A Krein-Milman type theorem for certain unbounded convex sets

A cone theoretic Krein-Milman theorem in semitopological cones

In this paper, a Krein-Milman  type theorem in $T_0$ semitopological cone is proved,  in general. In fact, it is shown that in any locally convex $T_0$ semitopological cone, every convex compact saturated subset is the compact saturated convex hull of its extreme points, which improves the results of Larrecq.

Unbounded convex sets for non-convex mixed-integer quadratic programming

This paper introduces a fundamental family of unbounded convex sets that arises in the context of non-convex mixed-integer quadratic programming. It is shown that any mixed-integer quadratic program with linear constraints can be reduced to the minimisation of a linear function over a set in the family. Some fundamental properties of the convex sets are derived, along with connections to some o...

A Representation Theorem for Bounded Convex Sets

Unbounded Convex Semialgebraic Sets as Spectrahedral Shadows

Recently, Helton and Nie [3] showed that a compact convex semialgebraic set S is a spectrahedral shadow if the boundary of S is nonsingular and has positive curvature. In this paper, we generalize their result to unbounded sets, and also study the effect of the perspective transform on singularities.

Finite Illumination of Unbounded Closed Convex Sets

If K is an unbounded closed convex subset of Ed having nonempty interior, we seek necessary and/or sufficient conditions to ensure that the boundary of K can be externally illuminated from a finite set of directions. This problem was stated as open in a recent book by Boltyanski. The tools used in this search are those developed by Visibility Theory such as the ideas of star, inner stem, visibi...