Based on the authors' previous work which established theoretical foundations of two, conceptual, successive convex relaxation methods, i.e., the SSDP (Successive Semide nite Programming) Relaxation Method and the SSILP (Successive Semi-In nite Linear Programming) Relaxation Method, this paper proposes their implementable variants for general quadratic optimization problems. These problems have...

Based on the authors' previous work which established theoretical foundations of two, conceptual, successive convex relaxation methods, i.e., the SSDP (Successive Semide nite Programming) Relaxation Method and the SSILP (Successive Semi-In nite Linear Programming) Relaxation Method, this paper proposes their implementable variants for general quadratic optimization problems. These problems have...

We give several different geometric characterizations of the situation in which the parallel set Fε of a self-similar set F can be described by the inner ε-parallel set T −ε of the associated canonical tiling T , in the sense of [16]. For example, Fε = T−ε ∪Cε if and only if the boundary of the convex hull C of F is a subset of F , or if the boundary of E, the unbounded portion of the complemen...

A gauge function f(.) is a nonnegative convex function that is positively homogeneous and satisfies f(O)=O. Norms and pseudonorms are specific instances of a gauge function. This paper presents a gauge duality theory for a gauge program, which is the problem of minimizing the value of a gauge function f(.) over a convex set. The gauge dual program is also a gauge program, unlike the standard La...

Let P be a set of n points in R and F be a family of geometric objects. We call a point x ∈ P a strong centerpoint of P w.r.t F if x is contained in all F ∈ F that contains more than cn points from P , where c is a fixed constant. A strong centerpoint does not exist even when F is the family of halfspaces in the plane. We prove the existence of strong centerpoints with exact constants for conve...

We consider a family of elliptic equations introduced in the context of traffic congestion. They have the form ∇·(∇F(∇u)) = f , where F is a convex function which vanishes inside some convex set and is elliptic outside. Under some natural assumptions on F and f , we prove that the function ∇F(∇u) is continuous in any dimension, extending a previous result valid only in dimension 2 [14]. Résumé....

We prove the existence of a class A of subsets of R of vc dimension 1 such that the symmetric convex hull F of the class of characteristic functions of sets in A is rich in the following sense. For any absolutely continuous probability measure μ on R, measurable set B ⊂ R and > 0, there exists a function f ∈ F such that the measure of the symmetric difference of B and the set where f is positiv...