Adi Ben-Israel

A modified newton-raphson method for the solution of systems of equations

An implicit function theorem and a resulting modified Newton-Raphson method for roots of functions between finite dimensional spaces, without assuming non-singularity of the Jacobian at the initial approximation.

Asymptotic duality over closed convex sets

Abstract The asymptotic duality theory of linear programming over closed convex cones [4] is extended to closed convex sets, by embedding such sets in appropriate cones. Applications to convex programming and to approximation theory are given.

Some Topics in Generalized Inverses of Matrices

Publisher Summary This chapter presents some topics in generalized inverse of matrices. By a generalized inverse of a given matrix A, one means a matrix X associated in some way with A that (1) exists for a class of matrices larger than the class of non-singular matrices, (2) has some of the properties of the usual inverse, and (3) reduces to the usual inverse when A is non-singular. The various generalized inverses that can be defined, differ widely in their properties, applicability, and computation. Generalized inverses extend to singular and rectangular matrices. It is, therefore, not surprising that generalized inverses play a role, although a minor one, in the spectral theory for rectangular matrices.

Linear inequalities, mathematical programming and matrix theory

A survey is made of solvability theory for systems of complex linear inequalities.This theory is applied to complex mathematical programming and stability and inertia theorems in matrix theory.

Linear equations over cones with interior: a solvability theorem with applications to matrix theory

Abstract The solvability of linear equations with solutions in the interior of a closed convex cone is characterized. Corollaries include Lyapunovs theorem characterizing stable matrices and a generalization of Stiemkes theorem of the alternative for complex linear inequalities.

Technical Note-Explicit Solutions of Interval Linear Programs

This note gives conditions for an explicit noniterative representation of the set of optimal solutions of interval programming problems: maximize {c, x: a âx89¦ Ax âx89¦ b} and standard programming problems: maximize {c, x: Ax = b, 0 âx89¦ x âx89¦ u}.

A Note on Pencils of Hermitian or Symmetric Matrices

Necessary and sufficient conditions are given for the pencil generated by two, or more, Hermitian matrices to contain a positive definite matrix.

On direct sum decompositions of hestenes algebras

In a *-linear Hestenes algebra, the elements with *-reciprocals are characterized by means of certain direct sum decompositions of the algebra.

Applications of Generalized Inverses to Programming, Games and Networks

Publisher Summary Generalized inverses are applicable to various engineering and management models where linear transformations are used to describe a process, its rate of change, or its cost. Often such applications are straightforward, but even then, using generalized inverses may result in simpler notation and theory or in a more efficient computation. Whether used for convenience or out of necessity, a generalized inverse most suitable for the problem at hand has to be chosen from the many available generalized inverses. This chapter presents an introduction to the Bott-Duffin inverse and its application to electrical network analysis. It discusses a special class of linear programs that are explicitly solvable by using {1}-inverses. It describes an application of generalized inverses to the solution of Diophantine linear equations. The results obtained from this application may prove very useful in integer programming. The chapter also presents a characterization of equilibrium points of bimatrix games in terms of certain sub-matrices and their Moore-Penrose inverses.

The geometry of solvability and duality in linear programming

Solvability and boundedness criteria for dual linear programming problems are given in terms of the problem data and the intersections of the nonnegative orthant with certain complementary orthogonal subspaces.