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Mathematical Foundation

Extracted documentation from src/mdotoolbox/core/utils.py.

assess_progress

Examples

Full hierarchy - feasibility improved

>>> assess_progress(h=0.01, h_best=0.1,
...                 J_i=2.0, J_i_best=1.0, f=20.0, f_best=10.0)
True

>>> # Jf mode - coupling improved, no h check
>>> assess_progress(h=0.0, h_best=0.0,
...                 J_i=0.5, J_i_best=1.0, f=20.0, f_best=10.0,
...                 mode="Jf")
True

>>> # f mode - strictly objective
>>> assess_progress(h=0.0, h_best=0.0,
...                 J_i=0.0, J_i_best=0.0, f=9.0, f_best=10.0,
...                 mode="f")
True

format_time_multi

Notes

  • All rows padded to same number of columns
    • Columns right-aligned for readability
    • Useful for timing comparisons in output tables

compute_Ji

Mathematical Formulation

J_i = ||z_bar - z_under_i||2 + ||x_bar_i - x_under_i||2 + ||y_bar_i - y_i||**2

Args: y_i_func: Callable Subsystem output function: y_i = f(z_under_i, x_under_i, y_bar_coupled) z_bar: ndarray System-level shared design variables x_bar_i: ndarray System-level target for subsystem local variables y_bar_i: ndarray System-level target for subsystem outputs y_bar_coupled: ndarray Target values for coupled variables from other subsystems z_under_i: ndarray Subsystem copy of shared variables (optimization variable) x_under_i: ndarray Subsystem local variables (optimization variable)

Returns: J_i: float Total discrepancy (sum of squared deviations) y_i: ndarray Actual subsystem outputs from y_i_func evaluation

Example: >>> def subsystem_output(z, x, y_coupled): ... return np.array([z[0]**2 + x[0]]) >>> >>> z_bar = np.array([2.0, 3.0]) >>> x_bar = np.array([1.0]) >>> y_bar = np.array([5.0]) >>> y_bar_coupled = np.array([]) >>> z_under = np.array([2.1, 3.0]) >>> x_under = np.array([0.9]) >>> >>> J_i, y_i = compute_Ji( ... subsystem_output, z_bar, x_bar, y_bar, y_bar_coupled, z_under, x_under ... )

Notes

  • Lower J_i indicates better consistency with system targets
    • J_i = 0 means perfect coupling consistency
    • Used in CO subsystem optimization objective

clip_to_bounds

Notes

  • Creates a copy, does not modify input
    • Useful for enforcing bounds after optimizer steps
    • Common in CO/BACO after subsystem optimization