cobyqa.subsolvers.tangential_byrd_omojokun#

cobyqa.subsolvers.tangential_byrd_omojokun(grad, hess_prod, xl, xu, delta, debug, **kwargs)[source]#

Minimize approximately a quadratic function subject to bound constraints in a trust region.

This function solves approximately

\[\begin{split}\min_{s \in \mathbb{R}^n} \quad g^{\mathsf{T}} s + \frac{1}{2} s^{\mathsf{T}} H s \quad \text{s.t.} \quad \left\{ \begin{array}{l} l \le s \le u\\ \lVert s \rVert \le \Delta, \end{array} \right.\end{split}\]

using an active-set variation of the truncated conjugate gradient method.

Parameters:
gradnumpy.ndarray, shape (n,)

Gradient \(g\) as shown above.

hess_prodcallable

Product of the Hessian matrix \(H\) with any vector.

hess_prod(s) -> `numpy.ndarray`, shape (n,)

returns the product \(H s\).

xlnumpy.ndarray, shape (n,)

Lower bounds \(l\) as shown above.

xunumpy.ndarray, shape (n,)

Upper bounds \(u\) as shown above.

deltafloat

Trust-region radius \(\Delta\) as shown above.

debugbool

Whether to make debugging tests during the execution.

Returns:
numpy.ndarray, shape (n,)

Approximate solution \(s\).

Other Parameters:
improve_tcgbool, optional

If True, a solution generated by the truncated conjugate gradient method that is on the boundary of the trust region is improved by moving around the trust-region boundary on the two-dimensional space spanned by the solution and the gradient of the quadratic function at the solution (default is True).

Notes

This function implements Algorithm 6.2 of [1]. It is assumed that the origin is feasible with respect to the bound constraints and that delta is finite and positive.

References

[1]

T. M. Ragonneau. Model-Based Derivative-Free Optimization Methods and Software. PhD thesis, Department of Applied Mathematics, The Hong Kong Polytechnic University, Hong Kong, China, 2022. URL: https://theses.lib.polyu.edu.hk/handle/200/12294.