A center of a polytope: An expository review and a parallel implementation

S K Sen,D W Fausett,Hongwei Du

doi:10.1155/s0161171293000262

S K Sen, D W Fausett + Show 1 more

Open Access

https://doi.org/10.1155/s0161171293000262

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Abstract

The solution space of the rectangular linear systemAx=b, subject tox≥0, is called a polytope. An attempt is made to provide a deeper geometric insight, with numerical examples, into the condensed paper by Lord, et al. [1], that presents an algorithm to compute a center of a polytope. The algorithm is readily adopted for either sequential or parallel computer implementation. The computed center provides an initial feasible solution (interior point) of a linear programming problem.

Highlights

The solution space w of a linear system Ax b, where A is an rn x n matrix of rank r, will (i) be empty if and only if the system is inconsistent, i.e., iff r rank (A,b), (ii) have a unique point iff r rank(A,b) m n, and (iii) have infinite points or, equivalently, an n- r parameter solution space iff r rank(A,b) _< m when m < n or r rank(A,b) < n when m _> n
The paper by Lord, et al [1] suggests a new definition of a ’center’ of a convex polytope in Euclidean space a point is a center of a convex polytope in Euclidean space if it is the center of a sphere that lies within the polytope and it touches a set of bounding hyperplanes that have no common intersection
If the polytope defined by Az b, : > 0 is nonexistent the centering algorithm detects that condition

Summary

INTRODUCTION

The solution space w of a linear system Ax b, where A is an rn x n matrix of rank r, will (i) be empty if and only if (iff) the system is inconsistent, i.e., iff r rank (A,b), (ii) have a unique point (solution) iff r rank(A,b) m n, and (iii) have infinite points (solutions) or, equivalently, an n- r parameter solution space iff r rank(A,b) _< m when m < n or r rank(A,b) < n when m _> n. In Case (i), the corresponding polytope (defined here by Ax b, :r >_ 0) is nonexistent, while in Case (ii) the polytope will be the (0-dimensional) unique point iff the point x satisfies the condition x > 0. In both these cases the proposed centering algorithm achieves the same result as that accomplished by an efficient linear equation solving algorithm. A center is usually defined uniquely as the extremum of a potential function that vanishes on the boundary [2,3,4] Such a definition adopts a nonlinear concept into what is essentially a linear system.

THE CENTERING ALGORITHM

GEOMETRY OF THE CENTERING ALGORITHM

EXAMPLES

PARALLEL IMPLEMENTATION

CONCLUDING REMARKS