Previous: Design of Experiments
Up: DOE/Opt: A System for Design of Experiments, Response Surface Modeling, and Optimization using Process and Device Simulation
Next: Examples
Previous Page: Response Surface Models
Next Page: Examples

Optimization

A key role of the DOE/Opt system is to provide an easy to use optimization capability. Real design and manufacturability optimization problems often have the following features. First, the optimization problems are often nonlinear, and multiple targets or objectives must be reconciled simultaneously. Second, numerous constraints apply in the optimization problem; these constraints may also be nonlinear in nature. Many design and optimization problems focus on choices of continuous parameter values. While discrete value or integer optimization is also often desirable (e.g. to make a choice in whether to include a design feature or not, or to decide between two alternative treatments), we have focused entirely on the continuous parameter optimization problem.

We have constructed a layered optimization capability as shown in Fig. 5. The fundamental optimization capability is provided by the NPSOL package [16]. On top of this a set of Tcl bindings provide interpretive access to the FORTRAN NPSOL routines. In addition to enabling one to dynamically ``call into'' the optimizer, the optimizer also ``calls out'' to the Tcl language layer for the evaluation of objective and constraint functions, as illustrated in Fig. 6. In this way it becomes possible to define an objective procedure and call the optimizer (which in turn evaluates the objective function several times), all dynamically and at run time. Layered on top of the basic access to NPSOL via Tcl, a generic weighted multiple target, multiple constraint optimization capability has been implemented. This layer is responsible for folding together and making use of all existing model and model gradient information (such as that provided by the DOE/Opt generated response surface models) to reduce optimization solution times. Through this interface, DOE/Opt is able to generate models and optimization problems, and solve them via NPSOL.

Several key issues have been identified and addressed in order to make the tool more appropriate for use in semiconductor process and device design. First, in practice optimization problems can become highly nonlinear and complex. No guarantees on global optimality are made. Instead, an approach to easily support the optimization from numerous starting points is used. In this case, we use the same design of experiments methods outlined in Section 4 to produce well distributed coverage of the parameter space in searching for potential optima. The Latin hypercube sample has proven particularly useful in generating optimization starting points.

Second, we have found that a selection of overall objective functions need to be provided. The current system includes the following choices: weighted sum of squares, weighted sum of normalized squares, weighted sum of absolute errors, weighted cumulative errors, and weighted normalized cumulative error. While some of these are not necessarily well-behaved (e.g. sum of absolute errors), they often map well onto the conceptual optimization problem at hand and work satisfactorily.

Third, we have found that it usually makes sense for the optimization to use transformed response surface model information rather than the ``pure'' values returned by models constructed under transformations. For example, a log transform selected when building a response surface model might fit , , and as a straight line internally. However, the model will still return exponentially large values when evaluated. For optimization, on the other hand, the linear underlying model is more effectively used. The ability to perform model regressions and optimizations in transformed space is important.