Convergence Criteria

Default Convergence Criteria

For sensitivity-based optimization, the optimization stops automatically when the following convergence criteria are fulfilled. More detailed explanations of the criteria can be found in the sections beneath.


OPT_PARAM
 ID_NAME                = NO_OPT_PARAM
 OPTIMIZE               = MY_OPTIMIZE
 STOP_CRITERION_LEVEL   = BOTH
 STOP_CRITERION_OBJ     = 0.001
 STOP_CRITERION_DENSITY = 0.005
 STOP_CRITERION_ITER    = 4
END_

Automatic Creation of Stop Conditions

For sensitivity-based optimization, some convergence stop criteria that consider the change of the objective function and the change in the design variables can be created. The creation of the convergence criteria can be influenced with the STOP_CRITERION_CONVERGENCE item in the OPT_PARAM command. Valid options are AUTO (default), NO, and YES. With AUTO, a stop criterion is created or neglected depending on the default of the installed version. If CREATE_DEFAULT_CRITERIA = NO is selected, no convergence stop criteria is created. Otherwise the optimization stops when convergence is detected.

Convergence Criteria

By default, the stop criterion measuring the change in the objective function and the criterion measuring the change in the design variables should both be fulfilled before the optimization algorithm stops. To specify that just one of the criteria must be fulfilled, set STOP_CRITERION_LEVEL equal to SINGLE. The default value for STOP_CRITERION_LEVEL is BOTH.

In the example below, the optimization algorithm should stop when just one of the criteria is fulfilled


OPT_PARAM
 ID_NAME              = opt_params
 OPTIMIZE             = id_of_optimize
 STOP_CRITERION_LEVEL = SINGLE
END_

Stop Criterion: Change in Objective Function

One of the available stop criteria is a measurement based on the change in the objective function from one optimization design cycle to the next design cycle. This stop criterion is defined by


$\frac{\| Θ_{n} - Θ_{n - 1} \|}{\| Θ_{n} \|} \leq STOP_CRITERION_OBJ$

where Θ is objective value, the index n is representing the present optimization iteration and is n-1 is the previous iteration. When the criterion is lower than STOP_CRITERION_OBJ, the stop criterion is fulfilled. The default for STOP_CRITERION_OBJ is 0.001. The value can be changed by the user; for example,


OPT_PARAM
 ID_NAME            = opt_params
 OPTIMIZE           = id_of_optimize
 STOP_CRITERION_OBJ = 0.01
END_

This could be desirable when the number of optimizations should be reduced.

Stop Criterion: Change in Density Values

Another stop criterion is based upon the change in the design variables from one optimization design cycle to the next optimization design cycle and is defined by:


$\frac{{\sum_{1}}^{N u m b e r o f e l e m e n t s} \| ρ_{n} - ρ_{n - 1} \|}{N u m b e r o f e l e m e n t s} \leq S T O P_C R I T E R I O N_D E N S I T Y$

where the index n is representing the present optimization iteration and n-1 is the previous iteration. When the criterion is lower than STOP_CRITERION_DENSITY, the stop criterion is fulfilled. The default for STOP_CRITERION_DENSITY is 0.005. The value can be changed by the user, for example,


OPT_PARAM
 ID_NAME                = opt_params
 OPTIMIZE               = id_of_optimize
 STOP_CRITERION_DENSITY = 0.0001
END_

Start Iteration for Convergence Check

The two convergence criteria above are not checked before a specified number of optimization iterations have been executed. This means that the optimization will always be executed until STOP_CRITERION_ITER is reached. The number is set by the number of STOP_CRITERION_ITER. To change STOP_CRITERION_ITER to, for example, 10, the following commands could be used:


OPT_PARAM
 ID_NAME             = opt_params
 OPTIMIZE            = id_of_optimize
 STOP_CRITERION_ITER = 10
END_

The default value for STOP_CRITERION_ITER is 4. Note that STOP_CRITERION_ITER must be greater than 2.