* Glossary of notation * Introduction * Preliminaries * Existence and Construction of Generalized Inverses * Linear Systems and Characterization of Generalized Inverses * Minimal Properties of Generalized Inverses * Spectral Generalized Inverses * Generalized Inverses of Partitioned Matrices * A Spectral Theory for Rectangular Matrices * Computational Aspects of Generalized Inverses * Miscellaneous Applications * Generalized Inverses of Linear Operators between Hilbert Spaces * Appendix A: The Moore of the Moore-Penrose Inverse * Bibliography * Subject Index * Author Index

Generalized inverses: theory and applications

The Hermite-Hadamard double inequality is the first fundamental result for convex functions defined on a interval of real numbers with a natural geometrical interpretation and a loose number of applications for particular inequalities. In this monograph we present the basic facts related to Hermite- Hadamard inequalities for convex functions and a large number of results for special means which can naturally be deduced. Hermite-Hadamard type inequalities for other concepts of convexities are also given. The properties of a number of functions and functionals or sequences of functions which can be associated in order to refine the result are pointed out. Recent references that are available online are mentioned as well.

Selected Topics on Hermite-Hadamard Inequalities and Applications

Mathematical programming problems dealing with functions, each of which can be represented as a difference of two convex functions, are called DC programming problems. The purpose of this overview is to discuss main theoretical results, some applications, and solution methods for this interesting and important class of programming problems. Some modifications and new results on the optimality conditions and development of algorithms are also presented.

DC programming: overview

Max-algebra: Two Special Features.- One-sided Max-linear Systems and Max-algebraic Subspaces.- Eigenvalues and Eigenvectors.- Maxpolynomials. The Characteristic Maxpolynomial.- Linear Independence and Rank. The Simple Image Set.- Two-sided Max-linear Systems.- Reachability of Eigenspaces.- Generalized Eigenproblem.- Max-linear Programs.- Conclusions and Open Problems.

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Max-linear Systems: Theory and Algorithms

The main advances regarding the use of the Choquet and Sugeno integrals in multi-criteria decision aid over the last decade are reviewed. They concern mainly a bipolar extension of both the Choquet integral and the Sugeno integral, interesting particular submodels, new learning techniques, a better interpretation of the models and a better use of the Choquet integral in multi-criteria decision aid. Parallel to these theoretical works, the Choquet integral has been applied to many new fields, and several softwares and libraries dedicated to this model have been developed.

A decade of application of the Choquet and Sugeno integrals in multi-criteria decision aid

espace lineaire norme # espace metrique # meilleure approximation # sous-espace lineaire # sous-espace lineaire de dimension finie # sous-espace lineaire ferme de codimension finie # element d'ensemble # element d'ensemble non-lineaire

Best Approximation in Normed Linear Spaces by Elements of Linear Subspaces

Convexity of Elements of a Complete Lattice. Convexity of Subsets of a Set. Convexity of Functions on a Set. Quasi-Convexity of Functions on a Set. Dualities Between Complete Lattices. Dualities Between Families of Subsets. Dualities Between Sets of Functions. Conjugations. v-Dualities and -|--Dualities. Subdifferentials. Notes and Remarks. References. Indexes.

Abstract convex analysis

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Cbms-nsf regional conference series in applied mathematics

The theory of best approximation and functional analysis

We consider convex sets and functions over idempotent semifields, like the max-plus semifield. We show that if $K$ is a conditionally complete idempotent semifield, with completion $\bar{K}$, a convex function $K^n\to\bar{K}$ which is lower semi-continuous in the order topology is the upper hull of supporting functions defined as residuated differences of affine functions. This result is proved using a separation theorem for closed convex subsets of $K^n$, which extends earlier results of Zimmermann, Samborski, and Shpiz.

Ivan Singer

Papers

Best Approximation in Normed Linear Spaces by Elements of Linear Subspaces

Abstract convex analysis

Cbms-nsf regional conference series in applied mathematics

The theory of best approximation and functional analysis

Max-plus convex sets and functions