Entropy functionals (i.e. convex integral functionals) and extensions of these functionals are minimized on convex sets. This paper is aimed at reducing as much as possible the assumptions on the constraint set. Dual equalities and characterizations of the minimizers are obtained with weak constraint qualifications.

In recent years we witnessed an impressive advance in the social networks field, which became a ”hot” topic and a focus of considerable attention. Also, the development of methods that focus on the analysis and understanding of the evolution of data are gaining momentum. In this paper we present an approach to visualize the evolution of dynamic social networks by using Tucker decomposition and ...

As modern high-resolution imaging devices allow to acquire increasingly large and complex volume data sets, their effective and compact representation for visualization becomes a challenging task. The Tucker decomposition has already confirmed higherorder tensor approximation (TA) as a viable technique for compressed volume representation; however, alternative decomposition approaches exist. In...

The purpose of this paper is to characterize the weak Pareto optimality for multiobjective pseudoconvex problem. In fact, it is a first order optimality characterization that generalize the Karush-Kuhn-Tucker condition. Moreover, this work is an extension of the single-objective case [6] to the multiobjective one with pseudoconvex continuous functions. Mathematics Subject Classification: 46N10,...

Multilinear algebra of the higher-order tensor has been proposed as a potential mathematical framework for machine learning to investigate the relationships among multiple factors underlying the observations. One popular model Nonnegative Tucker Decomposition (NTD) allows us to explore the interactions of different factors with nonnegative constraints. In order to reduce degeneracy problem of t...

This paper addresses the solution of instationary parabolic evolution equations simultaneously in space and time. As a model problem we consider the heat equation posed on the unit cube in Euclidean space of moderately high dimension. An a priori stable minimal residual Petrov-Galerkin variational formulation of the heat equation in space-time results in a generalized least squares problem. Thi...

These notes characterize maxima and minima in terms of first derivatives. I focus primarily on maximization. The problem of minimizing a function f has the same solution (or solutions) as the problem of maximizing −f , so all of the results for maximization have easy corollaries for minimization. The main result of these notes is the Karush-Kuhn-Tucker (KKT) Theorem, recorded as Theorem 3 in Se...

The main purpose of these lectures is to familiarize the student with the basic ingredients of convex analysis, especially its subdifferential calculus. This is done while moving to a clearly discernible end-goal, the Karush-Kuhn-Tucker theorem, which is one of the main results of nonlinear programming. Of course, in the present lectures we have to limit ourselves most of the time to the Karush...