply this m...

Approximating Largest Convex Hulls for Imprecise Points

Assume that a set of imprecise points is given, where each point is specified by a region in which the point will lie. Such a region can be modelled as a circle, square, line segment, etc. We study the problem of maximising the area of the convex hull of such a set. We prove NP-hardness when the imprecise points are modelled as line segments, and give linear time approximation schemes for a var...

POINTS FOR m-CONVEX SETS

Let S be closed, m-convex subset of R d S locally a full ddimensional, with Q the corresponding set of Inc points of S If q is an essential inc point of order k then for some neighborhood U of q Q u is expressible as a union of k or fewer (d2)-dimenslonal manifolds, each containing q For S compact, if to every q E Q there corresponds a k > 0 such that q is an essential inc point of order k then...