Functional Dependence and the Method of Alternate Formulations in Optimal Design

Abstract

In an earlier article in this journal we introduced the Method of Alternate Formulations (MAF). MAF is a nonnumerical approach to constrained optimal design implemented with symbolic mathematics. The MAF problem formulation is the same as is used by the generalized reduced gradient method. There are usually many ways to partition the design vector into decision variable and state variable components and so there are usually many different alternate formulations for the objective function and constraints. Each alternate formulation contains all of the information about the physical system. Yet all other mathematical properties (e.g., convexity, linearity, scaling, etc.) can change. It has been observed that some of the alternate formulations that should exist based strictly on the theory of combinations cannot be obtained. In this paper, we show that this phenomenon occurs whenever there is functional dependence in the system model. Several examples are used to show how functional dependence affects the search for the solution by MAF. Prediction of functional dependence at the outset informs the designer which formulations cannot exist. This allows the designer to concentrate effort (more productively) on other formulations of the problem.