On convexity properties of homogeneous functions of degree one

We provide an explicit example of a function that is homogeneous of degree one, rank-one convex, but not conve

Parity violating target asymmetry in electron - proton scattering

We analyze the parity-violating (PV) components of the analyzing power in elastic electron-proton scattering and discuss their sensitivity to the strange quark contributions to the proton weak form factors. We point out that the component of the analyzing power along the momentum transfer is independent of the electric weak form factor and thus compares favorably with the PV beam asymmetry for a determination of the strangeness magnetic moment. We also show that the transverse component could be used for constraining the strangeness radius. Finally, we argue that a measurement of both components could give experimental information on the strangeness axial charge.Comment: 24 pages, Latex, 5 eps figures, submitted to Phys.Rev.

Interior Point Methods 25 Years Later

Interior point methods for optimization have been around for more than 25 years now. Their presence has shaken up the field of optimization. Interior point methods for linear and (convex) quadratic programming display several features which make them particularly attractive for very large scale optimization. Among the most impressive of them are their low-degree polynomial worst-case complexity and an unrivalled ability to deliver optimal solutions in an almost constant number of iterations which depends very little, if at all, on the problem dimension. Interior point methods are competitive when dealing with small problems of dimensions below one million constraints and variables and are beyond competition when applied to large problems of dimensions going into millions of constraints and variables. In this survey we will discuss several issues related to interior point methods including the proof of the worst-case complexity result, the reasons for their amazingly fast practi-cal convergence and the features responsible for their ability to solve very large problems. The ever-growing sizes of optimization problems impose new requirements on optimizatio

Some Numerical Methods For The Study Of The Convexity Notions Arising In The Calculus Of Variations

. We propose numerical schemes to determine whether a given function is convex, polyconvex, quasiconvex, and rank one convex. These notions are of fundamental importance in the vectorial problems of the calculus of variations. 1. Introduction One of the most important problems in the calculus of variations deals with the integral I(u) = Z \Omega f(ru(x)) dx (1) where 1.\Omega ae R n is a bounded open set, 2. u :\Omega ae R n \Gamma! R m belongs to a Sobolev space, 3. f : R m\Thetan \Gamma! R is a continuous function. Usually one wants to minimize (1) subject to some constraints, e.g. certain boundary conditions, isoperimetric constraints, etc : : : The only general method to deal with these problems consists in proving the sequential weak lower semicontinuity of I(u). When m = 1 or n = 1, this property is equivalent to the convexity of f . However, when m;n ? 1, it is equivalent to the so called quasiconvexity of f , a notion introduced by Morrey [23]. Definition 1.1. ..