Higher-level modeling formulations in optimal design.

Optimal design models contain parameters that are fixed during the optimization process. When these parameters change values, the mathematical properties of the model and the underlying engineering system may change drastically. We consider the special case when a single parameter is embedded in the problem statement. While the necessary and sufficient conditions for optimality are local in nature, i.e., they provide information only in a neighborhood of a solution, it is possible to impose certain regularity conditions that allow us to study solution behavior when the model deforms under finite ( and large) changes in the parameter. For a one parameter family of models, such deformation generates a solution curve along which one may encounter singularities, such as limit points and bifurcations. The characterization of these singularities leads to a continuation-type algorithm for computing regular segments of the solution curve. This algorithm is novel in that no more information is required than what is a normal by-product during iterations of a typical nonlinear programming solver. Beyond the obvious utility in discovering singularities, it is argued that this theory be used as a model analysis tool as well. The proposed numerical method, for example, allows the designer to experiment with different variable-parameter partitions that may be consistent with the original problem intent. In addition, the functional form and numerical values of parameters reflect decisions made during the formulation of the problem, hence the resultant characteristics of the solution curve, singular or otherwise, are a direct manifestation of prior modeling decisions. Motivation for studying modeling in this way is provided by another instance of model analysis where monotonicity principles are used to generate facts about boundedness and constraint activity in design optimization problems. A production system is implemented that automatically extracts such facts from models. Two design problems illustrate the usefulness of this system.PhDMechanical engineeringUniversity of Michiganhttp://deepblue.lib.umich.edu/bitstream/2027.42/162514/1/9013994.pd