The Method of Alternate Formulations Applied to Optimization Problems With Functionally Dependent State Equations

The Method of Alternate Formulations (MAF) is a non-numerical approach to constrained optimal design. MAF requires that the problem statement be transformed into an objective function, a set of equality-constraints (i.e. state equations), and a set of upper and lower bounds on the variables. In this format, the design vector can be partitioned into decision variable and state variable components. This is the same format as used in the solution of such problems by the generalized reduced gradient method. The fact that there are usually several ways to effect the partition of the design vector gives rise to the existence of alternate formulations. Each alternate formulation contains all of the information about the physical system — and the constrained optima are invariant under the transformation from form to form. Yet all other mathematical properties (e.g., convexity, linearity, scaling, etc.) can change. In this paper, we consider the special case when the state equations are functionally dependent, hence some of the expected constraint intersections do not exist. Several examples are used to demonstrate the concept of functional dependence and to show how functional dependence affects the search for the solution.