Minimization of Energy Functionals Applied to Some Inverse Problems

We consider a general class of problems of the minimization of convex integral functionals subject to linear constraints. Using Fenchel duality, we prove the equality of the values of the minimization problem and its associated dual problem. This equality is a variational criterion for the existence of a solution to a large class of inverse problems entering the class of generalized Fredholm integral equations. In particular, our abstract results are applied to marginal problems for stochastic processes. Such problems naturally arise from the probabilistic approaches to quantum mechanics. Accepted 26 March 2001. Online publication 19 July 2001.     

The ``hot spots' conjecture for domains with two axes of symmetry

Consider a Neumann eigenfunction with lowest nonzero eigenvalue of a convex planar domain with two axes of symmetry. We show that the maximum and minimum of the eigenfunction are achieved at points on the boundary only. We deduce J. Rauch's ``hot spots' conjecture: if the initial temperature distribution is not orthogonal to the first nonzero eigenspace, then the point at which the temperature achieves its maximum tends to the boundary. This was already proved by Bañuelos and Burdzy in the case in which the eigenspace is one dimensional. We introduce here a new technique based on deformations of the domain that applies to the case of multiple eigenvalues.

     

Convex structures induced by Chebyshev systems

Recent developments have clarified that some tools of Convex Geometry are closely related to separation theorems obtained in the field of Functional Inequalities. This phenomenon has motivated the investigation of convex structures induced by Chebyshev systems. The present note characterizes such a possible structure, completely describing its combinatorial invariants.     

Errata corring ∨-dualities and ⊥-dualities

We correct an error in the proof of theorem 5.4 of [1]     

On a class of algorithms from experimental design theory

The theory of optimal (approximate) linear regression design has produced several iterative methods to solve a special type of convex minimization problems. The present paper gives a unified and extended theoretical treatment of the methods. The emphasis is on the mathematical structures relevant for the optimization process, rather than on the statistical background of experimental design. So the main body of the paper can be read independently from the experimental design context. Applications are given to a special class of extremum problems arising in statistics. The numerical results obtained indicate that the methods are of practical interest     

A class of generalised minimum effort problems in locally convex linear topological spaces

In this paper a class of generalised minimum effort problem has been formulated in locally convex linear topological spaces. Some necessary conditions has been obtained. Existence and uniqueness of the solutions are discussed     

On solving convex quadratic semi-infinite programming probelms

In this paper we study a from of convex quadratic semi-infinite programming problems with finitely many variables and infinitely many constraints over a compact metric space. An entropic path-following algorithum is introduced with a convergence proof. Some practical implementations and numerical experiments are also included     

Objective function and logarithmic barrier function properties in convex programming: level sets,solution attainment and strict convexity *

We give several results, some new and some old, but apparently overlooked, that provide useful characterizations of barrier functions and their relationship to problem function properties. In particular, we show that level sets of a barrier function are bounded if and only if feasible level sets of the objective function are bounded and we obtain conditions that imply solution existence, strict convexity or a positive definite Hessian of a barrier function. Attention is focused on convex programs and the logarithmic barrier function. Such results suggest that it would seem possible to extend many recent complexity results by relaxing feasible set compactness to the feasible objective function level set boundedness assumption.     

Constrained Nonparametric Maximum Likelihood via Mixtures

Abstract

This article discusses a new technique for calculating maximum likelihood estimators (MLEs) of probability measures when it is assumed the measures are constrained to a compact, convex set. Measures in such sets can be represented as mixtures of simple, known extreme measures, and so the problem of maximizing the likelihood in the constrained measures becomes one of maximizing in an unconstrained mixing measure. Such convex constraints arise in many modeling situations, such as empirical likelihood and estimation under stochastic ordering constraints. This article describes the mixture representation technique for these two situations and presents a data analysis of an experiment in cancer genetics, where a partial stochastic ordering is assumed but the data are incomplete.     

Learning the ECSI from binary signals for cognitive multicast secure beamforming

This paper investigates cognitive single-group multicast secure beamforming (SGMC-SBF) in multicast scenario where an eavesdropper who acts as a regular user seeks to intercept the multicast service without authorization. This study emphasizes that the transmitter iteratively learns the eavesdropper’s spatial correlation matrix from the accumulated binary feedback on received signal-to-noise-ratio. In each iteration, the transmitter learns the eavesdropper’s spatial correlation matrix based on the historical beamformings and the historical binary feedback information, which is then used to design the optimal beamforming that will be used to learn eavesdropper’s spatial correlation matrix in the next iteration. Without loss of generality, it is assumed that the transmitter knows instantaneous channel state information (CSI) of the legitimate users (LCSI), but not the instantaneous or statistical CSI of the eavesdropper (ECSI). For comparison, we also established the corresponding genie-aided SGMC-SBF with perfect ECSI and two traditional robust schemes with erroneous and statistical ECSI, respectively. The numerical results verify that the proposed cognitive SGMC-SBF are feasible solutions that provide excellent performance.