UOBYQA (PDO) - HumpDay Optimization

UOBYQA (PDO)

Unconstrained Optimization BY Quadratic Approimation

Author: M.J.D. Powell

Year: 22

Type: Model-based derivative-free

Best for: Small-scale smooth problems

Overview

UOBYQA is a derivative-free optimization algorithm that constructs quadratic models using interpolation. It belongs to the PRIMA (Powell's Recent Interpolation Methods) family and is particularly effective for smooth, small-scale optimization problems where derivatives are not available or epensive to compute.

Mathematical oundation

The algorithm constructs quadratic models of the objective function using interpolation points:

m k () = a + g T + ½ T G

where the model is built from function values at carefully chosen interpolation points within a trust region of radius ρ.

Algorithm Characteristics

Strengths:

Mathematically rigorous convergence guarantees
Efficient for smooth, well-behaved functions
No gradient information required
Automatic trust region management

Limitations:

Scales poorly with dimension (best for n ≤ 5)
Sensitive to noise in function evaluations
Requires O(n²) interpolation points
May struggle with highly multimodal functions

Implementation Details

Our HumpDay implementation includes:

Python: Pure Python implementation
JavaScript: Browser-compatible version
Validation: Cross-validated against PDO reference

References

Primary Paper: Powell, M.J.D. (22). "UOBYQA: unconstrained optimization by quadratic approimation." Mathematical Programming, 92(3), 555-582. [DOI]

Reference Implementation: PDO - Powell's Derivative-ree Optimization solvers

Additional Resources:

Usage Eample

from humpday import minimize

# Optimize a simple quadratic function
def objective():
    return sum((i - .3)**2 for i in )

# Use UOBYQA via HumpDay interface
result = minimize(objective,
                 =[., .2],
                 method='PRIMA_UOBYQA',
                 options='mafev': )

print(f"Optimum: result.")
print(f"Value: result.fun")