cobyqa.subsolvers.cauchy_geometry

cobyqa.subsolvers.cauchy_geometry(const, grad, curv, xl, xu, delta, debug)

Maximize approximately the absolute value of a quadratic function subject to bound constraints in a trust region.

This function solves approximately

\[\begin{split}\max_{s \in \R^n} \quad \bigg\lvert c + \transpose{g} s + \frac{1}{2} \transpose{s} H s \bigg\rvert \quad \text{s.t.} \quad \left\{ \begin{array}{l} \xl \le s \le \xu,\\ \lVert s \rVert \le \Delta, \end{array} \right.\end{split}\]

by maximizing the objective function along the constrained Cauchy direction.

Parameters:

constfloat

Constant \(c\) as shown above.

gradnumpy.ndarray, shape (n,)

Gradient \(g\) as shown above.

curvcallable

Curvature of \(H\) along any vector.

curv(s) -> float

returns \(\transpose{s} H s\).

xlnumpy.ndarray, shape (n,)

Lower bounds \(\xl\) as shown above.

xunumpy.ndarray, shape (n,)

Upper bounds \(\xu\) 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 \(s\).

Notes

This function is described as the first alternative in Section 6.5 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, The Hong Kong Polytechnic University, Hong Kong, China, 2022.