## 4.4.3 Convergence

The convergence of the optimization tools are often defined or proven for rather simple assumptions such as that the input parameter space is a convex domain or hyper-cube, that the objective function is a piecewise continuous function, or even that the second derivative of the object function exists4.12. However, these assumptions are rough estimations, which enable the engineer to decide whether a particular optimization algorithm is more suitable than others.

Since most simulation tools cannot provide all the desirable and advantageous properties of an ideal objective function, standard optimization methods are of only limited usability. Moreover, many problems of industrial interest produce no results with certain simulation tools because the simulation tool does not converge or would take too many resources. This might happen when hidden constraints are overseen, for instance if certain input parameters are correctly bound separately, but together they generate constellations that are physically not feasible.

The decision of how well a particular optimization method anticipates the current behavior of the objective score function has to be made by the design engineer who sets up the optimization task for a certain problem and who should know how the simulator behaves.

The convergence speed of standard optimization algorithms is often given by theorems but cannot be guaranteed for a general optimization task in the TCAD environment due to the guarantee that the simulation tool will converge for a finite number of parameter evaluations. However, if the behavior of the simulation tools can be appropriately estimated and a suitable optimization algorithm is applied to this problem, the probability is rather high that the optimization converges within the expected order of the convergence speed.

Optimization frameworks offer the possibility to apply additional limits. For instance, the CPU time for a single parameter evaluation or the total CPU time elapsed during the whole optimization can be limited. Another common method is to limit the number of parameter evaluations and the number of iterations of the optimizer. With these measures, the optimization often yields suboptimal results but it is guaranteed that the optimizer returns either the best result so far, or no result if all parameter evaluations have not been successfully terminated.

A critical point has to be noted here. The change and the manipulation of material boundaries, for instance if geometry parameters are optimized, requires an automatic remeshing. However, many simulation tools are very sensitive to misaligned meshes and require therefore appropriately tuned mesh kernels to provide good results. Therefore, the meshing tools have to be well configured in advance to provide automatic remeshing with the appropriate quality for the needs of the simulation tools [286].

Stefan Holzer 2007-11-19