Motivated by applications in financial mathematics, [3] showed that, although L(R+; Ω,F ,P) fails to be locally convex, an analogue to the classical bipolar theorem can be obtained for subsets of L(R+; Ω,F ,P) : if we place this space in polarity with itself, the bipolar of a set of non-negative random variables is equal to its closed (in probability), solid, convex hull. This result was extend...

Motivated by applications in financial mathematics, [3] showed that, although L(R+; Ω,F ,P) fails to be locally convex, an analogue to the classical bipolar theorem can be obtained for subsets of L(R+; Ω,F ,P) : if we place this space in polarity with itself, the bipolar of a set of non-negative random variables is equal to its closed (in probability), solid, convex hull. This result was extend...

In this report we treat nonlinear programs Pro(f, h;K) having an objective function f , a finite number of equality constraints h(x) = (h1(x), · · · , hl(x)) = 0, and an abstract convex constraint x ∈ K with its convex set K. Our particular interest is an algebraic criterion for a locally isolated stationary solution to be strong stable, in the sense of Kojima, under a Linear Independence Const...

Given a set of polyhedral cones C1, · · · , Ck ⊂ R, and a convex set D, does the union of these cones cover the set D? In this paper we consider the computational complexity of this problem for various cases such as whether the cones are defined by extreme rays or facets, and whether D is the entire R or a given linear subspace R. As a consequence, we show that it is coNP-complete to decide if ...

This note announces that a simple approximation property, known to hold in large classes of partially ordered locally convex spaces whose duals are lattice ordered (e.g., [Banach or] Fréchet spaces with closed, normal, generating positive cones possessing the Riesz decomposition property), actually characterizes spaces with lattice-ordered duals. Some very mild and natural assumptions about the...

The class of POPs (Polynomial Optimization Problems) over cones covers a wide range of optimization problems such as 0-1 integer linear and quadratic programs, nonconvex quadratic programs and bilinear matrix inequalities. This paper presents a new framework for convex relaxation of POPs over cones in terms of linear optimization problems over cones. It provides a unified treatment of many exis...

We consider two notions for the representations of convex cones: G-representation and liftedG-representation. The former represents a convex cone as a slice of another; the latter allows in addition, the usage of auxiliary variables in the representation. We first study the basic properties of these representations. We show that some basic properties of convex cones are invariant under one noti...

Using the T -algebra machinery we show that the only strictly convex homogeneous cones in R with n ≥ 3 are the 2-cones, also known as Lorentz cones or second order cones. In particular, this shows that the p-cones are not homogeneous when p 6= 2, 1 < p <∞ and n ≥ 3, thus answering a problem proposed by Gowda and Trott.