cobyqa.subsolvers.spider_geometry

cobyqa.subsolvers.spider_geometry(const, grad, curv, xpt, 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 given straight lines.

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

xptnumpy.ndarray, shape (n, npt)

Points defining the straight lines. The straight lines considered are the ones passing through the origin and the points in xpt.

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 second 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.