cobyqa.subproblems.bound_constrained_xpt_step

subproblems.bound_constrained_xpt_step(const, grad, hess_prod, xpt, xl, xu, delta, debug)

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

This function solves approximately

\[\begin{split}\begin{aligned} \max_{d \in \R^n} & \quad \abs[\bigg]{c + g^{\T}d + \frac{1}{2} d^{\T}Hd}\\ \text{s.t.} & \quad l \le d \le u,\\ & \quad \norm{d} \le \Delta, \end{aligned}\end{split}\]

by maximizing the objective function along the straight lines through the origin and the rows in xpt.

Parameters:

constfloat

Constant \(c\) as shown above.

gradnumpy.ndarray, shape (n,)

Gradient \(g\) as shown above.

hess_prodcallable

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

hess_prod(d) -> numpy.ndarray, shape (n,)

returns the product \(Hd\).

xptnumpy.ndarray, shape (npt, n)

Points defining the straight lines as shown above.

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 \(d\).

Notes

This function is described as the second alternative in p. 115 of [1]. It is assumed that the origin is feasible with respect to the bound constraints xl and xu, and that delta is finite and positive.

References

[1]

T. M. Ragonneau. “Model-Based Derivative-Free Optimization Methods and Software.” Ph.D. thesis. Hong Kong: Department of Applied Mathematics, The Hong Kong Polytechnic University, 2022.