Scaling Methods¶
This section describes scaling utility functions and methods.
Context¶
Creating well scaled models is important for increasing the efficiency and reliability of solvers. Depending on property package units of measure and process scale, variables and constraints are often badly scaled.
Scaling factors can be specified for any variable or constraint. Pyomo and many
solvers support the scaling_factor
suffix. To eliminate the possibility of
defining conflicting scaling factors in various places in the model, the IDAES
standard is to define the scaling_factor
suffixes in the same block as the
variable or constraint that they are scaling. This ensures that each scale
factor is defined in only one place, and is organized based on the model block
structure.
Scaling factors in IDAES (and Pyomo) are multiplied by the variable or constraint they scale. For example, a Pressure variable in Pa units may be expected to have a magnitude of around \(10^6\) for a specific process. To scale the variable to a more reasonable magnitude, the scale factor for the variable could be defined to be \(1 \times 10^{-5}\).
While many scaling factors should be give good default values in the property
packages, some (e.g. flow rates or material holdups) must be given scale factors
by the user for a specific process model. Still other scale factors can be
calculated from supplied scale factors, for example, mass balance scale factors
could be determined from flow rate scale factors. To calculate scale factors,
models may have a standard calculate_scaling_factors()
method. For more
specific scaling information, see the model documentation.
For much of the core IDAES framework, model constraints are automatically scaled
via a simple transformation where both sides of the constraint are multiplied by
a scale factor determined based on supplied variable and expression scaling
factors. The goal of this is to ensure that solver tolerances are meaningful for
each constraint. A constraint violation of \(1 \times 10^{-8}\) should be
acceptable, but not too tight to achieve given machine precision limits. IDAES
model constraints should conform approximately to this guideline after the
calculate_scaling_factors()
method is executed. Users should follow this
guideline for constraints they write. The scaling of constraints for reasonable
residual tolerances is done as a constraint transformation independent of the
scaling factor suffix. Scaling factors for constraints can still be set based
on other methods such as reducing very large Jacobian matrix entries.
Specifying Scaling¶
Suffixes are used to specify scaling factors for IDAES models. These suffixes
are created when needed by calling the set_scaling_factor()
function. Using
the set_scaling_factor()
, get_scaling_factor()
, and
unset_scaling_factor()
eliminates the need to deal directly with scaling
suffixes, and ensures that scaling factors are stored in the IDAES standard
location.
- idaes.core.util.scaling.set_scaling_factor(c, v, data_objects=True, overwrite=True)[source]¶
Set a scaling factor for a model component. This function creates the scaling_factor suffix if needed.
- Parameters
c – component to supply scaling factor for
v – scaling factor
data_objects – set scaling factors for indexed data objects (default=True)
overwrite – whether to overwrite an existing scaling factor
- Returns
None
- idaes.core.util.scaling.get_scaling_factor(c, default=None, warning=False, exception=False, hint=None)[source]¶
Get a component scale factor.
- Parameters
c – component
default – value to return if no scale factor exists (default=None)
warning – whether to log a warning if a scaling factor is not found (default=False)
exception – whether to raise an Exception if a scaling factor is not found (default=False)
hint – (str) a string to add to the warning or exception message to help locate the source.
- Returns
scaling factor (float)
Constraint Transformation¶
As mentioned previously, constraints in the IDAES framework are transformed such that \(1 \times 10^{-8}\) is a reasonable criteria for convergence before any other scaling factors are applied. There are a few utility functions for scaling transformation of constraints. When transforming constraints with these functions, the scaling applies to the original constraint, not combined with any previous transformation.
- idaes.core.util.scaling.constraint_scaling_transform(c, s, overwrite=True)[source]¶
This transforms a constraint by the argument s. The scaling factor applies to original constraint (e.g. if one where to call this twice in a row for a constraint with a scaling factor of 2, the original constraint would still, only be scaled by a factor of 2.)
- Parameters
c – Pyomo constraint
s – scale factor applied to the constraint as originally written
overwrite – overwrite existing scaling factors if present (default=True)
- Returns
None
- idaes.core.util.scaling.constraint_scaling_transform_undo(c)[source]¶
The undoes the scaling transforms previously applied to a constraint.
- Parameters
c – Pyomo constraint
- Returns
None
- idaes.core.util.scaling.get_constraint_transform_applied_scaling_factor(c, default=None)[source]¶
Get a the scale factor that was used to transform a constraint.
- Parameters
c – constraint data object
default – value to return if no scaling factor exists (default=None)
- Returns
The scaling factor that has been used to transform the constraint or the default.
Calculation in Model¶
Some scaling factors may also be calculated by a call to a model’s
calculate_scaling_factors()
method. For more information see specific model
documentation.
Sometimes a scaling factor may be set on an indexed component and prorogated to it’s data objects later can be useful for example in models that use the DAE transformation, not all data objects exist until after the transformation.
- idaes.core.util.scaling.propagate_indexed_component_scaling_factors(blk, typ=None, overwrite=False, descend_into=True)[source]¶
Use the parent component scaling factor to set all component data object scaling factors.
- Parameters
blk – The block on which to search for components
typ – Component type(s) (default=(Var, Constraint, Expression, Param))
overwrite – if a data object already has a scaling factor should it be overwrittten (default=False)
descend_into – descend into child blocks (default=True)
Constraint Auto-Scaling¶
Constraints can be scaled to automatically reduce very large entries in the Jacobian
matrix with the constraint_autoscale_large_jac()
function.
- idaes.core.util.scaling.constraint_autoscale_large_jac(m, ignore_constraint_scaling=False, ignore_variable_scaling=False, max_grad=100, min_scale=1e-06, no_scale=False, equality_constraints_only=False)[source]¶
Automatically scale constraints based on the Jacobian. This function imitates Ipopt’s default constraint scaling. This scales constraints down to avoid extremely large values in the Jacobian. This function also returns the unscaled and scaled Jacobian matrixes and the Pynumero NLP which can be used to identify the constraints and variables corresponding to the rows and comlumns.
- Parameters
m – model to scale
ignore_constraint_scaling – ignore existing constraint scaling
ignore_variable_scaling – ignore existing variable scaling
max_grad – maximum value in Jacobian after scaling, subject to minimum scaling factor restriction.
min_scale – minimum scaling factor allowed, keeps constraints from being scaled too much.
no_scale – just calculate the Jacobian and scaled Jacobian, don’t scale anything
equality_constraints_only – Include only the equality constraints in the Jacobian
- Returns
unscaled Jacobian CSR from, scaled Jacobian CSR from, Pynumero NLP
Inspect Scaling¶
Models can be large, so it is often difficult to identify where scaling is needed
and where the problem may be poorly scaled. The functions below may be helpful
in inspecting a models scaling. Additionally constraint_autoscale_large_jac()
described above can provide Jacobian information at the current variable values.
- idaes.core.util.scaling.extreme_jacobian_columns(m=None, scaled=True, large=10000.0, small=0.0001, jac=None, nlp=None)[source]¶
Show very large and very small Jacobian columns. A more reliable indicator of a badly-scaled variable than badly_scaled_var_generator.
- Parameters
m – model
scaled – if true use scaled Jacobian
large – >= to this value is considered large
small – <= to this is considered small
- Returns
(list of tuples), Column norm, Variable
- idaes.core.util.scaling.extreme_jacobian_rows(m=None, scaled=True, large=10000.0, small=0.0001, jac=None, nlp=None)[source]¶
Show very large and very small Jacobian rows. Typically indicates a badly- scaled constraint.
- Parameters
m – model
scaled – if true use scaled Jacobian
large – >= to this value is considered large
small – <= to this is considered small
- Returns
(list of tuples), Row norm, Constraint
- idaes.core.util.scaling.badly_scaled_var_generator(blk, large=10000.0, small=0.001, zero=1e-10, descend_into=True, include_fixed=False)[source]¶
This provides a rough check for variables with poor scaling based on their current scale factors and values. For each potentially poorly scaled variable it returns the var and its current scaled value.
Note that while this method is a reasonable heuristic for nonnegative variables like (absolute) temperature and pressure, molar flows, etc., it can be misleading for variables like enthalpies and fluxes.
- Parameters
blk – pyomo block
large – Magnitude that is considered to be too large
small – Magnitude that is considered to be too small
zero – Magnitude that is considered to be zero, variables with a value of zero are okay, and not reported.
- Yields
variable data object, current absolute value of scaled value
- idaes.core.util.scaling.unscaled_variables_generator(blk, descend_into=True, include_fixed=False)[source]¶
Generator for unscaled variables
- Parameters
block –
- Yields
variables with no scale factor
- idaes.core.util.scaling.unscaled_constraints_generator(blk, descend_into=True)[source]¶
Generator for unscaled constraints
- Parameters
block –
- Yields
constraints with no scale factor
- idaes.core.util.scaling.map_scaling_factor(iter, default=1, warning=False, func=<built-in function min>, hint=None)[source]¶
Map get_scaling_factor to an iterable of Pyomo components, and call func on the result. This could be use, for example, to get the minimum or maximum scaling factor of a set of components.
- Parameters
iter – Iterable yielding Pyomo components
default – The default value used when a scaling factor is missing. The default is default=1.
warning – Log a warning for missing scaling factors
func – The function to call on the resulting iterable of scaling factors. The default is min().
hint – Paired with warning=True, this is a string to indicate where the missing scaling factor was being accessed, to easier diagnose issues.
- Returns
The result of func on the set of scaling factors
- idaes.core.util.scaling.min_scaling_factor(iter, default=1, warning=True, hint=None)[source]¶
Map get_scaling_factor to an iterable of Pyomo components, and get the minimum scaling factor.
- Parameters
iter – Iterable yielding Pyomo components
default – The default value used when a scaling factor is missing. If None, this will raise an exception when scaling factors are missing. The default is default=1.
warning – Log a warning for missing scaling factors
hint – Paired with warning=True, this is a string to indicate where the missing scaling factor was being accessed, to easier diagnose issues.
- Returns
Minimum scaling factor of the components in iter
- idaes.core.util.scaling.get_jacobian(m, scaled=True, equality_constraints_only=False)[source]¶
Get the Jacobian matrix at the current model values. This function also returns the Pynumero NLP which can be used to identify the constraints and variables corresponding to the rows and columns.
- Parameters
m – model to get Jacobian from
scaled – if True return scaled Jacobian, else get unscaled
equality_constraints_only – Only include equality constraints in the Jacobian calculated and scaled
- Returns
Jacobian matrix in Scipy CSR format, Pynumero nlp
- idaes.core.util.scaling.jacobian_cond(m=None, scaled=True, ord=None, pinv=False, jac=None)[source]¶
Get the condition number of the scaled or unscaled Jacobian matrix of a model.
- Parameters
m – calculate the condition number of the Jacobian from this model.
scaled – if True use scaled Jacobian, else use unscaled
ord – norm order, None = Frobenius, see scipy.sparse.linalg.norm for more
pinv – Use pseudoinverse, works for non-square matrices
jac – (optional) previously calculated Jacobian
- Returns
(float) Condition number
Applying Scaling¶
Scale factor suffixes can be passed directly to a solver. How the scale factors are used may vary by solver. Pyomo also contains tools to transform a problem to a scaled version.
Ipopt is the standard solver in IDAES. To use scale factors with Ipopt, the
nlp_scaling_method
option should be set to user-scaling
. Be aware that
this deactivates any NLP automatic scaling.