Bayesian Algorithm for Collaborative Optimization Framework

Mathematical Formulation

System Level:

\[ \begin{align*} &\max_{ \overline{\mathbf{z}}, \overline{\mathbf{x}}, \overline{\mathbf{y}} }\ && \alpha_f( \overline{\mathbf{z}}, \overline{\mathbf{x}}, \overline{\mathbf{y}} ) \tag{\(\widetilde{P}_\text{sys}\)} \\ &\text{subject to:}\ &&\mathbf{\mu_c}( \overline{\mathbf{z}}, \overline{\mathbf{x}}, \overline{\mathbf{y}}, ) &&\geq \mathbf{0}\\ % & && \mu_{J_i}( \overline{\mathbf{z}}, \overline{\mathbf{x}}_i, \overline{\mathbf{y}}, \underline{\mathbf{z}}_i^*, \underline{\mathbf{x}}_i^* ) &&= 0\ \text{for all}\ i\in\{1\cdots N\} \end{align*} \]

Subsystem \(i\) level:

\[ \begin{equation*} \max_{ \underline{\mathbf{z}}_i, \underline{\mathbf{x}}_i }\ \alpha_{J_i}( \overline{\mathbf{z}}, \overline{\mathbf{x}}_i, \overline{\mathbf{y}}, \underline{\mathbf{z}}_i, \underline{\mathbf{x}}_i ) \tag{\(\widetilde{P}_i\)}\ \text{subject to:}\ \mathbf{\mu}_{\mathbf{g}_i}( \overline{\mathbf{y}}_{j\neq i}, \underline{\mathbf{z}}_i, \underline{\mathbf{x}}_i )\geq\mathbf{0} \end{equation*} \]

The discrepancy function:

\[ \begin{equation*} J( \overline{\mathbf{z}}, \overline{\mathbf{x}}_i, \overline{\mathbf{y}}, \underline{\mathbf{z}}_i, \underline{\mathbf{x}}_i ) = J_i = \bigl\|\overline{\mathbf{z}}-\underline{\mathbf{z}}_i\bigr\|^2 + \bigl\|\overline{\mathbf{x}}_i-\underline{\mathbf{x}}_i\bigr\|^2 + \bigl\|\overline{\mathbf{y}}_i-\mathbf{y}_i\bigl( \overline{\mathbf{y}}_{j\neq i}, \underline{\mathbf{z}}_i, \underline{\mathbf{x}}_i \bigr)\bigr\|^2 \end{equation*} \]

Key Innovations

• GP surrogates for subsystem objectives (J_i) and constraints (g_i)
• Acquisition function optimization (Expected Improvement)
• Latin Hypercube Sampling for efficient DoE initialization
• Reduced subsystem evaluations through surrogate modeling

Advantages

• Reduced subsystem calls
• Better for expensive analysis codes
• Uncertainty quantification via GP variance
• Adaptive sampling focuses on promising reg_ions

Implementation

BACOSubsystem

Attribute	Type	Description
problem	Problem	Local subsystem optimization problem with objective and constraints.
z_idxs	np.ndarray	Indices of shared design variables in the global variable vector.
x_idxs	np.ndarray	Indices of local design variables in the global variable vector.
y_idxs	np.ndarray	Indices of subsystem outputs in the global output vector.
y_coupled_idxs	np.ndarray	list of index arrays for coupled variables (targets) from other subsystems.
surrogate_config	SMTGPConfig or TorchGPConfig	GP configuration.

Workflow:

1. initialize_doe(): LHS sampling to create initial DoE
2. build_surrogate_J_i(), build_surrogate_g_i(): Train GP surrogates
3. solve_acquisition(): Optimize acquisition function
4. Update DoE and retrain (repeat)

BACOSystem

Attribute	Type	Description
problem	Problem	System-level optimization problem with objective and constraints.
subsystems	list[COSubsystem]	All subsystems in the MDO problem.
surrogate_config	SMTGPConfig or TorchGPConfig	GP configuration.

BayesianCollaborativeOptimization

Attribute	Type	Description
system	COSystem	System-level coordinator with subsystems.
subsystem_optimizer	str or Callable	Optimizer for subsystem minimization (e.g., 'cobyqa', 'cobyla')
system_optimizer	str or Callable	Optimizer for system-level coordination.
n_initial	int or Callable	Size of the initial Design of Experiments.
acq_func	Callable	Choice of acquisition function.
n_multistart	int	Number of multi-starts when solving \(\widetilde{P}_\text{sys}\) and \(\widetilde{P}_i\).
epsilon_J	float	Convergence tolerance for coupling discrepancy \(J_\text{total}\).
epsilon_h	float	Convergence tolerance for constraint violation \(h_\text{total}\).
max_eval	int	Evaluation budget.
cache_dir	Path	Directory to save intermediate progress in case code crashes.
name	str	Field to define problem name.
solver	str	Field to define solver name.