In this paper we address the basic geometric question of when a given convex set is the image under a linear map of an affine slice of a given closed convex cone. Such a representation or “lift” of the convex set is especially useful if the cone admits an efficient algorithm for linear optimization over its affine slices. We show that the existence of a lift of a convex set to a cone is equival...

The formulation of interior point algorithms for semide nite programming has become an active research area, following the success of the methods for large{ scale linear programming. Many interior point methods for linear programming have now been extended to the more general semide nite case, but the initialization problem remained unsolved. In this paper we show that the initialization strate...

We consider nonlinear systems dx=dt = f(x(t)) where Df(x(t)) is known to lie in the convex hull of L matrices A1, : : : , AL 2 R n . For such systems, quadratic Lyapunov functions can be determined using convex programming techniques [1]. This paper describes an algorithm that either nds a quadratic Lyapunov function or terminates with a proof that no quadratic Lyapunov function exists. The alg...

It is a classical inequality that the minimum of the ratio of the (weighted) arithmetic mean to the geometric mean of a set of positive variables is equal to one, and is attained at the center of the positivity cone. While there are numerous proofs of this fundamental homogeneous inequality, in the presence of an arbitrary subspace, and/or the replacement of the arithmetic mean with an arbitrar...

It is a classical inequality that the minimum of the ratio of the (weighted) arithmetic mean to the geometric mean of a set of positive variables is equal to one, and is attained at the center of the positivity cone. While there are numerous proofs of this fundamental homogeneous inequality, in the presence of an arbitrary subspace, and/or the replacement of the arithmetic mean with an arbitrar...