Let H1 and H2 be real Hilbert spaces. Suppose H2 is partially 2 ordered, a, b ε H2, c ε H1 and A :H1→H2is a continuous linear 21 12 map. We consider the following interval linear program: Maximize &lt;c, x&gt; subject to a ≤Ax ≤ b. Conditions under which explicit solutions to the above problem can be found are studied. The solutions are represented in terms of generalized inverses of A. Several examples are given to illustrate the main results. © 1995, Taylor &amp; Francis Group, LLC. All rights reserved.

K C Sivakumar

Department of Mathematics

IIT Madras is a public technical and research university located in Chennai, Tamil Nadu. Founded in 1959, it is recognised as an Institute of National Importance.

IIT Madras has been ranked as the top engineering institute in India for four years in a row by the National Institutional Ranking Framework of the MHRD

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IIT Madras

Applications of generalized inverses to interval linear programs in hilbert spaces

Numerical Functional Analysis and Optimization

This monograph covers the theory of finite and infinite matrices over the fields of real numbers, complex numbers and over quaternions. Emphasizing topics such as sections or truncations and their relationship to the linear operator theory on certain specific separable and sequence spaces, the authors explore techniques like conformal mapping, iterations and truncations that are used to derive precise estimates in some cases and explicit lower and upper bounds for solutions in the other cases. Most of the matrices considered in this monograph have typically special structures like being diagonally dominated or tridiagonal, possess certain sign distributions and are frequently nonsingular. Such matrices arise, for instance, from solution methods for elliptic partial differential equations. The authors focus on both theoretical and computational aspects concerning infinite linear algebraic equations, differential systems and infinite linear programming, among others. Additionally, the authors cover topics such as Bessel’s and Mathieu’s equations, viscous fluid flow in doubly connected regions, digital circuit dynamics and eigenvalues of the Laplacian © Springer International Publishing Switzerland 2016.

Infinite Matrices and Their Recent Applications

Infinite matrices and their recent applications

Infinite matrices, the forerunner and a main constituent of many branches of classical mathematics (infinite quadratic forms, integral equations, differential equations, etc.) and of the modern operator theory, is revisited to demonstrate its deep influence on the development of many branches of mathematics, classical and modern, replete with applications. This review does not claim to be exhaustive, but attempts to present research by the authors in a variety of applications. These include the theory of infinite and related finite matrices, such as sections or truncations and their relationship to the linear operator theory on separable and sequence spaces. Matrices considered here have special structures like diagonal dominance, tridiagonal, sign distributions, etc. and are frequently nonsingular. Moreover, diagonally dominant finite and infinite matrices occur largely in numerical solutions of elliptic partial differential equations.The main focus is the theoretical and computational aspects concerning infinite linear algebraic and differential systems, using techniques like conformal mapping, iterations, truncations etc. to derive estimates based solutions. Particular attention is paid to computable precise error estimates, and explicit lower and upper bounds. Topics include Bessel's, Mathieu equations, viscous fluid flow, simply and doubly connected regions, digital dynamics, eigenvalues of the Laplacian, etc. Also presented are results in generalized inverses and semi-infinite linear programming. © 2008 Elsevier Inc. All rights reserved.

Fulltext

Linear Algebra and Its Applications

A review of infinite matrices and their applications

Let X be a partially ordered real Banach space, let a,b X with ab. Let φ be a bounded linear functional on X. We say that X satisfies the box-optimization property (or X is a BOP space) if the box-constrained linear program: max aφ,x, s.t. aaxb, has an optimal solution for any φ,a and b. Such problems arise naturally in solving a class of problems known as interval linear programs. BOP spaces were introduced (in a different language) and systematically studied in the first author's doctoral thesis. In this paper, we identify new classes of Banach spaces that are BOP spaces. We present also sufficient conditions under which answers are in the affirmative for the following questions: (i) When is a closed subspace of a BOP space a BOP space? (ii) When is the range of a bounded linear map a BOP space? (iii) Is the quotient space of a BOP space a BOP space? © 2008 Springer Science+Business Media, LLC.

Journal	Numerical Functional Analysis and Optimization
ISSN	01630563
Open Access	No