A Test Problem Generator for Discrete Linear L ₁ Approximation Problems

Abstract

Described here are the theoretmal development and computer implementation of a procedure that generates test problems for L~ esth-nation of the linear model y= Xfl + u. The generation procedure allows the user flembdlty m specifying the problem dimensions, the LI solution vector fl*, the distribution of the observed residuals ~, as well as the column rank, row repetitions, and degree of degeneracy of the matrix X The user can also specify the distributional form, mean, and variance for each independent variable An lraportant feature of the generator is that any problem it creates is guaranteed to have a umque solutmn ~ whenever X has full rank

An assumption‐free convergence analysis for a perturbation of the scaling algorithm for linear programs, with application to the L 1 estimation problem

“…But this implies from (10) that the limit {nk}K + n* is well defined and from (12) and part (ii) that {Ok}K -+ 0. Consequently, (x*,n*) satisfies (18) and this completes the proof.…”

“…(iii) For any convergent subsequence { x~}~ + x*, we have x* > 0. Moreover, IT^}^ converges to some r* such that (x*,r*) satisfies (18), and hence yields an optimal pair of primal and dual solutions to BP.…”

A collection of test problems for discrete linear L? data fitting

“…Since an optimal solution can be obtained with 3j " 0 for j > r, the original Lâ pproximation problem can be formulated [16] as the following linear program: These conditions can be considerably simplified and tlien applied to judge whether a given 3 = represents an optimal solution to (1).…”

“…Controlled experiments with these codes are facilitated by the availability of well-defined classes of test problems so that results obtained by sampling a finite number of test problems from a class will possess some measure of validity. For this reason, generators for problem classes with controllable characteristics have been developed [8,16] and used in performing comparisons of Lj codes [12].…”

“…In addition, the problems described here may suggest types of problem structures to be mimicked by problem generators [8,16,17,19].…”