The inexact fixed matrix iteration for solving large linear inequalities in a least squares sense

Abstract

A fixed matrix iteration algorithm was proposed by A. Dax (Numer. Algor. 50, 97-114 2009) for solving linear inequalities in a least squares sense. However, a great deal of computation for this algorithm is required, especially for large-scale problems, because a least squares subproblem should be solved accurately at each iteration. We present a modified method, the inexact fixed iteration method, which is a generalization of the fixed matrix iteration method. In this inexact iteration process, the classical LSQR method is implemented to determine an approximate solution of each least squares subproblem with less computational effort. The convergence of this algorithm is analyzed and several numerical examples are presented to illustrate the efficiency of the inexact fixed matrix iteration algorithm for solving large linear inequalities.

Discrete-Time Zeroing Dynamics With Quadruplicate Error Pattern for Time-Varying Linear Inequality

“…whereė(t) denotes the time derivative of e(t), and γ > 0 ∈ R is the design parameter that affects the solution convergence. By expanding (4), the following computational model is obtained:…”

“…where coefficient matrix A ∈ R n×n and vector b ∈ R n are constant, and x ∈ R n is the unknown vector to be obtained. To solve (1), many numerical algorithms and neural networks have been developed and investigated [4]- [13]. For example, in [8], different iterative methods were presented by Yang et al to solve the LI system.…”

Best Approximate Solutions of Inconsistent Linear Inequality Systems

“…Consider the deterministic counterpart of the uncertain system in R 2 f 2x 1 0; 2x 1 0g ; with uncertainty intervals of the form " for each coe¢ cient ; with " > 0; formulated as P R in (17). This counterpart is the inconsistent continuous system a > t x b t ; t 2 T ; where T = T 1 [ T 2 ; with T j = ( 1) j 2; 0; 0 + C; j = 1; 2; C = [ "; "] 3 ; a t = (t 1 ; t 2 ) > and…”

“…x k 0; k = 1; :::; n; (17) which is a linear SIP problem whose mixed constraint system can be written…”

“…Numerical methods to compute best least squares solutions (see, e.g., the recent paper [22] on variants of Han's algorithm [12], whose fundamentals are the existence and characterization theorems for discrete inconsistent system proved in the latter paper; the recent work [5] provides two new proofs of Han's existence theorem; two additional works, [6] and [17], on the so-called hybrid algorithm). To the best of our knowledge, no extension of these results and methods to continuous and mixed linear inconsistent systems is still available.…”