Modular Design, Generalized Inverses, and Convex Programming

It is shown that the modular design problem \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}$$\begin{array}{l@{\quad}l@{\quad}l}\mbox{minimize} & \sum^{i=m}_{i=1}y_{i} \sum^{j=n}_{j=1} z_{j},\\ \mbox{subject to} & y_{i}z_{j}\geq r_{ij},\\ & y_{i}z_{j} > 0, & (i=1, \ldots , m,\ j=1, \ldots , n)\end{array}$$\end{document} can be transformed into a problem of minimizing a separable convex function subject to linear equality constraints and nonnegativities. This transformation is effected by using a generalized inverse of the constraint matrix. Moreover the nature of the functional and the constraints of the separable problem are such that a good starting point for its solution can be obtained by solving a particular transportation problem. Several possible methods for solving the separable problem are discussed, and the results of our computational experience with these methods are given. It is also shown that the modular design problem can be viewed as a special case of a large class of general engineering design problems that have been discussed in the literature.