An Explicit Solution of a Special Class of Linear Programming Problems

The linear programs considered here are of the form: \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mbox{maximize}\ c, x\quad \mbox{subject to }a\leq Ax \leq b,$$ \end{document} where A is of full row rank, and LP is feasible with bounded optimal solutions. The main result is an explicit representation of the general optimal solution of LP in terms of a generalized inverse of A. This explicit solution of LP-explicit in the sense that A-1b is an explicit solution of Ax = b-has obvious theoretical and possibly computational advantages over the well-known iterative methods of linear programming. The results are illustrated by a simple example, and extensions to general linear programs are discussed.