About the Approximation Loop Strategy

You can use the Approximation Loop strategy to solve a standard single objective function problem.

The following is the sequence of events that occur during the Approximation Loop strategy’s execution when solving a standard single objective function problem:

1. A local design space is created around the starting design point.
2. A linear response surface model (RSM) approximation is created and initialized by executing a suitable DOE design matrix using an orthogonal array at the smallest suitable size, such that at least N + 1 design points are included. N is the number of inputs of the approximation (the design variables).
3. The starting point is selected as the best point from all of the initialization design points.
4. Optimization is executed inside the local design space using the initial linear approximation.
5. The optimum design is analyzed using the actual simulation process flow.
6. If the exact analysis yields no improvement in the value of the ObjectiveAndPenalty parameter, the new approximate optimum design is rejected and the next sub-optimization is set to start from the previous design point.
7. Convergence is checked with respect to the value of the ObjectiveAndPenalty parameter. If the strategy is converged or the maximum number of sub-optimization runs is reached, the strategy execution is terminated.
8. Approximation errors are compared to the acceptable levels.

   If the errors are within the acceptable levels and there was an improvement in the value of the ObjectiveAndPenalty parameter, another sub-optimization is executed using the same approximation (no re-initialization).

   If the errors were too high or there was no improvement in the ObjectiveAndPenalty parameter, the last design point is added to the approximation and its coefficients are re-calculated before the next sub-optimization is executed. A quadratic RSM is generated if there are enough design points. Otherwise, term selection is used to calculate the maximum possible number of coefficients.

   In both cases if the Always re-initialize approximation before sub-optimization option is selected, all previous design points are filtered by the current local design bounds and additional points are generated using a Latin Hypercube DOE matrix, such that there are enough points for a full quadratic RSM approximation. Then, the approximation is re-initialized.

9. If the last approximate optimum point was rejected or the approximation errors were too high, the relative size of the local design space is reduced by multiplying it by the reduction factor. If the last approximate optimum was an improvement and the relative size of the local design space is at its minimum allowed value, the relative size of the local design space is increased by multiplying it by the expansion factor.
10. The process is repeated with the next sub-optimization run inside the newly constructed local design space, until convergence is reached or the maximum number of sub-optimizations is reached.

You can use the Approximation Loop strategy to solve a multi-objective optimization problem.

The following is the sequence of events that occur during the Approximation Loop strategy's execution when solving a multi-objective optimization problem:

1. A local design space is created around the starting design point.
2. A linear response surface model (RSM) approximation is created and initialized by executing a suitable DOE design matrix using an orthogonal array at the smallest suitable size, such that at least N + 1 design points are included. N is the number of inputs of the approximation (the design variables).
3. The starting point is selected as the best point from all of the initialization design points by comparing the weighted sum of all objective parameters (this step is similar to the case with a single objective function).
4. The multi-objective optimization technique AMGA is executed inside the local design space using the initial linear approximation. This technique produces a Pareto set of design points, called the "approximate Pareto set" because it is produced using the internal approximation.
5. The approximate Pareto set is sorted using one of two criteria: 1) best weighted sum of all objective parameters (similar to the single objective function), or 2) minimum Chebyshev distance from the desired levels of all objective parameters. The first point of the sorted approximate Pareto set is selected as the best point of the sub-optimization.
6. A single point (the best point described above) or several points are selected from the sorted approximate Pareto set for exact analysis and inclusion into the internal approximation. If the selected point falls within less than 1% of any design variable value from any previously selected point, it is dropped and the next point is checked. This prevents ill-conditioning of the approximation data set. The optimum design is analyzed using the actual simulation process flow.
7. Convergence is checked with respect to the value of the ObjectiveAndPenalty parameter. If the strategy is converged or the maximum number of sub-optimization is reached, the strategy execution is terminated.
8. The selected approximate Pareto set design points are added to the internal Approximation data set, and its coefficients are re-calculated before the next sub-optimization run is executed. A quadratic RSM is generated if there are enough design points. Otherwise, term selection is used to calculate the maximum possible number of coefficients. In addition to the selected approximate Pareto set points, if the Always re-initialize approximation before sub-optimization option is selected, all previous design points are filtered by the current local design bounds and additional points are generated using a Latin Hypercube DOE matrix, such that there are enough points for a full quadratic RSM approximation. Then, the approximation is re-initialized.
9. If the approximation errors were too high, the relative size of the local design space is reduced by multiplying it by the reduction factor.
10. The process is repeated with the next sub-optimization run inside the newly constructed local design space, until convergence is reached or the maximum number of sub-optimizations is reached.
11. At the end of the Approximation Loop execution, a new Pareto set is selected from all of the exact subflow executions. This so-called exact Pareto set is reported in the Summary of the Approximation Loop and saved in the Exploration component's special file parameter called Pareto Set.